Paper Title: Low-depth, compact and error-tolerant photonic matrix-vector multiplication beyond the unitary group.





















Russian Quantum Center, Bolshoy Boulevard 30, building 1, Moscow, 121205, Russia
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

saygin@physics.msu.ru

Sber Quantum Technology Center, Kutuzovski prospect 32, Moscow, 121170 Russia
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

Laboratory of Quantum Engineering of Light, South Ural State University (SUSU), Prospekt Lenina 76, Russia, Chelyabinsk, 454080 Russia 



Sber Quantum Technology Center, Kutuzovski prospect 32, Moscow, 121170 Russia

Russian Quantum Center, Bolshoy Boulevard 30, building 1, Moscow, 121205, Russia
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia


§ ABSTRACT 

Large-scale programmable photonic circuits are opening up new possibilities for information processing providing fast and energy-efficient means for matrix-vector multiplication. Here, we introduce a novel architecture of photonic circuits capable of implementing non-unitary transfer matrices, usually required by photonic neural networks, iterative equation solvers or quantum samplers. Our architecture exploits compact low-depth beam-splitter meshes rather than bulky fully connected mixing blocks used in previous designs, making it more compatible with planar integrated photonics technology.  We have shown that photonic circuits designed with our architecture have lower depth than their standard counterparts and are extremely tolerant to hardware errors. 






                              
Low-depth, compact and error-tolerant photonic matrix-vector multiplication beyond the unitary group
    Stanislav S. Straupe
    Received ...; accepted ...
====================================================================================================










§ INTRODUCTION



Programmable photonic circuits are indispensable ingredients of classical <cit.> and quantum information processing devices <cit.>. For these devices to achieve the ultimate efficiencies promised by photonics, namely, accelerated speed and energy-efficiency of computation, it is necessary to use programmable interferometers capable of implementing large-scale and high-quality linear transformations. However, fabrication of programmable interferometers with the required scale and quality is challenging due to hardware errors <cit.>. Therefore, developing interferometer architectures that are less susceptible to errors, while  optimally exploiting the resources of a specific photonic platform is very important and timely <cit.>.



The fundamental mathematical operation of matrix-vector multiplication (MVM) can be efficiently performed by programmable photonics. In particular, this operation is essential for data transformation, weight adjustments, and learning processes in photonic neural networks <cit.>. In recent years, the development of novel architectures has been primarily focused on programmable interferometers that implement multiplications by unitary transfer matrices. In particular, the planar meshes of Mach-Zehnder interferometers (MZIs) proposed by Reck <cit.> and Clements <cit.> can implement arbitrary unitary transfer matrices. These architectures are well suited for fabrication by the mature integrated photonics technology enabling massive production of sophisticated optical circuits <cit.>. Sadly, errors in the constituent directional couplers introduce errors in the implemented multimode transformations <cit.>. The effect of errors can be alleviated by adding extra elements into optical circuits that correct the errors, however, this makes the circuits larger and harder to program <cit.>. Several error-tolerant architectures for programmable unitaries that do not require extra phase-shift elements have been proposed <cit.> and demonstrated in experiments <cit.>.

A class of linear transformations broader than unitaries is usually required for photonic information processing, in particular, in classical photonic neural networks <cit.>, iterative equation solvers <cit.> or quantum graph problem solvers <cit.>. Fortunately,  one can use programmable unitary circuits to implement these generic linear transformations. Common approaches embed the transformation in a larger unitary leading to an overhead in the circuit depth that is typically two times larger than the number of modes in the desired transformation. It turns out that this depth scaling is suboptimal <cit.>.

Recently, Tang et al. <cit.> have shown that arbitrary transfer matrices can be implemented by programmable circuits with twice lower depths than in the aforementioned approaches. To achieve that, the authors considered a low-depth version of the architecture <cit.>, known to be universal and exhibiting high tolerance to hardware errors. This architecture relies on static fully-connected mixing blocks requiring interaction between all participating modes at once, which is costly to accomplish, especially by planar integrated photonics technologies.

Here we propose an architecture of programmable interferometers that are simultaneously low-depth and error-tolerant. The architecture relies on partially mixing static blocks conveniently realized by a minor modification of the beam-splitter (BS) mesh, which we have studied previously as means to implement programmable unitaries <cit.>. The proposed programmable circuits are more compatible with planar manufacturing technologies, since they do not require multimode mixing contrary to <cit.>. As a result, the overall footprint, that includes both static and programmable elements, is more compact than that of the known counterparts.

The paper is organized as follows. In Sec. <ref> we review the methods of constructing programmable multimode optical circuits relevant for understanding our results that follow. In Sec. <ref> we introduce our architecture of programmable circuits and study its performance. We conclude in Sec. <ref>.








§ PROGRAMMABLE MULTIMODE OPTICS


The  transformation of a linear N-mode optical device can be described by its complex-valued N×N transfer matrix U linking the input, 𝐚=(a_1,…,a_N)^T, and output, 𝐛=(b_1,…,b_N)^T  amplitude vectors:

    
    𝐛=U𝐚.


Multimode interferometers performing linear transformations have the form of sequence of layers, each consisting of tunable phase-shifts and static multimode blocks, as illustrated in Fig. <ref>a. The transfer matrices of such interferometers can be written as

    
    U=Φ^(L+1)(φ^(L+1))∏_l=1^LV^(l)Φ^(l)(φ^(l)),


where Φ^(l)(φ)=diag(e^iφ_1,…,e^iφ_N-1,1) is the diagonal matrix of a tunable  layer with index l (l=1,L+1) with φ_j being tunable phase-shifts, V^(l) – the transfer matrix of a static layer with index l. In the following, we consider interferometers with programmable transfer matrices, which is achieved by properly setting the tunable phase-shifts.



    


The transformation (<ref>) with N-1 phase-shifts per layer corresponds to the circuit with densest possible packing of these elements. In this work, we are interested in low-depth architectures of programmable circuits and, thus, from the very beginning consider interferometers of this kind. However, it should be noted that not all programmable circuits can be reduced to the form. For example, the unitary architectures <cit.>  can be compactified to form (<ref>) using the method proposed in <cit.>, whereas the  architecture proposed in <cit.> cannot. 




 §.§ Unitary transformations





In real programmable photonic devices tunable phase-shifts are relatively large and lossy. This hinders the scaling of the programmable circuits and motivates the development of low-depth architectures. Therefore, an important parameter that should be kept in mind at designing a programmable circuit is its depth D, which we define as the number of phase-shift layers, i.e. D=L+1. For universal unitary architectures the minimal depth of an N-mode circuit is D=N+1, which corresponds to the total of N^2-1 phase-shifts — the number of real parameters necessary to describe an arbitrary N×N unitary matrix up to a global phase.

In (<ref>), the choice of particular static transfer matrices V^(l) specified by a chosen architecture, has a strong effect on the circuit functionality. For example, according to our previous work <cit.>, error-tolerance of programmable unitaries is achieved by choosing static multimode mixing linear optics for which V^(l) are dense matrices. However, the experimental realizations of such transfer matrices requires organization of interference between all participating modes in each layer. As a result, the area required to allocate the circuit  scales unfavorably with the size of the target transformations N, thereby limiting the size of achievable circuits. To illustrate this issue, Fig. <ref>b shows an example of a multimode mixing block implemented by coupled waveguide arrays  <cit.>. The transformation performed by the coupled waveguides is governed by the coupling taking place between neighbouring waveguides, which necessitates increasing the interaction length l_int as N grows in order to achieve a fully mixing matrix V^(l). Moreover, the bends required to couple the waveguides should have longer length l_bend with increasing N (see Fig. <ref>b). Another way to implement fully mixing matrices V^(l) by integrated photonics is using the multimode interferometers (MMI) <cit.>, shown in Fig. <ref>c. Similarly to the coupled waveguides arrays, the MMIs' length l_mmi scales with the size N of the programmable circuits. More sophisticated methods of designing mixing blocks in planar integrated photonics do not aid in precluding the scaling of the static blocks either <cit.>.

As an alternative, compact circuits can be realized by another programmable unitary architecture that uses meshes of beam-splitters (BSs), which has been proposed in our previous work <cit.>. The corresponding static block used in this architecture is depicted in Fig. <ref>d. The block consists of two BS layers, so that the static block transfer matrices read:

    
    V^(l)_BS=∏_k∈Ω_2T_BS,k^(l,2)∏_j∈Ω_1T_BS,j^(l,1),  (l=1,L),


where 


    T^(l,m)_BS,j=(
    		[          1          ⋯          ⋯          ⋯          ⋯          0;          ⋮          ⋱                                           ⋮;          ⋮             ρ_j^(l,m) iτ_j^(l,m)                     ⋮;          ⋮            iτ_j^(l,m)  ρ_j^(l,m)                     ⋮;          ⋮                                           ⋱          ⋮;          0                     ⋯          ⋯                     1 ])


designate the transfer matrix of a single BS acting on modes j and j+1 in the sub-layer with index m (m=1,2). Ω_1 and Ω_2 denote the ordered sequence of BSs in the first and second sub-layer, respectively. The BS matrix (<ref>) has all diagonal elements equal to one except those labeled by reflectivity ρ_j^(l,m), and all off-diagonal elements equal to zero except those labeled by transmissivity τ_j^(l,m) (τ_j^(l,m)2+ρ_j^(l,m)2=1). Obviously, matrices (<ref>) are radically different from the ones of the multimode mixing blocks, e.g. implemented by coupled waveguide arrays, in that they provide much lesser connectivity.

At the same time the BS-based architecture with static blocks depicted Fig. <ref>c is advantageous compared to the architecture based on multimode mixing blocks, such as those shown in Fig. <ref>b. This is due to the fact that the length of the static block l_BS is independent on the scale of the multimode transformation N. In addition,  the BS-based architecture is better suited for fabrication by intergated photonics technology, wherein it enables the creation of deeper programmable circuits as demonstrated recently in <cit.>. Despite having a lower degree of connectivity and experimental complexity than the architectures based on fully mixing static blocks, the BS-based architecture still offers ultimate error-tolerance <cit.>. 




 §.§ Transformations beyond the unitary group


Numerous applications require programmable interferometer architectures that can implement arbitrary transfer matrices beyond the unitary group. Hereinafter, we will designate such non-unitary matrices by A, in order to distinguish them.

It is known that programmable unitaries can be utilized to implement non-unitary matrices. The architectures capable of performing this task can be divided into two categories as illustrated in Fig. <ref>. Firstly, using a singular value decomposition (SVD) method, an N×N matrix A can be represented as a product of three N×N matrices:

    
    A=WΣU,


each of which can be implemented by optics, as shown in Fig. <ref>a. Here, U and W are N×N unitary matrices that can be implemented by any programmable unitary interferometer. The realization of the non-unitary diagonal matrix Σ=diag(σ_1,…,σ_N), comprised of singular values σ_j, requires individual modulation of field amplitudes. In Fig. <ref>a this is attained by N parallel Mach-Zehnder modulators (MZIs) programmed by phase-shifts located in their arms. It should be noted that such amplitude modulation is limited to singular values  |σ_j|≤1. However, it does not diminish the applicability of the SVD-based interferometers, since in photonic information processing devices, in particular, in photonic neural networks, scaled target transfer matrices A (e.g., divided by the maximum singular value to meet the requirements) have the same utility as the original ones. The circuit depth of the SVD-based interferometers scales as:

    
    D=2N+3.




    


The second approach is based on considering a target N×N non-unitary matrix A as being a submatrix a larger M×M unitary one (M>N). This embedding becomes obvious when we use the fact that any A with the singular values obeying |σ_j|≤1 can be always represented as a part of a 2N×2N unitary matrix U of the form:

    
    U=([             A √(𝕀_N×N-AA^†); √(𝕀_N×N-A^†A)          -A^† ]),


where 𝕀_N×N is an N×N identity matrix, e.g. <cit.>. In this approach, a straightforward method to implement arbitrary non-unitary matrices A (with singular values |σ_j|≤1) is using a universal  programmable 2N×2N interferometer, as shown in Fig. <ref>b. The circuit depth of the optical scheme scales as D=2N+2.

The circuit depth can be made substantially shallower than (<ref>). As has been demonstrated in the recent paper by Tang et al. <cit.>, arbitrary non-unitary N×N matrices can be embedded into 2N×2N programmable unitary circuits with depth scaling as D=N+2 — a two-fold improvement over the previously known results for complex-valued matrices. The architecture by Tang et al. <cit.> uses the unitary architecture depicted in Fig. <ref>a with L=N+1 multimode mixing blocks. In this work, we propose an even more compact and practical architecture. 



    





§ LOW-DEPTH ERROR-TOLERANT ARCHITECTURE


The proposed architecture is illustrated in Fig. <ref>. Here, the constituent static blocks V^(l) take the form of BS meshes acting on 2N modes, which are more compact than the previously used multimode mixing blocks <cit.>. This blocks resemble the planar meshes used in the unitary programmable architecture <cit.> shown in Fig. <ref>d. However,  there is an important modification making the circuit non-planar. Namely, there is an additional BS coupling two distant modes, thereby the whole mesh has the circular topology shown in Fig. <ref>b, with the transfer matrices having the following form: 

    
    V^(l)_circ=∏_k∈Ω_circT_BS,k^(l,2)∏_j∈Ω_1T_BS,j^(l,1)=T^(l,2)_BS,NV^(l)_BS,(l=1,L),


where Ω_circ is a sequence of BSs acting on pairs of neighbouring modes in a circular manner (see Fig. <ref>b). The sequence Ω_circ is simply the sequence Ω_2 from (<ref>) with an added BS described by the following transfer matrix:


    T^(l,2)_BS,N=(
    		[  ρ_N^(l,2)          0          ⋯          0 iτ_N^(l,2);          0          1                     0          0;          ⋮                     ⋱                     ⋮;          0          0                     1          0; iτ_N^(l,2)          0          ⋯          0  ρ_N^(l,2) ]).



As will be shown below, the seemingly minor modification of the planar BS mesh has a strong effect on the capability of the programmable circuits, especially on their tolerance to hardware errors. We notice that the modification of the planar mesh does not present an issue from the hardware perspective, since the waveguide crossings, necessary to implement the circular mesh on a planar chip, is the standard element accessible in mature integrated photonic technologies <cit.>. Moreover, modern  facilities offer fabrication of multilayer photonic integrated circuits, making the proposed architecture even easier to implement <cit.>. 

To study the capabilities of the architecture in implementing non-unitary transfer matrices, we consider N×N submatrices A of the transfer matrix of the 2N-mode interferometer corresponding to the field amplitudes, a_n and b_n, occupying the inputs and outputs alternately, as illustrated in Fig. <ref>a. To quantify the fidelity of the programmable circuits, we use the normalized square error (NSE):

    
    NSE(A^(0),A)=1/N∑_i,j=1^N|A^(0)_ij-A_ij|^2,


which compares the target  matrix A^(0) consisting of elements A_ij^(0) and the actual transfer matrix A consisting of elements A_ij realized by the interferometer, where N is the size of the matrices. Provided that the matrices A^(0) and A are equal, (<ref>) gets its minimal value of NSE=0.

Until now, we did not specify the required parameters of the programmable circuits, namely, the depth D and the values of BS's transmissivities τ_j^(l,m), corresponding to high-fidelity implementation of non-unitary transfer matrices. To find the minimal depth D=L+1 necessary to implement target matrices of given size N, we performed numerical simulations at various values of D. In parallel with this, we analyze the dependence of (<ref>) on τ_j^(l,m)'s that allowed us to investigate the error-tolerance capabilities of the circuit at specified N and D. First, we study the effect of coherent errors where all BSs are identical τ_j^(l,m)=τ, but τ can take various values.




 §.§ Non-unitary transfer matrices



We investigate the capabilities of our architecure to implement randomly generated non-unitary matrices. We used SVD (<ref>) to generate a target matrix A^(0), wherein the unitary complex-valued matrices U and V were drawn from the Haar random distribution using the method based on the QR-decomposition of random matrices from the Ginibre ensemble <cit.>. The diagonal matrix Σ was filled with independently generated values from a uniform distribution in the [0,1] range. 

There is no known analytical solution for the values of phase shifts that minimize the fidelity measure (<ref>), except for some specific architectures <cit.>, thereby necessitating the use of a numerical optimization algorithm (see Appendix <ref> for details). Given a target matrix A^(0), the algorithm was searching towards a global minimum of NSE over the space of phase-shifts φ^(l) (l=1,D). To decrease the chance of falling into local minima, we used multiple runs of optimization (∼100) with random initial values of the
phases. Each numerical experiment involved optimization over a series of target matrices A^(0).



    


Using the optimization method, we studied the effect of different values of τ on attained NSE values. In addition to the circuits with the circular BS mesh, we also conducted simulations for the circuits based on the  traditional planar BS meshes (see Fig. <ref>c).  Fig. <ref> shows the obtained dependencies of NSE on τ for the proposed circular programmable BS meshes (Fig. <ref>a,b) and the planar programmable BS meshes (Fig. <ref>c,d). For the circuits built with the circular BS meshes, two dependencies are plotted  that correspond to 

    
  * a value of depth D_ET at which the circuit exhibits a high-fidelity plateau with NSE≈0, where a particular value of τ does not matter, and
    
  * a value of depth D_ET-1, to demonstrate the threshold-like appearance of the NSE≈0 plateau.

Specifically, for N=5 and N=10 we arrive at D_ET=7 and D_ET=12, respectively. Therefore, this suggests that D_ET=N+2, which is the same as shown in <cit.> for the architecture based on the multimode mixing blocks, implemented as coupled waveguides (Fig. <ref>b) or MMIs (Fig. <ref>c). 

Obviously, the appearance of a wide plateau of τ's with NSE≈0 suggests that the programmable circuits are tolerant to hardware errors that comes in the form of deviations in the constituent static BS elements. Also notice that the NSE≈0 plateau is so wide that the corresponding static errors in BS elements, that should fall well in this plateau, are far beyond what is likely to be encountered in practice. 

We used the obtained depth values D_ET and D_ET-1 to plot the corresponding dependencies for the planar BS meshes in Fig. <ref>c,d. One can see that, in strike contrast to the circular BS meshes, the  planar BS meshes do not exhibit the high-fidelity plateau, necessary for tolerance to hardware errors. 

In addition to the coherent errors, assuming correlated deviations of the BS parameters,  we studied the influence of incoherent BS errors. For this, we generated τ_j^(l,m) independently from the gaussian random distribution for various  distribution widths. The high-fidelity plateau, similar to the case of coherent errors, is observed, thereby proving the tolerance to hardware errors of our architecture.






 §.§ Special cases




    


The analysis of the capabilities of the proposed architecture on randomly sampled non-unitary matrices by no means prove its operation for arbitrary transfer matrices that can be encountered in real uses. Therefore, we have also considered a special case of transfer matrices that fall out of the analysis conducted above. 


In particular, we analyse the implementation of generic unitary transfer matrices, which is also a case very unlikely to be caught by sampling non-unitary matrices of the larger size.
We generated the target matrix A^(0) to be a unitary matrix sampled from

the Haar random distribution using the method based on the QR-decomposition of random matrices from the Ginibre ensemble. As before, we used the fidelity measure (<ref>).  Fig. <ref>a and Fig. <ref>b demonstrate the obtained results for unitary transfer matrices of N=5 and N=10, respectively. As can be seen from the figures, the circular interferometer does not work in error-tolerant way, as it did for non-unitary target matrices. In addition to generic unitary matrices, we also studied the implementation of permutation matrices which are unitary, but present a more narrower class. The obtained results turned out to be similar to the ones illustrated by Fig. <ref>.






 §.§ Relaxed requirements for target matrices





    

To achieve high-fidelity matrix-vector multiplication by programmable photonic circuits in general and our circular architecture, in particular, we suggest the following trick. In many applications, matrix-vector multiplication is allowed to be performed up to a global complex value s. In other words, multiplication by the matrix sA^(0) can be as useful as  multiplication by the specified one A^(0). For example, this is usually the case in opto-electronic devices <cit.>, wherein the vector result of the photonic multiplication is immediately measured and converted to electronic format for further processing. The scaling operation present a simple routine, which can be easily performed by electronics.  

We used the modified fidelity measure that takes into account the freedom of scaling factor:

    NSE_s(A^(0),A)=1/|s|^2NSE(sA^(0),A),

which we optimize in the space of phase-shifts {φ^(l)} and the scaling factor s (see Appendix <ref> for details). It should be noted that, because lower  values of |s| are associated with higher losses introduced to the transformation, higher |s| are usually preferable. Our optimization algorithm used to minimize (<ref>) opts for higher values of |s|.

We challenge our architecture again in the task of implementing unitary transfer matrices. The results obtained in optimization of (<ref>) are shown in Fig. <ref>. Comparing these results with those shown in Fig. <ref>, we see the significant effect of using the relaxed fidelity measure (<ref>) on the quality of implemented unitary matrices. The high-fidelity plateau with NSE_s≈0 is observed, which suggest the tolerance to hardware errors.





§ CONCLUSION


In this work we proposed a novel architecture of programmable circuits capable of implementing transfer matrices beyond the unitary group. Our architecture has the advantage of having a lower depth than the previously known counterparts and possess tolerance to hardware errors. In addition, the programmable circuits designed with the architecture is compatible with planar integrated photonics. All these properties, simultaneously offered by our architecture, facilitate the fabrication of large-scale programmable circuits performing photonic matrix-vector multiplication by mature integrated photonics technologies. Taking into account the typical scale of matrix-vector multiplications  required by modern computing, in particular, deep neural networks, the advantages of our 
circuits are decisive for practical implementation in real-world computing systems. This makes us believe that the results of this work are of interest for both fundamental and applied fields.









§ OPTIMIZATION ALGORITHM FOR NSE


The optimization algorithm used to find the phase-shift values corresponding to a minimum of the NSE (<ref>) works as follows. 

Consider one phase-shift layer with index l, so that the interferometer transfer matrix  A = CΦ^(l)B, where B and C are the matrices describing the parts of the interferometer acting  before and after the phase-shift layer, respectively (see Fig. <ref>a).

Then, the NSE  may be rewritten as

    NNSE(A^(0),A)=
    Tr(A^(0)†-B^†Φ̅^(l)C^†)(A^(0)-CΦ^(l)B)=∑_i,j=1^N|A^(0)_ij|^2+
    
        +Tr(Φ̅^(l)C^†CΦ^(l)BB^†)-2[Tr(Φ̅^(l)C^†A^(0)B^†)].

Next, it is possible to determine the NSE minimum with respect to a single phase-shift ϕ^(l)_k, 0<k<N in layer l, using the condition:

    
    ∂NSE(A^(0),A)/∂ϕ^(l)_k=0,


which yields

    
    [e^-iϕ^(l)_k(∑_j=1
    j ≠ k^N(C^†C)_kje^iϕ^(l)_j(BB^†)_jk
        -(C^†A^(0)B^†)_kk)]=0,


assuming ϕ^(l)_N=0 and is not changeable. From (<ref>) follows the analytical solution corresponding to a minima of the NSE with respect to the phase-shift parameter:

    ϕ^(l)∗_k=((C^†A^(0)B^†)_kk-∑_j=1
    j ≠ k^N(C^†C)_kje^iϕ^(l)_j(BB^†)_jk).



Our algorithm iteratively chooses a programmable phase-shift layer and a phase-shift within it, to which the analytical solution (<ref>) is applied. Namely, one iteration updates phase-shifts ϕ^(l)_k one-by-one using (<ref>) by sequentially selecting  layer index l and phase-shifts index k. We run this process until one of the following conditions is met: 1) NSE<10^-10, 2) the relative difference between successive iterations (NSE_t+1-NSE_t)/NSE_t<10^-8, or 3) the maximum number of iterations 10^6 has been reached. To find optimal phase-shift values corresponding to a global minimum of (<ref>), the optimization procedure was conducted 100 times, each time initialized by random starting values.







§ OPTIMIZATION ALGORITHM FOR NSE_S




The optimization algorithm used to find the phase-shift values and scaling factor s simultaneously corresponding to the highest possible quality and lowest level of introduced losses was as follows.


For a target matrix A^(0) and actual transfer matrix A the scaling factor s minimizing NSE_s can be found analytically. For this,  NSE_s is written explicitly

    NNSE_s(A^(0),A) =∑_i,j=1^N|A^(0)_ij|^2-
    
        -2[e^-iarg(s)Tr[A^†A^(0)]]/|s| + ∑_i,j=1^N|A_ij|^2/|s|^2,

which readily yields the optimal value 

    s = ∑_i,j=1^N|A_ij|^2/Tr[A^†A^(0)].



To optimize the NSE_s, we modify the procedure for NSE by adding updating the s value after each iteration of the algorithm. If NSE_s<10^-10 was reached, then the next optimization procedure starting from a new random point had a lower bound on |s|. This bound was gradually increased as the iterations proceeded. This was done in order to determine the scaling factor with the maximum possible amplitude, which is preferable to minimize losses.









 Paper Title: Generating three-photon Rabi oscillations without a large-detuning condition.


	
	
	
	Department of Physics, Fuzhou University, Fuzhou, 350116, China
	Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350116, China
	
	
	Department of Physics, Fuzhou University, Fuzhou, 350116, China
	Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350116, China
	
	
	Department of Physics, Fuzhou University, Fuzhou, 350116, China
	Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350116, China
	
	yehong.chen@fzu.edu.cn
	Department of Physics, Fuzhou University, Fuzhou, 350116, China
	Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350116, China
	Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
	
	xia-208@163.com
	Department of Physics, Fuzhou University, Fuzhou, 350116, China
	Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350116, China
	
	
	
	
	
	§ ABSTRACT 
		It is well known that in the quantum Rabi model, a three-photon resonance occurs when the cavity field bare frequency is about 1/3 of the atomic transition frequency. In this manuscript, we show that the resonance can also be generated in the absence of the “1/3 condition" by employing an artificial atom with tunable transition frequency. To realize the protocol, the modulation frequency should be comparable to the cavity frequency in order to induce a counter-rotating interaction in the effective Hamiltonian. In this way, three-photon Rabi oscillations can be observed in a small-detuning regime, thus avoiding the excitation of high-energy states. We derive an effective Hamiltonian (equivalent to the anisotropic Rabi model Hamiltonian) to determine the magnitude of the energy splitting and the resonance position. Numerical simulations results show that the protocol not only generates a three-photon resonance, but also has a detectable output photon flux. We hope the protocol can be exploited for the realization of Fock-state sources and the generation of multiparticle entanglement.
	
	
	Generating three-photon Rabi oscillations without a large-detuning condition
    Yan Xia
    August 5, 2024
============================================================================



§ INTRODUCTION

	The quantum Rabi model (QRM) <cit.>, describing the interaction of a two-level quantum system and a bosonic mode, is one of the fundamental physical models.
	Despite its simplicity, the QRM exhibits a rich set of physical properties. It has been used to describe the dynamics of various physical setups <cit.>, ranging from quantum optics <cit.> to condensed matter physics <cit.>. The full Hamiltonian of the QRM includes the counter-rotating (CR) terms, the presence of which leads to non-conservation of the system excitation number <cit.>.
	This excitation-number-nonconserving process produces virtual photons, enabling high-order atom-field resonant transitions <cit.>. Hence, one can realize analogs of many nonlinear-optics effects <cit.>, such as various frequency-conversion processes <cit.>, parametric amplification <cit.>, multiphoton absorption <cit.>, Kerr effect <cit.>, and other nonlinear processes <cit.>. 
	
	In general, the high-order resonant transition between system states requires a strong light-matter coupling to be noticeable <cit.>. When the atomic transition frequency and the cavity frequency satisfy a certain relation, this transition is also to be visible even if the coupling strength is weak.
	For instance <cit.>, a three-photon-resonance transition can be realized in a weak coupling regime as long as the frequency of the bosonic cavity is close to one third of the atomic transition frequency (i.e. “1/3 condition") through a virtual excitation process. This is an anomalous Rabi oscillation induced by the higher-order processes, in which three photons are co-emitted by the excited atom into a resonator and vice versa. Such a high-order resonance process can be used to realize Fock-state sources <cit.> and to generate multiparticle entanglement <cit.>.
	
	However, the “1/3 condition" <cit.> may lead to problems in experimental observation of the three-photon resonance. With such a large detuning, the anharmonicity of the artificial atom is too small (compared to detuning) to avoid excitation to the higher energy states. Addressing this problem, we propose a protocol to generate the three-photon resonance without such a large detuning. Specifically, by introducing a modulation to the energy separation of the artificial atom <cit.>, dynamics of the system can be described by an effective anisotropic Rabi Hamiltonian. By tuning the modulation frequency, the effective cavity frequency and the effective atomic transition frequency can satisfy the “1/3 condition" resulting in the three-photon resonance. The calculations indicate that under such a longitudinal drive <cit.>, the artificial atom does not jump to the third excited state or higher energy states, even if the energy level anharmonicity is much smaller than the atom transition frequency. In contrast to the protocol, the scheme <cit.> using the large-dutuning condition fails to generate three-photon Rabi oscillations in superconducting circuits when considering the influence of the energy level anharmonicity.
	
	The rest of the paper is organized as follows: In Sec. <ref>, we introduce the model and the Hamiltonian. In Sec. <ref>, the evidence of the three-photon resonance in an effective anisotropic Rabi model is shown by analysing the energy spectrum of the system. At the same time, we also give theoretical formulas to calculate the resonant frequency and energy splitting. In Sec. <ref>, we calculate the output photon flux rate of the system in the presence and absence of frequency modulation. In Sec. <ref>, the influence of the high energy levels of the artificial atom on the three-photon resonance is discussed. In Sec. <ref>, the experimental parameters are provided. At last, the work is summarized in Sec. <ref>. 
	

§ MODEL
 
	The QRM we consider (see Fig. <ref>) consists of an artificial atom with transition frequency Ω_0 and a cavity with resonant frequency Ω_c, where the energy levels  separation of the atom are modulated by an external flux bias. The Hamiltonian (ħ=1) of this model is given by

	
    H_1(t)   =H_R+H_M(t),

	where
	
    H_R=Ω_ca^†a+Ω_0/2σ_z+λ(a^†+a)σ_x

	is the Hamiltonian of the QRM, and

	
    H_M(t)=Acos(ω_ft)/2σ_z

	describes a sinusoidal modulation applied to the artificial atom. Here, a^† and a are the creation and annihilation operators, respectively. The coupling strength between the cavity and the artificial atom is λ. The Pauli matrices are defined as σ_z=|e⟩⟨e|-|g⟩⟨g| and σ_x=|e⟩⟨g|+|g⟩⟨e|, where the excited state and the ground state of the atom are |e⟩ and |g⟩, respectively. The modulation amplitude and frequency are A and ω_f, respectively. 
	  

    To study the dynamics of the model, we can transform the Hamiltonian H_R to the rotating frame defined by 
    
    U_1(t)= exp{-iΩ_cta^†a-i[Ω_0/2t+A/2ω_fsin(ω_ft)]σ_z}.
  
    Thus, using the Jacobi-Anger expansion
    
    e^ixsin(ω_ft)=∑_n=-∞^+∞J_n(x)e^inω_ft,

 the Hamiltoian in Eq. (<ref>) is written as

    H_2(t)   =   iU̇^†(t)U(t)+U^†(t)H_1(t)U(t)
       =   ∑_n=-∞^+∞λ J_n(x) a^†σ_+e^i(Ω_0+Ω_c+nω_f)t
          +∑_m=-∞^+∞λ J_m(x) aσ_+e^i(Ω_0-Ω_c+mω_f)t+ H.c.,

where x=A/ω_f. The transition operators of the atom are defined as σ_+=σ_-^†=|e⟩⟨g|. J_n(x) is the nth Bessel function of the first kind. For further discussion, we set δ=Ω_0-Ω_c and Δ=Ω_0+Ω_c-ω_f, as well as J_n≡ J_n(x), resulting in

    H_2(t)   =   λ(J_-1a^†σ_+e^iΔ t+ J_0aσ_+e^iδ t)      +∑_n≠-1λ J_n a^†σ_+e^i[Δ+(n+1)ω_f]t      +∑_m≠0λ J_m aσ_+e^i(δ+mω_f)t+ H.c..

Under the conditions

    ω_f≫{δ,Δ,λ| J_n(x) |},

we discard the fast oscillating terms in Eq. (<ref>) by performing the rotating wave approximation (RWA). Then the Hamiltonian H_2(t) in Eq. (<ref>) can be simplified as

    H_2(t)≈λ_1a^†σ_+e^iΔ t
    	+λ_2aσ_+e^iδ t+ H.c.,

where λ_1 = λ J_-1 and λ_2 = λ J_0 are the effective coupling strengths.

The Hamiltonian H_2(t) in Eq. (<ref>) is equivalent to the following anisotropic Rabi Hamiltonian <cit.>

    H_aR=ω_ca^†a+ω_0/2σ_z
    		+(λ_1a^†σ_++λ_2aσ_++ H.c.),

where ω_c=(Δ-δ)/2 and ω_0=(Δ+δ)/2 are the effective cavity frequency and effective atomic transition frequency, respectively. 



§ THREE-PHOTON RESONANCE
 
In this section, we give the parameter conditions for the three-photon resonance described by the anisotropic Rabi model Hamiltonian in Eq. (<ref>), and derive the effective Hamiltonian which describes an effective three-photon coupling between |e,0⟩ and |g,3⟩.



 §.§ Avoided Crossing and Resonance

 We show that there is also a three-photon resonance in the anisotropic Rabi model. Figure <ref> depicts the transition between states |e,0⟩ and |g,3⟩, and this transition is completely caused by excitation-number-nonconserving processes. In the parameter region of x=0.5 and λ=0.01ω_0≪δ, we numerically calculate a portion of the energy spectrum for the Hamiltonian H_aR as a function of ω_c and the results are plotting in Fig. <ref>. Here we focus on the third excited state and the fourth excited state of H_aR, i.e. H_aR|ψ_n⟩=E_n|ψ_n⟩ with n=3,4. In Fig. <ref>, we can find that the two energy levels exhibit an avoided crossing, which is a signature of the resonance between |e,0⟩ and |g,3⟩.  

 At this point, |ψ_3⟩ and |ψ_4⟩ can be well approximated by |B⟩=(|e,0⟩+|g,3⟩)/√(2) and |C⟩=(|e,0⟩-|g,3⟩)/√(2), respectively. By numerical calculations, the overlap, F_B=| ⟨ B|ψ_3⟩| and F_C=| ⟨ C|ψ_4⟩|, at the resonance point as functions of λ/ω_0 shows in Fig. <ref>. Both F_B and F_C are approximately equal to 1 for λ<0.01ω_0. We note that the overlaps decrease as the coupling strength increases. That is because when the interaction strength is too strong, eigenstates of the Hamiltonian H_aR are highly dressed so that cannot be described by |B⟩ and |C⟩.
 
  Figure <ref> demonstrates that three-photon Rabi oscillations are generated in the parametric regime. It also shows that the approximate result agrees [Fig. <ref>] well with the exact dynamics [Fig. <ref>] when the conditions in Eq. (<ref>) are satisfied. We see that the probability of the three-photon state |g,3⟩ can reach 99.18%. In the protocol, the parameter x is the ratio of the modulation amplitude A and the modulation frequency ω_f, i.e. x=A/ω_f. At a certain modulation frequency, the magnitude of the modulation amplitude A affects the value of the parameter x, which in turn changes the fidelity of the three-photon state. To find out how the fidelity depends on the parameter x, we plot the fidelity of the state |g,3⟩ at time t ≈ 1.1× 10^6/ω_0 as a function of x (see Fig. <ref>). We can find that when x is taken as 0.5, the fidelity of |g,3⟩ can reach nearby 1. In this case, the modulation amplitude A is less than half of the modulation frequency ω_f <cit.>, ensuring that the present protocol is physically reasonable.
 




 
   
    

 §.§ Effective Hamiltonian

    We derive the effective Hamiltonian of H_aR which determines the magnitude of energy splitting and the position of the resonance. To further understand how the three-photon resonance occur, we derive effective Hamiltonian using the high-order time-averaging method <cit.>.
    
    For the Hamiltonian H_2(t) in Eq. (<ref>), it can be written as 
    
    H_I=∑_n=-∞^+∞h_1,ne^iω_1,nt+∑_m=-∞^+∞h_2,me^iω_2,mt+ H.c.,
 
    where the identification h_1,n=λ J_na^†σ_+, h_2,m=λ J_maσ_+, ω_1,n=Δ+(n+1)ω_f, and ω_2,m=δ+mω_f. Under the condition of the three-photon resonance: ω_c≈ω_a/3, we have Δ≈2δ (i.e., ω_1,-1≈2ω_2,0). When |λ J_n|≪δ,
    we can get the second-order effective Hamiltonian <cit.>
    
    H_ eff^(2)=λ_2^2/δ(a^†aσ_z+σ_+σ_-)+λ_1^2/Δ(a^†aσ_z-σ_-σ_+)

    and the third-order effective Hamiltonian <cit.>
    
    H_ eff^(3)=-λ_2^2λ_1/δ^2[a^3σ_++(a^†)^3σ_-],

    where λ_1 = λ J_-1 and λ_2 = λ J_0. Therefore, the total effective Hamiltonian is given by
    
    H_ eff   =   ω_ca^†a+ω_0/2σ_z+H_ eff^(2)+H_ eff^(3)
       =   ω_ca^†a+ω_0/2σ_z+λ_2^2/δ(a^†aσ_z+σ_+σ_-)
          +λ_1^2/Δ(a^†aσ_z-σ_-σ_+)-λ_2^2λ_1/δ^2[a^3σ_++(a^†)^3σ_-],

    which descrbes the transition between |e,0⟩ and |g,3⟩, with Rabi frequency Ω_ eff,
    
    Ω_ eff=√(6)λ_2^2λ_1/δ^2.

    To check the validity of the obtained result, we then write the matrix form of the effective Hamiltonian H_ eff in the subspace formed by |e,0⟩ and |g,3⟩:
    
    H_ eff=[              ω_0/2+λ_2^2/δ          -√(6)λ_2^2λ_1/δ^2;          -√(6)λ_2^2λ_1/δ^2 3ω_c-ω_0-3λ_2^2/δ-4λ_1^2/Δ ].

    By equating the diagonal elements of this matrix, we can get a solution of the cavity field frequency ω_c=ω_c^',
    
    ω_c^'/ω_0=1/3+2(J_0^2+1/2J_-1^2)(λ/ω_0)^2+O(λ/ω_0)^4.

    According to the effective Hamiltonian in Eq. (<ref>), 2Ω_ eff can be understand as the energy splitting at the avoided crossing <cit.>. We compare the energy splitting Δ E=2Ω_ eff with the numerical simulation results obtained from the diagonalization of the Hamiltonian H_aR when a resonant transition occurs between |e,0⟩ and |g,3⟩. Figure <ref> shows the comparison between the analytical and numerical results for the energy splitting Δ E as a function of λ/ω_0. It can be seen that the percentage difference is less than 3% for λ/ω_0<0.05. 
    
    

§ OUTPUT PHOTON FLUX
 
    In this section, we will use the master equation to study the output photon flux of the cavity. Cavity field damping and atomic decay caused by environmental noise inevitably have an impact on the evolution of the research object. We consider a system that the artificial atom and the cavity field are connected respectively to different baths. We assume that the temperature of both baths is zero. By using Born-Markov approximation, the master equation <cit.> for the reduced density matrix of the system ρ(t) in the Schrödinger picture can be written as
    
    dρ(t)/dt=-i[H(t),ρ(t)]+κ D[X_1]ρ+γ D[X_2]ρ,

    where H(t) is the Hamiltonian of the system given by Eq.(<ref>), the decay rates of the artificial atom and the bosonic mode are γ and κ, respectively, and X_1(t) and X_2(t) are defined by
    
    X_1   =∑_E_n>E_m⟨ψ_m|(a+a^†)|ψ_n⟩|ψ_m⟩⟨ψ_n|,
    
        		X_2   =∑_E_n>E_m⟨ψ_m|σ_x|ψ_n⟩|ψ_m⟩⟨ψ_n|,

    with |ψ_n⟩ being the eigenvector of the Rabi Hamiltonian, i.e., H_R|ψ_n⟩=E_n|ψ_n⟩. The jumping operators are X_1 and X_2, respectively, which only contain transitions from high-energy eigenstates to low-energy eigenstates. The standard Lindblad superoperator D is defined by
    
    D[O]ρ=1/2(2Oρ O^†-O^† Oρ-ρ O^† O).

    The output photon flux rate <cit.> from the cavity field is defined by
    
    Φ_ out(t)=κ Tr[ρ(t)X_1^†X_1].

    Note that the forms of X_1 and X_2 have taken the CR terms into account. If the Hamiltonian H(t) is replaced by the Jaynes-Cummings (JC) Hamiltonian under the RWA, X_1 and X_2 are reduced to the forms a and a^† respectively appearing in the standard Lindblad form master equation <cit.>.
         
    We assume that the system is initially prepared in the state |e,0⟩, and obtain the reduced density operator ρ(t) of the system by numerically solving the master equation. In Fig. <ref>, we plot the time dependence of the output bosonic excitation flux rate Φ_ out(t). For the driving frequency ω_f=0, there is no steady output flux. After applying the drive with frequency ω_f=1.98Ω_0 and amplitude A=0.99Ω_0, we can see that the value of Φ_ out(t) approaches a stationary value when t≫100/ω_0. When the modulation frequency is increased to 5Ω_0, the steady output photon flux rate disappears. By analyzing the form of the Hamiltonian describing the system, we can explain why such a stable output bosonic excitation flux occurs only when ω_f∼ 2Ω_0. When there is no external driving, the system can be well described by the JC model in the near-resonance regime (i.e. the transition frequency of the atom and the cavity frequency are almost equal). If the system is initially prepared in the state |e,0⟩, the dissipation induced by the environment (the zero-temperature baths) will lead the system to the lowest eigenstates |g,0⟩, and the system will always stay in this state without emitting stable output photon flux. As the appropriate modulation is applied, the system is described by the quantum Rabi model, and the lowest energy eigenstate is no longer the state |g,0⟩. Instead, the state |g,0⟩ is a superposition of many eigenstates of the Rabi model. The influence of cavity field damping and atomic decay will induce transitions from high-energy states to low-energy states resulting in a stable output photon flux rate. 
    
    We show that when the modulation frequency is ω_f=1.98Ω_0, the system generates a three-photon Rabi oscillation. At this frequency, the dynamics of the system is well described by an effective anisotropic Rabi model, so that the system continues to radiate a stream of photons outward when dissipation is considered. However, when the modulation frequency is equal to 5Ω_0 or higher, the CR terms in Eq. <ref> can be neglected, at which point the dynamics of the system is described by the JC model, so no steady output flux is produced.
    
    

§ DISCUSSION ON THE INFLUENCE OF ENERGY LEVEL ANHARMONICITY
 
    For simplicity, we consider the lowest three energy levels of the artificial atom that is biased by a sinusoidal flux (see Fig. <ref>). The system dynamics is described by the Hamiltonian (ħ=1)
    
    H_3(t)=[   Ω_0+Acos(ω_ft)]|e⟩⟨e|
    
        		+   [Ω_b+2Acos(ω_ft)]|f⟩⟨f|
    
        		+   Ω_ca^†a+λ(a^†+a)(|e⟩⟨g|+|f⟩⟨e|+ H.c.).

    In the rotating frame defined by
    
    U_2(t)= exp   [-i(Ω_ca^†a+Ω_0|e⟩⟨e|+Ω_b|f⟩⟨f|)t
       -ixcos(ω_ft)(|e⟩⟨e|+2|f⟩⟨f|)] ,

    the Hamiltonian in Eq. (<ref>) is written as 
    
    H_3^'(t)=
        		λ   [a^†|e⟩⟨g|e^i(Ω_0+Ω_c)t+ixsin(ω_ft)
    
        		+   a|e⟩⟨g|e^i(Ω_0-Ω_c)t+ixsin(ω_ft)
    
        		+   a^†|f⟩⟨e|e^i(Ω_b-Ω_0+Ω_c)t+ixsin(ω_ft)
    
        		+   a|f⟩⟨e|e^i(Ω_b-Ω_0-Ω_c)t+ixsin(ω_ft)+ H.c.],

    where x=A/ω_f. By setting δ_b=Ω_b-2Ω_0 as the anharmonicity of atomic energy levels, and using Eq. (<ref>), resulting in
    
    H_3^”(t)=λ   (J_-1a^†|e⟩⟨g|e^iΔ t+ J_0a|e⟩⟨g|e^iδ t)
        		
        		+   ∑_n≠-1λ J_n a^†|e⟩⟨g|e^i[Δ+(1+n)ω_f]t
    
        		+   ∑_m≠0λ J_m a|e⟩⟨g|e^i(Δ+mω_f)t
    
        		+   ∑_lλ J_l
        		a^†|f⟩⟨e|e^i[δ_b+Δ+(1+l)ω_f]t
    
        		+   ∑_kλ J_k|f⟩⟨e|e^i[δ_b+δ+kω_f]t
        		+ H.c.

    Under the conditions
    
    λ| J_n(x) |≪δ≪{ω_f,δ_b},

    the Hamiltonian H_3^”(t) in Eq. (<ref>) can be simplified as
    
    H_3(t)≈λ J_1a^†|e⟩⟨g|e^iΔ t
        		+λ J_0a|e⟩⟨g|e^iδ t+ H.c.,

    which is equivalent to the Hamiltonian in Eq. (<ref>). Note that, in the protocol, the detuning δ can be very small, such that the anharmonicity δ_b can be much greater than δ. It shows that even considering the influence of high energy levels, the protocol can also generate the three-photon resonance.
    
    Without the frequency modulation (see Fig. <ref>), in the rotating frame, the Hamiltonian of the QRM is given by
    
    H_4(t)=
        		λ   [a^†|e⟩⟨g|e^iΔ t+a|e⟩⟨g|e^iδ^' t
    
        		+   a^†|f⟩⟨e|e^i(Δ+δ_b)t+a|f⟩⟨e|e^i(δ^'+δ_b)t+ H.c.].

    To avoid the excitation of the second-excited state |f⟩, according to Eq. (<ref>), the condition δ_b≫δ^' must be well satisfied. Due to the large-detuning regime, the anharmonicity δ_b and the detuning δ^' are on the same order of magnitude, which makes the second row in Eq. (<ref>) unable to be omitted by performing RWA. Therefore, the existence of the third excited state |f⟩ causes the three-photon resonance not to occur.
      
    

§ CIRCUIT IMPLEMENTATION AND THE EXPERIMENTAL PARAMETERS
 
    
    The protocol has the potential to be implemented in the circuit in Fig. <ref>. After quantisation (see Appendix <ref>), the LC oscillator can be represented as a bosonic field and the Kerr parametric oscillator (KPO) as an artificial atom with modulated energy levels separation. The applied flux bias is Φ_ ex=Acos(ω_ft) with A=0.5ω_f and ω_f=9.93 GHz. As reported in Ref. <cit.>, the modulation frequency ω_f/2π can range from 0 to 500 MHz in a circuit-QED, leading the modulation frequency close to the same order of magnitude as the atomic transition frequency. Therefore, it is possible that the modulation frequency may approach twice the artificial atom transition frequency. In the protocol, the frequency of the cavity field is Ω_c=5 GHz, the atomic transition frequency is Ω_0=5.066 GHz, and the coupling strength λ=0.198 MHz. The artificial atom decaying rate γ and the cavity field decaying rate κ are both 1.98 MHz. The above parameters are all experimentally reasonable <cit.>. By simulating the parameters in the protocol, we can observe the steady photon flux rate Φ_ out=0.06 kHz that can be detected in experiments.
      
    

§ CONCLUSIONS
 
    We propose a protocol to generate three-photon Rabi oscillations in the small-detuning regime. Specifically, by introducing a modulation to the artificial atom in the Rabi model, we can obtain an effective anisotropic Rabi model with tunable parameters. Through numerical simulations, clear evidence is shown that the three-photon resonance occurs when the modulation frequency is close to twice the transition frequency of the artificial atom. We derive the three-photon coupling effective Hamiltonian between |e,0⟩ and |g,3⟩, which determines the resonance position and the energy splitting in the coupling regime λ<0.05ω_0. The discussion of energy level anharmonicity indicates that the protocol can work well with small anharmonicity. Additionally, we calculate the output photon flux of the entire system. It is worth noting that there exists a nonzero stationary value of the output bosonic excitation flux rate as we apply a modulation to the artifical atom, indicating the occurrence of the three-photon resonance. The protocol will provide a way for investigating the excitation-number-nonconserving processes and have a more profound impact in the near future.
     
    

§ ACKNOWLEDGEMENTS

    Y.X. was supported by the National Natural Science Foundation of China under Grants No. 11575045 and No. 11874114, the Natural Science Funds for Distinguished Young Scholar of Fujian Province under Grant No. 2020J06011, and a project from Fuzhou University under Grant No.  JG2020001-2.  Y.-H.C. was supported by the National Natural Science Foundation of China under Grant No. 12304390.
	
	

§ DERIVATION OF THE HAMILTONIAN
 
	In this appednix, we derive the Hamiltonian of the circuit in Fig. <ref>. The Lagrangian of the circuit can be written as 
	
    ℒ=𝒯-𝒰,

	where
	
    𝒯   =(Φ_0/2π)^2[(C_ J+C_ JK/2N)ϕ̇^2+C_ res/2θ̇^2+C_ i/2(ϕ̇+θ̇)^2], 
    𝒰   =-E_ Jcos(ϕ+φ_ ex/2)-E_ Jcos(ϕ-φ_ ex/2)
           -NE_ JKcos(ϕ/N)+(Φ_0/2π)^21/2L_ resθ^2.

	Here, Φ_0 is the magnetic flux quantum and φ_ ex≡ 2πΦ_ ex/Φ_0; E_ J is the Josephson energy and C_ J is the capacitance of each junction in the DC SQUID part of the KPO; the energy and capacitance for each junction in the junction array are E_ JK and C_ JK; L_ res is the inductance and C_ res is the capacitance of the LC resonator; and C_ i is the coupling capacitance between the KPO and the LC resonator. By defining the conjugate number operators as 
	
    ħN̂_ϕ   =∂ℒ/∂ϕ̇=C_ KPOϕ̇-C_ intθ̇,
    ħN̂_θ   =∂ℒ/∂θ̇=-C_ intϕ̇+C_ anθ̇,

	where
	C_ KPO≡ (Φ_0/2π)^2(2C_ J+C_ JK/N+C_ int), C_ an≡ (Φ_0/2π)^2(C_ res+C_ int), and C_ int≡ (Φ_0/2π)^2C_ i, the resulting Hamiltonian is 
	
    ℋ̂=
    			4   E_ C^ϕN̂_ϕ^2+4E_ C^θN̂_θ^2+8E_ C^ intN̂_ϕN̂_θ+E_ L/2θ̂^2
       -2E_ Jcos(φ_ ex/2)cos(ϕ̂)
    			-NE_ JKcos(ϕ̂/N),

	where 
	
    E_ C^ϕ≡ħ^2C_ an/8(C_ KPOC_ an-C_ int^2), E_ C^θ≡ħ^2C_ KPO/8(C_ KPOC_ an-C_ int^2),   
       E_ C^ int≡ħ^2C_ int/8(C_ KPOC_ an-C_ int^2), E_ C^ int≡ħ^2C_ int/8(C_ KPOC_ an-C_ int^2),   
       E_ L≡ (Φ_0/2π)^21/L_ res.

	We move to the occupation-number representation by defining
	
    N̂_ϕ=iN_0^ϕ(â^†-â), ϕ̂=ϕ_0(â^†+â),   
       N̂_θ=iN_0^θ(b̂^†-b̂),
    		θ̂=θ_0(b̂^†+b̂),

	where N_0^ϕ=√(E_ JK/32NE_ C^ϕ), ϕ_0=√(2NE_ C^ϕ/E_ JK), N_0^θ=√(E_ L/32NE_ C^θ), and θ_0=√(2E_ C^θ/E_ L), we have 
	
    ℋ̂=ħ   ω_ KPOâ^†a+ħω_ LCb̂^†b+ħλ(â^†b̂+âb̂^†-âb̂-â^†b̂^†)
    
    			-   NE_ JKcos(ϕ̂/N)-E_ JK/2Nϕ̂^2-2E_ Jcos(φ_ ex/2)cos(ϕ̂),

	where ħω_ KPO≡√(8E_ C^ϕE_ JK/N), ħω_ LC≡√(8E_ C^θE_ L), and ħλ≡ E_ C^ int√(4E_ JKE_ L/(NE_ C^ϕE_ C^θ)). 
	
	Using the Taylor series cos(ϕ̂)=1-ϕ̂^2/2+ϕ̂^4/24..., we have 
	
    ℋ̂_ JJ   =   
    		-NE_ JK[1+1/24(ϕ̂/N)^4-...]
          -2E_ Jcos(φ_ ex/2)(1-ϕ̂^2/2+ϕ̂^4/24-...),

	which can lead to energy level anharmonicity and energy level modulation. Utilizing the Eq. (<ref>), we can obtain the Hamiltonian for the circuit as follows,
	
    ℋ̂   ≈   [ħω_ KPO-E_ C^ϕ/N^2+ħω_ KPONE_ J/E_ JKcos(φ_ ex/2)]â^†â
          -E_ C^ϕ/2N^2â^†â^†ââ+ħω_ LCb̂^†b+ħλ(â^†b̂+âb̂^†-âb̂-â^†b̂^†)
       =   ħ[Ω_0+Acos(ω_ft)]â^†â+ħΩ_cb̂^†b̂-ħδ_bâ^†â^†ââ
          +ħλ(â^†b̂+âb̂^†-âb̂-â^†b̂^†),

    where ħΩ_0=ħω_ KPO-E_ C^ϕ/N^2, ħ A=ħω_ KPONE_ J/E_ JK, ω_ft=φ_ ex/2, ħΩ_c=ħω_ LC, and ħδ_b=E_ C^ϕ/N^2. By controlling the value of the inharmonicity δ_b, we can consider the KPO as an artificial atom with only the lowest three energy levels. Therefore, we obtain the Hamiltonian in Eq. (<ref>).
	

	


 Paper Title: Rank Is All You Need: Estimating the Trace of Powers of Density Matrices.



















hanwoolmj@kaist.ac.kr
School of Computing, KAIST, Daejeon 34141, Korea


harris.junseo@gmail.com
Quantum AI Team, Norma Inc., Seoul 04799, Korea


smilelee9@kaist.ac.kr
School of Computing, KAIST, Daejeon 34141, Korea


kgjeong6@snu.ac.kr
Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea

Rank Is All You Need: Estimating the Trace of Powers of Density Matrices
    Kabgyun Jeong
    
========================================================================



§ ABSTRACT 
Estimating the trace of powers of identical k density matrices (i.e., (ρ^k)) is a crucial subroutine for many applications such as calculating nonlinear functions of quantum states, preparing quantum Gibbs states, and mitigating quantum errors. Reducing the requisite number of qubits and gates is essential to fit a quantum algorithm onto near-term quantum devices. Inspired by the Newton-Girard method, we developed an algorithm that uses only 𝒪(r) qubits and 𝒪(r) multi-qubit gates, where r is the rank of ρ. We prove that the estimation of {(ρ^i)}_i=1^r is sufficient for estimating the trace of powers with large k > r. With these advantages, our algorithm brings the estimation of the trace of powers closer to the capabilities of near-term quantum processors. We show that our results can be generalized for estimating (Mρ^k), where M is an arbitrary observable, and demonstrate the advantages of our algorithm in several applications.






§ INTRODUCTION




 §.§ Trace of Powers

Estimation task for the trace of the product of identical density matrices, which is represented as

    (ρ^k)   `trace of powers'

given access to copies of the quantum state ρ, is a core subroutine for many algorithms and applications in quantum information theory. We refer to this quantity as the `trace of powers,' which is used to calculate the value of integer Rényi entropy <cit.>, nonlinear functions of quantum states <cit.>, and deducing the eigenvalues of the quantum state, a process known as entanglement spectroscopy <cit.>.

We focus on estimating (ρ^k) for large integer k. The main applications are calculating the nonlinear functions of quantum states, which need estimation of the trace of large powers. Precisely, Yirka and Subaşı <cit.> proved that the trace of `well-behaved’ polynomials g(ρ), such as g(x) = (1+x)^α and log(1+x), can be efficiently estimated using the trace of powers. Moreover, (e^βρ) is an example with applications in thermodynamics.

The preparation of quantum Gibbs states <cit.> is an essential part of quantum computation, used in various applications such as quantum simulation, quantum optimization, and quantum machine learning. The truncated Taylor series

    S_k(ρ) = ∑_i=1^k ((ρ-I)^iρ)

is exploited as the cost function for variational quantum Gibbs state preparation <cit.>, which can be calculated by {(ρ^i)}_i=1^k.

Several methods for the estimation of the trace of powers have been proposed, such as the generalized swap test <cit.>, entanglement spectroscopy via Hadamard test <cit.>, two-copy test <cit.>, qubit-efficient entanglement spectroscopy <cit.>, multivariate trace estimation <cit.>, and methods using randomized measurement such as classical shadows <cit.>. An analysis of these methods is performed in Section <ref>.

Our work is inspired by the Newton-Girard method, as demonstrated in Section <ref>, we only uses the quantum devices for estimating {(ρ^i)}_i=1^r, where r=rank(ρ). (Here, r always refers to the rank of the quantum state ρ.) Subsequently, we make use of a classical computer with a recursive formula to calculate (ρ^k) for large k. The Newton-Girard method and recursion are used in the proof. Furthermore, we argue that combining our work with previous ones <cit.> improves its algorithmic performance. The number of needed qubits (i.e., width of the circuit) and the required multi-qubit gates are reduced, and in many cases, the copy complexity is also reduced. We support our work with numerical simulations. To emphasize the importance of our work, we demonstrate advantages when applying our method to applications such as calculating nonlinear functions of quantum states, preparation of quantum Gibbs states, and quantum error mitigation <cit.>.

Our paper is structured as follows. In Section <ref>, we provide a review of the related research conducted so far and explain the Newton-Girard Method, which provided the intuition for our study, in Section <ref>. Next, in Section <ref>, we present the main theorem and proof that the core idea of this paper, `rank is sufficient' for estimating the trace of powers. Section <ref> provides the numerical simulation results and interpretations to explain our ideas. In Section <ref>, we discuss examples where efficient computation of (ρ^k) for large integers k can be applied and the efficiency gains brought by our method. Finally, we present the conclusions and suggest directions for future research in Section <ref>. Detailed theoretical proofs or additional information not covered in the main text are included in the appendix.




 §.§ Literature Review

The swap test (ST) <cit.> estimates (ρσ), the trace of the product of two matrices ρ and σ:

    (S(ρ⊗σ)) = (ρσ),

where S denotes the swap operator. The ST can be performed using 1 ancilla qubit with 1 controlled-SWAP () operation and 2 Hadamard gates as shown in Figure <ref>. The ST can be thought of as performing the observable S on equation (<ref>). The observation that quantities like (ρσ) can be estimated without the need for full-state tomography was a significant development.


Following this line of thinking, Ekert et al. <cit.> proposed a cyclic shift permutation operator W^π for a generalized ST:

    (W^π(ρ_1⊗⋯⊗ρ_k)) = (ρ_1⋯ρ_k).


By using equation (<ref>) above, the trace of powers (ρ^k) can be easily calculated. Note that regardless of the dimension d=dim(ρ_i) and the number of quantum states k, the generalized ST needs only 𝒪(ϵ^-2) runs on a quantum device for ϵ error prediction. Thus, 𝒪(kϵ^-2) copies are needed for the estimation of (ρ^k). This method requires 𝒪(k) qubits, a quantum circuit of 𝒪(k) depth, and 𝒪(k) multi-qubit gates, where n is the number of qubits in ρ.

Various methods have been proposed for better estimation <cit.> of the trace of powers. A comparison of these methods is shown in Table <ref>.

The entanglement spectroscopy via hadamard test (HT) <cit.> is a generalized algorithm that estimates the expectation value of an arbitrary unitary operator or observable M. Specifically, ST can be thought of as a special case of HT when M=S. The HT has linear depth 𝒪(k) and uses 𝒪(k) copies of the state. A more improved algorithm, the entanglement spectroscopy via two-copy test (TCT) <cit.>, achieves constant depth and uses 𝒪(k) copies of the state. Thus, both use 𝒪(k) qubits in the estimation circuit. That is, both HT and TCT are improved algorithms but need the original entangled pure state |ψ⟩_AB for the estimation of (ρ_A^k), where ρ_A = _B(|ψ⟩⟨ψ|_AB).

Qubit-efficient entanglement spectroscopy <cit.> employs qubit-reset strategies to reduce the number of qubits in the quantum circuit. This method requires only n qubits, constant in terms of power k. When combined with TCT, it requires a linear circuit depth 𝒪(k). Also, the reference <cit.> defines the notion of `effective depth,' and TCT with qubit-reset strategy requires only a constant effective circuit depth 𝒪(1). However, this qubit-reset strategy still demands 𝒪(k) copies of the original entangled pure state |ψ⟩_AB, and qubit-reset could lead to more vulnerability to noise.

Without the need for the entangled pure state |ψ⟩_AB, multivariate trace estimation <cit.> (ρ_1ρ_2⋯ρ_k), a general case of the trace of powers, with constant quantum depth <cit.> has been proposed. Inspired by the method of Shor error correction <cit.>, this method requires only constant circuit depth, using 𝒪(k) multi-qubit gates and 𝒪(k) qubits. This result establishes and contributes many applications for multivariate trace and trace of powers estimation.

Combining our work with previous ones provides advantageous solutions for estimating (ρ^k) with large k. In combination with multivariate trace estimation <cit.>, we can reduce the number of required qubits from 𝒪(k) to 𝒪(r) and multi-qubit gates from 𝒪(k) to 𝒪(r) for (ρ^k) estimation. Additionally, in many cases (when the largest eigenvalue of ρ is not close to 1), our method offers improvement in terms of copy complexity.




§ INTUITION: NEWTON-GIRARD METHOD

The main idea of entanglement spectroscopy demonstrates that the trace of powers of density matrices can be used to estimate the largest eigenvalues <cit.>. The k largest eigenvalues can be estimated using {(ρ^i)}_i=1^k. The Newton-Girard method <cit.> provides the mathematical foundation of entanglement spectroscopy and serves as an important component in our method. Therefore, we describe the details of the Newton-Girard method and explain the inspiration that leads to the notion that `rank is sufficient' for estimating the trace of powers.

Let r=rank(ρ), and the eigenvalues of ρ are {p_i}_i=1^r, sorted in descending order. We prove, using the Newton-Girard method, that exactly knowing the trace of powers  {(ρ^i)}_i=1^r is equivalent to knowing {p_i}_i=1^r.

Consider the equation having these eigenvalues as root in the form of

    ∏_m=1^r(x-p_m) = 0.

The values of Tr(ρ^i) are now the i-th power sum of the roots. Denote the power sum as

    P_i := ∑_m=1^rp_m^i = Tr(ρ^i).



Here, Simply expanding the terms of the equation (<ref>) above as follows:

    ∏_m=1^r(x-p_m) = ∑_k=0^r (-1)^k a_k x^r-k,

where a_k is the elementary symmetric polynomial, defined as the sum of all distinct products of k distinct variables, such as:

    a_0    = 1, 
    
        a_1    = p_1 + p_2 + ⋯ + p_r = ∑_1 ≤α≤ r p_α, 
    
        a_2    = p_1 p_2 + p_1 p_3 + ⋯ + p_r-1 p_r = ∑_1 ≤α < β≤ r p_α p_β, 
    
        a_3    = ∑_1 ≤α < β < γ≤ r p_α p_β p_γ, 
    ⋮
    
        a_r    = p_1 p_2 ⋯ p_r.


The Newton-Girard method states the relationship of the elementary symmetric polynomials and the power sums recursively as follows. For all r≥ k ≥ 1,

    a_k = 1/k∑_i=1^k (-1)^i-1 a_k-i P_i.

Given P_i for 1 ≤ i ≤ r, we can determine the values of a_k and the full equation as

    ∏_m=1^r(x-p_m)

uniquely. Therefore, the set of eigenvalues is uniquely determined as the roots of the equation (<ref>).

Unfortunately in real-world situations, we cannot exactly calculate the trace of powers; instead, we can obtain the estimation with errors using previous strategies. Then, it is natural to ask the following question: `If the error of estimated power sums P_i=(ρ^i) is small, are the roots obtained by the Newton-Girard method close to the eigenvalues of ρ?' No, the statement is not always true. A counterexample is Wilkinson's polynomial <cit.>, which shows that the location of the roots can be very sensitive to perturbations in the coefficients of a polynomial. Generally, to obtain the eigenvalues, the estimation error of the trace of powers should be exponential, causing the copy and time complexity to be exponential <cit.>. Therefore, estimating the eigenvalues with the estimated values of {(ρ^i)}_i=1^r is unfeasible. 

However, we get an intuition from the Newton-Girard method that estimating {(ρ^i)}_i=1^r contains valuable information about the quantum states. In the next section, we prove that estimating the trace of powers {(ρ^i)}_i=1^r is sufficient for estimating the trace of larger powers (ρ^i) for i>r. The error of each eigenvalue obtained by the Newton-Girard method is large, but as the power of the eigenvalues is summed up, the error diminishes to a smaller extent.





§ RANK IS ALL YOU NEED

Our method for estimating (ρ^i) for large i > r is a direct consequence of the Newton-Girard method. For estimation with error less than 𝒪(ϵ), proceed with this Algorithm 1.




See Figure <ref> for a visualization of our (hybrid quantum-classical) algorithm. Even though the method is simple, we argue that it is advantageous in terms of the number of needed qubits and multi-qubit gates. It also uses fewer copies of the quantum states in some cases. We believe that our work is the first to prove that the trace of at most rank powers {(ρ^i)}_i=1^r is sufficient for the estimation of (ρ^k) when k is large. Now, we demonstrate our proof.

Let ϵ_i := Q_i - P_i and let a_k, b_k be the elementary symmetric polynomials corresponding to each P_i and Q_i. Here, P_i = (ρ^i) and Q_i are defined as the estimated value of P_i on a quantum device for i ≤ r and otherwise defined by equation (<ref>).


Let t_k := b_k - a_k, then the following holds:

    |t_k| ≤∑_j=1^k |ϵ_j|/j

where ϵ_j = Q_j - P_j is the error that occurred by the estimation of P_j = (ρ^j).




Let's try to find some properties of |t_i|.

    
  * If i=1,

    t_1    = b_1 - a_1 = Q_1 - P_1 = ϵ_1,

this gives |t_1|≤|ϵ_1|.
    
  * And if i=2,

    t_2    = b_2 - a_2 = Q_1^2 - Q_2/2 - P_1^2 - P_2/2
       = (Q_1 + P_1)(Q_1 - P_1) - (Q_2 - P_2)/2
       = ϵ_1 - 1/2ϵ_2,

this gives |t_2|≤|ϵ_1| + |ϵ_2|/2.
    
  * And if i=3,

    t_3    = (b_2 - a_2)Q_1 - (b_1 - a_1)Q_2 + (Q_3 - P_3)/3
           + -a_1(Q_2 - P_2) + a_2(Q_1 - P_1)/3.

this gives,

    |t_3|    ≤|t_2| P_1 + |t_1| P_2 + |ϵ_3| + a_1 |ϵ_2| + a_2 |ϵ_1|/3
       ≤ |ϵ_1| + |ϵ_2|/2 + |ϵ_3|/3.



Now, we suppose |t_k| ≤∑_i=1^k |ϵ_i|/i. Then,

    t_k+1   = ∑_i=1^k+1 (-1)^i-1 b_k+1-i Q_i - ∑_i=1^k+1 (-1)^i-1 a_k+1-i P_i 
       = ∑_i=1^k+1 (-1)^i-1 (b_k+1-i - a_k+1-i) Q_i 
           - ∑_i=1^k+1 (-1)^i-1 a_k+1-i (Q_i - P_i).

Taking the absolute value, and by using |Q_i|≤ 1 and |a_k+1-i|≤ 1,

    |t_k+1|    ≤∑_i=1^k |b_k+1-i - a_k+1-i| + ∑_i=1^k+1 |Q_i - P_i| 
        = ∑_i=1^k ∑_j=1^k+1-jϵ_j/j + ∑_i=1^k+1 |ϵ_i| 
        = ∑_i=1^k+1(k+1-j/j + 1 ) |ϵ_j| 
        = (k+1) ∑_j=1^k+1|ϵ_j|/j.

So, we can conclude that ∀ k∈ℕ, |t_k|≤∑_i=1^k |ϵ_i|/i by strong mathematical induction logic.




Suppose that

    ε_i:=|ϵ_i| = |P_i - Q_i| < ϵ/2T ln r

holds for i=1, 2, ⋯, r, where T is defined as:

    T = ∑_i=1^r p_i(1 - p_i^k)(1 - p_i^r)/(1 - p_i)^2≤ kr.

Then the following relation always holds:

    |ϵ_i| = |P_i - Q_i| ≤ϵ

for i=1, 2, ⋯, k.


Consider the following polynomials,

    x^r-a_1x^r-1+a_2x^r-2-⋯+(-1)^ra_r
          =(x-p_1)(x-p_2)⋯(x-p_r),
       y^r-b_1y^r-1+b_2y^r-2-⋯+(-1)^rb_r
          =(y-q_1)(y-q_2)⋯(y-q_r).

Then,

    p_i^r+k   = ∑_j=1^r (-1)^j-1 a_j p_i^r+k-j, 
    
    q_i^r+k   = ∑_j=1^r (-1)^j-1 b_j q_i^r+k-j.

The difference of the two terms becomes:

    ∑_j=1^r   (q_i^r+k-p_i^r+k)
       =∑_j=1^r(-1)^j-1b_j∑_i=1^r(q_i^r+k-j-p_i^r+k-j)
           +∑_i=1^r(-1)^j-1(b_j-a_j)∑_i=1^rp_i^r+k-j

Here, we notice that

    P_r+k-j = ∑_i=1^r p_i^r+k-j.

Then,

    ϵ_r+k   =∑_j=1^r {(-1)^j-1(b_j ϵ_r+k-j+t_jP_r+k-j)},
    ϵ_r+k-1   =∑_j=1^r {(-1)^j-1(b_j ϵ_r+k-j-1 + t_jP_r+k-j-1)}.

By exploiting b_1 = 1, we can sum up the above expressions in the form of

    ϵ_r+k   =∑_j=2^r (-1)^j-1 b_j ϵ_r+k-j + ∑_j=1^r (-1)^j-1 b_j ϵ_r+k-j-1
           + ∑_j=1^r (-1)^j-1 t_j (P_r+k-j+P_r+k-j-1) 
        = ∑_j=1^r-1 (-1)^j (b_j-b_j+1)ϵ_r+k-j-1 + (-1)^r-1 b_r ϵ_k 
           + ∑_j=1^r (-1)^j-1 t_j (P_r+k-j+P_r+k-j-1).

Since b_j ≥ b_j+1 (see the details in Appendix <ref>),

    |ϵ_r+k|≤   ∑_j=1^r-1 (b_j-b_j+1) |ϵ_r+k-j-1| + b_r |ϵ_k| 
        + ∑_j=1^r |t_j| (P_r+k-j+P_r+k-j-1).

Let ϵ' := max_1≤ j≤ r |ϵ_j|. Suppose that

    ϵ_r+m = ϵ' + ∑_j=1^r {|t_j| (∑_i=1^m P_r-j+i)}

holds for m=1, 2, ⋯, k-1. Then,

    |ϵ_r+k| 
       ≤∑_j=1^r-1(b_j - b_j+1) {ϵ' + ∑_j=1^r |t_j| (∑_i=1^k-2 P_r-j+i)}
           + b_r {ϵ' + ∑_j=1^r |t_j| (∑_i=1^k-2 P_r-j+i)}
           + ∑_j=1^r |t_j| (P_r+k-j + P_r+k-j-1) 
       ≤ b_1 {ϵ' + ∑_j=1^r |t_j| (∑_i=1^k-2 P_r-j+i)}
           + ∑_j=1^r |t_j| (P_r+k-j + P_r+k-j-1)
       = ϵ' + ∑_j=1^r {|t_j| (∑_i=1^k P_r-j+i)}.

Moreover, m = k also holds. Since m = 0 trivially holds, by strong mathematical induction logic, for every m, equation (<ref>) holds.

By the assumption of the theorem, we have

    ϵ' ≤ϵ/2T ln r.

By applying Lemma <ref>, we get: 

    |t_k| ≤∑_i=1^k ϵ'/i≤ϵ' ln r ≤ϵ/2T.


Next, with elementary sequence summation we address that

    ∑_i=1^r ∑_j=1^k P_r-j+i = T,

Finally,

    ϵ_r+k≤ϵ' + ϵ/2T× T ≤ϵ,

which completes the proof.






To estimate (ρ^i) for all i ≤ k within an additive error of ϵ and with a success probability of at least 1 - δ, where δ∈ (0,1), it is necessary to estimate each (ρ^j) for j ≤ r within an additive error of ε_j, as defined in Theorem <ref>. This can be achieved by using

    𝒪(T^2/ϵ^2ln^2 r ln(1/δ))

runs on a constant-depth quantum circuit consisting of 𝒪(j) qubits and 𝒪(j)  operations. Here, T is defined in Theorem <ref>. The parameter T scales from a constant to kr and increases as the maximum eigenvalue of ρ approaches 1.


Using the multivariate trace estimation method <cit.>, it is known that with 𝒪(ϵ^-2ln(δ^-1)) runs on a constant-depth quantum circuit consisting of 𝒪(i) qubits and 𝒪(i)  operations, we can estimate each (ρ^i) within an ϵ additive error and with a success probability of no less than 1-δ. Note that only the maximum error ϵ' := max_1≤ j≤ r |ϵ_j| affects the error of our algorithm. Thus, with 𝒪(T^2 ln^2 rϵ^-2ln(δ^-1)) runs, we can satisfy the assumption in Theorem <ref>. Hence, (ρ^i) (∀ i ≤ k) can be estimated within an ϵ error and with a success probability of no less than 1-δ.


Note that T scales from a constant to kr, mainly depending on the largest eigenvalue p_1 of ρ. If p_1 is not close to 1 (e.g., p_1 = 1/2), then T = 𝒪(1). This implies that if ρ is far from a pure state, then T is a constant value. In Section <ref>, we examine how the value of T changes as the largest eigenvalue p_1 increases. We emphasize that, in most cases, T is very small compared to the achievable maximum kr. Hence, in most cases, we achieve copy complexity advantage using our method. A comparison with previous methods is elaborated in Table <ref>.

As mentioned, our work provides an advantage in terms of the number of needed qubits and multi-qubit gates. Since we only need to estimate {(ρ^i)}_i=1^r, nr qubits and 𝒪(r)  operations are sufficient for the estimation. We emphasize that reducing the number of qubits and  operations used in the quantum circuit is an important improvement because it is less sensitive to noise, and having fewer qubits is advantageous for implementation on near-term quantum devices <cit.>.

Similarly, we can generalize our result to estimating (Mρ^k), the trace of powers with arbitrary observables M. Let P_i,M := (Mρ^k) and Q_i,M be the estimated value of P_i,M on a quantum device using the method of <cit.>.


Suppose that

    ε_i,M:=|ϵ_i,M| = |P_i,M-Q_i,M| < ϵ/2,

and

    ε_i=|ϵ_i| = |P_i-Q_i| < ϵ/2T‖ M‖_∞ln r,

holds for i=1,2,⋯,r, 
where the operator norm ‖ M ‖_∞ is defined corresponding to the ∞-norm for vectors ‖ x ‖_∞, as ‖ M ‖_∞ = sup_x ≠ 0‖ Mx ‖_∞/‖ x ‖_∞,

    T = ∑_i=1^r p_i(1-p_i^k)(1-p_i^r)/(1-p_i)^2≤ kr.

Then the following holds:

    |ϵ_i,M| = |P_i,M-Q_i,M| ≤ϵ

for i=1,2,⋯,k.


The idea of the proof is similar to Theorem <ref>. Detailed proof can be found in Appendix <ref>.   



Suppose there is an efficient decomposition M = ∑_ℓ=1^N_M a_ℓ P_ℓ, where a_ℓ∈ℝ and P_ℓ = σ_ℓ_1⊗⋯⊗σ_ℓ_n are tensor products of Pauli operators σ_ℓ_1, ⋯, σ_ℓ_n∈{σ_x, σ_y, σ_z, I}. The quantity ∑_ℓ=1^N_M |a_ℓ| = 𝒪(c) is bounded by a constant c. To estimate (Mρ^i) for all i ≤ k within an additive error of ϵ and with a success probability of at least 1 - δ, where δ∈(0,1), it is necessary to estimate each (Mρ^j) for j ≤ r within an additive error of ε_j,M as defined in Theorem <ref>. This can be achieved by using

    𝒪(c^2N_M/ϵ^2ln(1/δ))

runs on a constant-depth quantum circuit consisting of 𝒪(j) qubits and 𝒪(j)  operations, and estimating each (ρ^j') for j' ≤ r within an additive error of ε_j' as defined in Theorem <ref>, by using

    𝒪(T^2/ϵ^2ln^2 r ln(1/δ))

runs on a constant-depth quantum circuit consisting of 𝒪(j') qubits and 𝒪(j')  operations. Here, T is defined in Theorem <ref>. The parameter T scales from a constant to kr and increases as the maximum eigenvalue of ρ approaches 1.


Using the multivariate trace estimation method <cit.>, it is known that with 𝒪(ϵ^-2ln(δ^-1)) runs on a constant-depth quantum circuit consisting of 𝒪(i) qubits and 𝒪(i)  operations, we can estimate each (ρ^i) within ϵ additive error and with success probability not smaller than 1-δ. Note that only the maximum error ϵ' := max_1 ≤ j ≤ r |ϵ_j| affects the error of our algorithm. So with 𝒪(T^2 ln^2 rϵ^-2ln(δ^-1)) runs for estimating (ρ^j') (j' ≤ r), and 𝒪(c^2N_Mϵ^-2ln(δ^-1)) runs for estimating (Mρ^j) (j ≤ r), we can satisfy the assumption in Theorem <ref> with success probability not smaller than 1-δ. Hence, (Mρ^i) (∀ i ≤ k) can be estimated within ϵ error and with success probability not smaller than 1-δ.






§ NUMERICAL SIMULATIONS




 §.§ Reduction of Required Runs





Now, We analyze the behavior of T defined in equation (<ref>) as the eigenvalues of ρ change. Since we don't know the exact distribution of the eigenvalues of an arbitrary density matrix ρ, we consider random eigenvalues from a uniform distribution, along with several typical cases where the eigenvalues are uniform except for the maximum, and cases where the eigenvalues decay arithmetically or geometrically. These examples are described in Figure <ref>.

Figure <ref> shows that T is mostly constant and that our method provides a copy complexity advantage over previous works (refer to Table <ref>). The value of T is significantly less than the upper bound, kr, in most cases. Especially when the state is mixed (e.g., the largest eigenvalue is small), the system is large (e.g., rank is large), or k grows large, the expected advantage becomes much more dramatic compared to the upper bound.



 §.§ Simulation Results

We conducted numerical simulations for three types of additive error bounds ϵ, two types of quantum state ranks r, four types of trace powers k, and three patterns D (distributions) of eigenvalues λ = {λ_i}. The parameter settings are as follows:

    
  * Additive error bound ϵ∈{10^-8, 10^-6, 10^-4}
    
  * Rank r ∈{4, 8}
    
  * Trace power k ∈{16, 32, 64, 128}
    
  * Eigenvalue patterns D(λ) ∈{, , }


For the eigenvalue patterns, detailed settings regarding the maximum eigenvalue λ_max and the minimum eigenvalue λ_min are as follows, based on their differences or ratios:

    
  * λ_max - λ_min∈{0, 0.25, 0.5} for (r, D(λ)) = (4,)
    
  * λ_max - λ_min∈{0, 0.06, 0.125} for (r, D(λ)) = (8,)
    
  * λ_max/λ_min∈{2^8, 2^10, 2^12} for (r, D(λ)) = (4,)
    
  * λ_max/λ_min∈{2^14, 2^16, 2^18} for (r, D(λ)) = (8,)








In Figure <ref>, results for eigenvalue distributions with arithmetic decay are presented. Furthermore, Figure <ref> shows results for eigenvalue distributions with geometric decay, and Figure <ref> displays results for random eigenvalue distributions. It can be observed from Figures <ref> and <ref> that for the same trace of powers, a larger rank value corresponds to a decrease in mean absolute error. This is demonstrated by the relative positions of markers with different shapes for lines of the same color in Figures <ref> and <ref>. In Figure <ref>, it is also evident that the different marker shapes distinctly show variations in the magnitude of the errors.

Additionally, as the power of the trace increases, both the scale of Tr(ρ^i) and the scale of the absolute error become very small, sometimes even dropping below the machine epsilon, which represents the upper bound of the error that a computer can handle for floating-point numbers. Therefore, to eliminate the impact of errors from floating-point calculations, we expressed the numbers from iterative estimations as fractions of integers instead of using floating-point types in the implementation of our algorithm.




§ APPLICATIONS



 §.§ Calculating the Nonlinear Functions of Quantum State

Applying Corollary <ref> to Theorem 5 in <cit.>, we can enhance the theorem.


Let ρ be a quantum state with rank r. Suppose there exist ϵ > 0 and a slowly-growing function C (as a function of m) such that g: ℝ→ℝ is approximated by a degree m polynomial f(x) = ∑_k=0^m c_k x^k on the interval [0,1], in the sense that

    sup_x∈ [0,1] |g(x)-f(x)| < ϵ/2r,

and

    ∑_k=0^m |c_k| < C.

Then estimating (g(ρ)) within an ϵ additive error and with a success probability of at least 1 - δ, where δ∈(0, 1) requires

    𝒪(T^2 C^2 r^2/ϵ^2ln^2 r ln(1/δ))

copies of ρ and

    𝒪(T^2 C^2 r/ϵ^2ln^2 r ln(1/δ))

runs on a constant-depth quantum circuit consisting of 𝒪(r) qubits and 𝒪(r)  operations.

In the original theorem mentioned in reference <cit.>, 𝒪(C^2m^2ϵ^-2log(δ^-1)) copies of ρ were required, and the circuit consisted of 𝒪(m) qubits and 𝒪(m)  operations. Typically, m is much larger than r, so our enhanced theorem offers advantages for estimating g(ρ). When g(x) = e^β x, C becomes e^|β|. We can efficiently estimate (e^βρ) using Theorem <ref>, which has applications in thermodynamics and the density exponentiation algorithm <cit.>.




 §.§ Preparation of Quantum Gibbs State

We highlight that our method improves the efficiency of preparing the quantum Gibbs state. The truncated Taylor series:

    S_k(ρ) = ∑_i=1^k ((ρ-I)^kρ)

is used as the cost function for variational quantum Gibbs state preparation <cit.>. It is shown that the fidelity F(ρ(θ_0), ρ_G) between the optimized state ρ(θ_0) and the Gibbs state ρ_G is bounded by

    F(ρ(θ_0), ρ_G) ≥ 1-√(2(βϵ + 2r/k+1(1-Δ)^k+1))
 
where β is the inverse temperature of the system, and Δ is a constant that satisfies -Δln(Δ) < 1/k+1(1-Δ)^k+1. By using the inequality

    D(ρ(θ_0), ρ_G) < √(1-F(ρ(θ_0), ρ_G)),

to achieve T(ρ(θ_0), ρ_G) < ϵ, we need to set K = 𝒪(r ϵ^-4), where T is the trace distance. Using previous methods, K = 𝒪(r ϵ^-4) qubits and  operations are required, which are impractical for near-term quantum devices. Our work significantly reduces the number of qubits and  operations to 𝒪(r), independent of the desired error bound ϵ. This demonstrates that our method makes the preparation of the quantum Gibbs state using the truncated Taylor series much more feasible.





 §.§ Quantum Error Mitigation

If we attempt to prepare an n-qubit pure state |ψ⟩, environmental noise can cause the actual prepared state to be a mixed state ρ = ℰ(|ψ⟩⟨ψ|), where ℰ represents a map including unitary evolution and noise channels. In the fault-tolerant case, the expected value of a Hermitian operator M is given by ⟨ M ⟩ = Tr(M|ψ⟩⟨ψ|). Due to noise, this value becomes ⟨ M ⟩_noise = Tr(Mρ) ≠⟨ M ⟩.

Virtual distillation protocol <cit.> offers a method to address this issue. The protocol involves using collective measurements of k copies of the mixed state ρ to suppress incoherent errors. Specifically, it measures expectation values with respect to the state ρ^k / Tr(ρ^k). This approach approximates the error-free expectation value of M by:


    ⟨ M ⟩_vd^(k) = Tr(Mρ^k)/Tr(ρ^k),

where k denotes the number of copies used. As k increases, the resulting estimator converges exponentially quickly toward the expectation value of the pure state closest to ρ. Thus, efficiently estimating values such as Tr(Mρ^k) and Tr(ρ^k) plays a crucial role as a subroutine in quantum error mitigation. 

Quantum error mitigation is needed when fault-tolerant error correction is not available. Therefore, if an additional quantum circuit is used during the error mitigation process, it is necessary to reduce the number of gates and qubits. Our Corollary <ref> and Corollary <ref> limit the needed number of gates and qubits to O(r). Consequently, our method enables us to achieve improvements over previous results <cit.>.



§ CONCLUDING REMARKS AND FUTURE WORK

In this work, we introduce a new efficient algorithm for estimating the trace of powers. Our main observation is that estimating {(ρ^i)}_i=1^r is sufficient for estimating (ρ^k) for large k>r. The calculation of (ρ^k) can be done recursively by applying the Newton-Girard method. The Newton-Girard method has already been widely used in entanglement spectroscopy <cit.>. Our main contribution lies in proving that the error increases linearly at most when applying the Newton-Girard method with a recursive strategy. We also generalize the result to traces of powers with observables M, which are represented as (Mρ^k).

Our work can enhance any previous algorithms that estimate traces of powers. Combining our method with the TCT and qubit-reset algorithm <cit.> reduces the depth to 𝒪(r) and the number of  operations to 𝒪(r). With multivariate trace estimation <cit.>, we can estimate the trace of powers with 𝒪(1)-depth, 𝒪(r)-width, and only 𝒪(r)- operations. Our method also provides advantages in copy complexity when estimating the trace of large powers with low-rank states or sufficiently mixed states.

We considered the application of our method to calculating nonlinear functions of quantum states, preparing quantum Gibbs states, and mitigating quantum errors. Our method significantly improves the circuit complexity in these applications, allowing for more feasible implementation on near-term quantum devices. Our findings open up possibilities for any applications that use the estimation of traces of powers, with or without observables. It remains open for future work to find more applications that can take advantage of our work.

Another extension for future work is to generalize our algorithm for multivariate trace estimation. We suggest an algorithm for efficiently estimating (ρ^kσ) for different density matrices ρ and σ. Generalizing this result to multivariate trace estimation, or even (ρ^kσ^l), can open up more possibilities, such as calculating functions that satisfy the data-processing inequality under unital quantum channels, which can be an alternative tool for distance measures <cit.>. Additionally, tightening the bounds on Theorems <ref> and <ref> is an interesting future research topic. Modifying the proof of Theorems <ref> and <ref> or utilizing the concept of effective rank (Appendix <ref>) could lead to a tighter bound, which could further emphasize the effectiveness of our “rank is all you need” idea.



§ DATA AVAILABILITY STATEMENT

The data generated and analyzed during the current study are available from the authors upon request.



§ ACKNOWLEDGMENTS

This work was supported by the National Research Foundation of Korea (NRF) through a grant funded by the Ministry of Science and ICT (NRF-2022M3H3A1098237; RS-2023-00211817).
This work was partially supported by the Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korean government (MSIP) (No. 2019-000003; Research and Development of Core Technologies for Programming, Running, Implementing, and Validating of Fault-Tolerant Quantum Computing Systems), and Korea Institute of Science and Technology Information (KISTI: P24021).

quantum




§ PROOF OF THEOREM <REF>


Let

    ρ   = ∑_i=1^r p_i |ψ_i⟩⟨ψ_i|,
    
    m_i    = ⟨ψ_i| M |ψ_i⟩.

Then we have

    P_k,M = (Mρ^k) = ∑_i=1^r m_i p_i^k.


Next, let Q_k,M be the estimated value of (Mρ^k) for k ≤ r. Define Q_r+k,M recursively:

    Q_r+k,M = ∑_j=1^r (-1)^j-1 b_j Q_r+k-j,M

for k ≥ 1. We also have:

    P_r+k,M = ∑_j=1^r (-1)^j-1 a_j P_r+k-j,M.

And the difference of the two terms becomes that

    Q_r+k,M-P_r+k,M
       =∑_i=1^r(-1)^j-1b_j(Q_r+k-j,M-P_r+k-j,M)
           +∑_i=1^r(-1)^j-1(b_j-a_j)P_r+k-j,M.

Note that ϵ_i,M := Q_i,M - P_i,M, then

    ϵ_r+k,M
       =∑_j=1^r ((-1)^j-1(b_j ϵ_r+k-j,M+t_jP_r+k-j,M)),
       ϵ_r+k-1,M
       =∑_j=1^r ((-1)^j-1(b_j ϵ_r+k-j-1,M + t_jP_r+k-j-1,M)).

Let us define

    ϵ_1    = max_1≤ j≤ r |ϵ_j,M|,
    ϵ_2    = max_1≤ j≤ r |ϵ_j|.

Using the same logic in the proof of Theorem <ref>, we conclude that, for every k,

    ϵ_r+k = ϵ_1 + ∑_j=1^r (|t_j| (∑_i=1^k P_r-j+i,M))

holds. Then, by the assumption of the theorem,

    ϵ_1    ≤ϵ/2,
    ϵ_2    ≤ϵ/2T ‖ M ‖_∞ln r.

By applying Lemma <ref>, we can obtain that

    |t_k| ≤∑_i=1^k ϵ_2/i≤ϵ_2ln r≤ϵ/2T‖ M‖_∞.

Next, with elementary sequence summation, we get the following sequential inequalities:

    |t_k| ≤∑_i=1^k ϵ_2/i≤ϵ_2 ln r ≤ϵ/2T ‖ M ‖_∞.

Finally, we can deduce that

    ϵ_r+k≤ϵ/2 + ϵ/2T ‖ M ‖_∞× T ‖ M ‖_∞≤ϵ.





§ CORRECTNESS OF B_J ≥ B_J+1

In the proof of Theorems <ref> and <ref>, we make use the fact that b_j ≥ b_j+1. If ∑_i=1^r q_i = 1 and q_i ≥ 0, then b_j ≥ b_j+1 is trivial. We know that (ρ) = 1 for every ρ, so ∑_i=1^r q_i = 1 obviously holds. However, q_i ≥ 0 could be an issue because q_i are the solutions of the estimated polynomial of the Newton-Girard method, and if the estimation error is large, the solutions could be negative. To make our proof more precise, we suggest a method that ensures q_i always satisfies the positivity condition while we estimate {(ρ^i)}_i=1^r.




§ EFFECTIVE RANK

We define the term effective rank with error ϵ as the minimum value r_ϵ, which satisfies that

    ∑_i=1^r_ϵ p_i > 1-ϵ.



For every integer k ≥ 2, the following holds:

    |∑_i=1^r p_i^k - ∑_i=1^r_ϵ p_i^k| < ϵ^2.


This lemma suggests that we only need {(ρ^i)}_i=1^r_ϵ for the estimation of the trace of powers. Compared to previous methods, the copy complexity of our method was disadvantageous when ρ is close to a pure state. The main reason was that polynomials of r were multiplied to the previous complexity. The concept of effective rank offers hope that our method can match the copy complexity of previous methods when ρ is close to a pure state. As the maximum eigenvalue of ρ approaches 1, the effective rank decreases. Future research aimed at finding the optimal complexity of our work can employ the concept of effective rank.



§ FINDING THE R=RANK(Ρ)

To proceed with our algorithm, we need to know the rank r of quantum state ρ. Due to the concept of effective rank, the estimated rank doesn't have to be precise. Hence, we can use a variational approach <cit.> to estimate the rank. The required quantum circuit complexity is 𝒪(n), which is usually smaller than 𝒪(r). We suggest an alternative approach for estimating r. Note that taking a larger r than the actual rank does not affect the performance of our algorithm in Section <ref>.




 Paper Title: Multi-Purpose Architecture for Fast Reset and Protective Readout of Superconducting Qubits.



APS/123-QED






These authors contributed equally to this work.
National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China


These authors contributed equally to this work.
Beijing Academy of Quantum Information Sciences, Beijing 100193, China


wanghe@baqis.ac.cn
Beijing Academy of Quantum Information Sciences, Beijing 100193, China


xuegm@baqis.ac.cn
Beijing Academy of Quantum Information Sciences, Beijing 100193, China


Beijing Academy of Quantum Information Sciences, Beijing 100193, China


Beijing Academy of Quantum Information Sciences, Beijing 100193, China


Beijing Academy of Quantum Information Sciences, Beijing 100193, China


Beijing Academy of Quantum Information Sciences, Beijing 100193, China


National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China


Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 101408, China


State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing, China


State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing, China


National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China



Beijing Academy of Quantum Information Sciences, Beijing 100193, China


yanfei@baqis.ac.cn
Beijing Academy of Quantum Information Sciences, Beijing 100193, China



             

§ ABSTRACT 
The ability to fast reset a qubit state is crucial for quantum information processing. However, to actively reset a qubit requires engineering a pathway to interact with a dissipative bath, which often comes with the cost of reduced qubit protection from the environment. Here, we present a novel multi-purpose architecture that enables fast reset and protection of superconducting qubits during control and readout. In our design, two on-chip diplexers are connected by two transmission lines. The high-pass branch provides a flat passband for convenient allocation of readout resonators above the qubit frequencies, which is preferred for reducing measurement-induced state transitions. In the low-pass branch, we leverage a standing-wave mode below the maximum qubit frequency for a rapid reset. The qubits are located in the common stopband to inhibit dissipation during coherent operations. We demonstrate resetting a transmon qubit from its first excited state to the ground state in 100 ns, achieving a residual population of 2.7%, mostly limited by the thermal effect. The reset time may be further shortened to 27 ns by exploiting the coherent population inversion effect. We further extend the technique to resetting the qubit from its second excited state. Our approach promises scalable implementation of fast reset and qubit protection during control and readout, adding to the toolbox of dissipation engineering.



                              
Multi-Purpose Architecture for Fast Reset and Protective Readout of Superconducting Qubits
    Fei Yan
    August 5, 2024
==========================================================================================







§ INTRODUCTION


Initialization of the quantum state is an important criterion for the physical implementation of quantum computation <cit.>. It can be realized through either passive waiting until reaching thermal equilibrium or active reset using a dissipative bath <cit.>. In addition to initialization, the reset operation is useful in quantum error correction <cit.>, preparation of stable quantum-correlated many-body states <cit.>, qubit reuse compilation <cit.> and simulation of open quantum systems <cit.>.

Implementing a fast reset with superconducting qubits is a challenge. One approach is through real-time feedback conditioned on the outcome of a projective measurement <cit.>. However, the performance of this approach is susceptible to measurement errors and feedback loop latency. Alternatively, the lossy readout resonator can drain excitation away from the qubits without the need for measurements or feedback operations <cit.>. To speed up the dissipative process while maintaining qubit coherence, an additional resonator known as the Purcell filter is often used as a spectral mask of the environment <cit.>. However, using the same resonator for both readout and reset imposes serious constraints. It requires a lower resonator frequency than the maximum qubit frequency to be compatible with the prevailing transmon qubit design <cit.>. Unfortunately, the resonator-below-qubit configuration is more vulnerable to measurement-induced state transitions (MIST) due to the hybridization among highly excited states <cit.>. In the otherwise configuration, the phenomenon becomes less probable. Reset techniques compatible with the resonator-above-qubit configuration include parametric flux modulation <cit.>, all-microwave driving <cit.>, bath engineering <cit.>, intrinsic Purcell filter <cit.>, quantum circuit refrigerator <cit.>, and other methods. However, some of these methods may involve extra hardware costs or control complexity.




To address the aforementioned issues in one design, we introduce a novel multi-purpose architecture that employs a carefully engineered transmission line mode (the dissipator mode) that can, in principle, couple to an arbitrary number of superconducting qubits and serve as a common dissipative channel for fast resetting the qubit states. The architecture features the use of on-chip diplexers that allow the physical separation of the readout and reset channels. This allows the readout resonator mode to be above the maximum qubit frequency and the dissipator mode to be below it, enabling both fast reset and less measurement-induced state transitions while preventing qubits from dissipating into external circuitry during coherent operations. The experimental results show that, when the transmon qubit frequency is tuned in resonance with the dissipator mode, the population of the qubit in the first excited state decays with a characteristic rate of Γ≈ 1/(21 ns). With a reset duration of 100 ns, the residual error is approximately 2.7%, due mainly to thermal activation at a device temperature of ∼50 mK. Our design promises a hardware-efficient implementation of fast unconditional reset of multiple qubits while protecting the qubits from radiation loss and measurement-induced errors.





§ DEVICE DESIGN




The key concept and circuit implementation of our design is illustrated in Fig. <ref>, which features the use of two symmetric on-chip diplexers, each composed of a low-pass filter with a cut-off frequency (-3 dB point) at 3.64 GHz and a high-pass filter with a cut-off frequency at 5.37 GHz. As shown in Fig. <ref>, connecting the two diplexers are two parallel transmission lines. The top high-pass line is coupled to multiple readout resonators for multiplexed readout. The bottom low-pass line (25 mm long) is coupled to multiple qubits. A lossy standing wave mode (4.23 GHz, full wavelength) is formed along the line, which we shall utilize for qubit reset. The diplexers are made up of lumped element capacitors and inductors [Fig. <ref>]. We use interdigitated capacitors and spiral inductors upon implementation, which produces a dimension of 3.7 mm × 0.8 mm in the final layout [Fig. <ref>]. 


The purpose of the design is multi-fold.
As illustrated in Fig. <ref>, the filters are designed such that the resonators (ω_r/2π∼ 7.27.4 GHz fall within the passband of the high-pass filter, leading to a uniform external coupling. Meanwhile, the qubits (ω_q/2π∼ 4.8 GHz) are allocated within the common stopband of the high-pass and low-pass filters. This way, the qubits are protected from dissipation into external circuitry as they see an unmatched impedance at the frequency of coherent operations. Unlike conventional bandpass or notch filters, both the qubits and resonators now have a large bandwidth for convenient frequency allocation, promising a greater readout multiplicity.

Over the low-pass line, a standing wave mode can be formed between the two low-pass filters above their cut-off frequency where reflection dominates, reminiscent of a Fabry-Pérot cavity formed by two reflective mirrors. With the finite transmission of the low-pass filter (S_21≈ -30 dB) near the mode frequency at ω_d/2π = 4.23 GHz (design value), one can expect enhanced dissipation around this mode. Therefore, tuning the frequency of a tunable transmon qubit down to the dissipator frequency shall deplete excitations in the qubit. In addition, the measured energy relaxation time T_1 of the qubits at the maximum frequency is 30-40 μs, suggesting that the dissipator has little influence on the qubits during their coherent manipulations.

In the qubit-dissipator system, the effective decay rate Γ of the qubit is modified by the lossy resonator - known as the Purcell effect - and can be given by <cit.>: 

    Γ = 1/2(κ_d-Re{√(-16 g^2+(κ_d-2 i Δ_qd)^2)})




where κ_d is the decay rate or linewidth of the dissipator, g is the coupling strength between the qubit and the dissipator, Δ_qd = ω_q - ω_d is the detuning between the qubit and the dissipator. The qubit relaxation rate peaks when resonant with the dissipator (Δ_qd = 0). 
Note that the maximum decay rate κ_d/2 can always be achieved at positions satisfying g ≥κ_d/4.






§ EXPERIMENTAL RESULTS







In Fig. <ref>, we plot the S_21 signal between the input and output ports of the full measurement chain using a network analyzer and compare them with finite-element simulations of the transmission between the two ports as depicted in Fig. <ref>. In both plots, the dissipator, the readout resonators, and the protected region can be identified. The line width of the dissipator extracted from the S parameters is κ_d/2π∼ 10-15 MHz.

We then calibrate the reset dynamics using the time sequence described in Fig. <ref>. We test the protocol on two transmon qubits coupled to the same transmission line. The qubits are first prepared in their first excited states |e⟩, and then frequency-tuned using a square flux pulse before projective measurement. By scanning the pulse amplitude and duration, it can be seen that the qubits dissipate about three orders of magnitude faster when biased near the dissipator at 4.3-4.4 GHz, which is about 200 MHz above the design value due to imperfect modeling and fabrication variations. Note that there is uncertainty in the nominal frequency as the qubit spectra are not visible near the dissipator. 
A finer scan in the vicinity of the dissipator resolves a fringing pattern [Fig. <ref>]. This is because the excitation can swap between the qubit and the dissipator before being finally dissipated given a relatively strong coupling g/2π≈ 10 MHz by design. 

As shown in Fig. <ref>, the qubit excitation oscillates while rapidly dropping to a baseline (2.7%) within approximately 100 ns. The envelop decay is in good agreement with the theoretical value Γ^-1 = 21 ns according to Eq. (<ref>). On the other hand, the reset gate can be further shortened by exploiting the excitation swapping effect, for example, to 27 ns at the first minimum <cit.>. 


Next, we benchmark the efficiency of the calibrated |e⟩|g⟩ reset gate of Q_0 by initializing the qubit in different states [Fig. <ref>].


The single-shot distribution plots [Figs. <ref>-<ref>] and assignment matrices [Figs. <ref>-<ref>] show that the majority of the excited state population is depleted.  

The reset gate reduces the excited-state population from 92.56% to 2.65%, while the ground-state population increases from 0.34% to 1.35%. This suggests that the residual population comes from thermal excitation when the qubit reaches equilibrium with the environment at a lower frequency. According to the Boltzmann distribution, the corresponding environmental temperature is approximately 50 mK, which is in agreement with the typical device temperature that we measured in the same setup.



We further test the efficiency of the reset gate applied to the qubit in its second excited state. Similar to the calibration of the |e⟩|g⟩ reset gate, we apply a flux pulse that adjusts the qubit frequency to let the |e⟩|f⟩ transition be resonant with the dissipator to depopulate the second excited state. To benchmark the |f⟩|e⟩ reset gate, we first perform the three-state discrimination and show the assignment probability matrix in [Fig. <ref>]. Then, we study the performance of a single |f⟩|e⟩ reset operation [Fig. <ref>]. The population of an initial |f⟩ state decreases from 65.8% to 2.92%, while the population of an initial |e⟩ state increases from 0.76% to 1.5%, as a consequence of the same thermal effect as in the |f⟩|e⟩ reset gate. We note that the |f⟩|e⟩ reset gate can be used for leakage reduction during quantum error correction, provided that the residual error can be further suppressed. We finally show the performance of concatenated |f⟩|e⟩ and |e⟩|g⟩ reset gates [Fig. <ref>]. The cascade of the two reset gates reduces the population of the |f⟩ state from 65.8% to 1.73%, showcasing the dissipator's capability to reset qubits from even higher excited states using the same mechanism. Since the transmon qubit has a negative anharmonicity, it is possible to go through a ladder of biases and step-by-step reset the qubit from a high-energy state to the ground state. The reset pulse can also be integrated into one pulse in piecewise steps. 




§ CONCLUSIONS

To summarize, we have introduced a novel architecture based on superconducting quantum circuits. The architecture can satisfy multiple functionalities simultaneously. By separating qubit reset and readout channels using on-chip diplexers, the architecture provides substantial flexibility in circuit design, enabling protection for qubits from dissipating into the lossy circuitry, convenient allocation of qubit and resonator frequencies, a flat passband for the resonators, the qubit-below-resonator configuration for less measurement-induced state transitions, and, most importantly, fast reset under these constraints. The residual error can be improved by reducing the device temperature and elevating the frequencies of the dissipator. Note that the dissipator channel may be terminated with a matched impedance separately without diplexing. However, that would require additional ports and wiring. The scheme is promising for scalable implementation in quantum error correction, enabling the resetting of syndrome qubits and reducing leakage in data qubits. 
Furthermore, the engineered dissipative bath is a valuable resource and could be integrated as an integral part of dissipation engineering, which attracts significant attention in quantum information science <cit.>, such as the preparation of dissipative stabilized quantum states <cit.>, the realization of autonomous quantum error correction <cit.> and the study of quantum transport <cit.>. Given the flexibility of the architecture, we believe our versatile design will find broader practical applications in superconducting quantum information processing.


The authors would like to thank Ji Chu and Yasunobu Nakamura for their valuable comments. This work was supported by the National Natural Science Foundation of China (Grants No. 92365206 and No. 12322413), Innovation Program for Quantum Science and Technology (Grant No.2021ZD0301802 and No. 2021ZD0301702), NSF of Jiangsu Province (Grants No. BE2021015-1 and No. BK20232002).

















 Paper Title: Multipartite Entanglement for Multi-node Quantum Networks.

  











C>c<[4]







apsrevCorresponding authorgabriel.aranedamachuca@physics.ox.ac.ukDepartment of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom§ ABSTRACT 
Scaling the number of entangled nodes in a quantum network is a challenge with significant implications for quantum computing, clock synchronisation, secure communications, and quantum sensing. In a quantum network, photons interact with matter qubits at different nodes, flexibly enabling the creation of remote entanglement between them. Multipartite entanglement among multiple nodes will be crucial for many proposed quantum network applications, including quantum computational tasks and quantum metrology. To date, experimental efforts have primarily focused on generating bipartite entanglement between nodes, which is widely regarded as the fundamental quantum resource for quantum networks. However, relying exclusively on bipartite entanglement to form more complex multipartite entanglement introduces several challenges. These include the need for ancillary qubits, extensive local entangling operations which increases the preparation latency, and increasingly stringent requirements on coherence times as the number of nodes grows. Here, we analyse various schemes that achieve multipartite entanglement between nodes in a single step, bypassing the need for multiple rounds of bipartite entanglement. We demonstrate that different schemes can produce distinct multipartite entangled states, with varying fidelity and generation rates. Additionally, we discuss the applicability of these schemes across different experimental platforms, highlighting their primary advantages and disadvantages.
Multipartite Entanglement for Multi-node Quantum Networks
    G. Araneda
    August 5, 2024
=========================================================



§ INTRODUCTION

The need for quantum networks is ubiquitous in the field of quantum technologies and fundamental quantum physics. Multi-node quantum networks have been proposed to compare and synchronise atomic clocks, or to create sensing networks that could enrich our understanding of the universe <cit.>. In fundamental quantum physics, they will permit testing concepts such as local realism to a new extent <cit.>. In quantum communications, multi-node networks will enable essentially secure multi-party key exchange <cit.>. In quantum computing, the ability to engineer complex and high-fidelity entangled states between separated processors will allow for the construction of modular quantum computers <cit.>. Increasing the number of processing nodes and qubits while maintaining a high degree of connectivity between all qubits is crucial for practical and useful quantum computation <cit.>.

So far, bipartite entanglement between nodes mediated by photons has generally been considered as the initial quantum network resource, across all experimental platforms <cit.>. Nevertheless, most applications require more complex multipartite entangled states, such as GHZ, W, graph, and cluster states <cit.>. Building multipartite entanglement from bipartite entanglement relies on using multiple entangled pairs, local mid-circuit measurements, and feed-forward control of the qubits concerned <cit.>; these operations are costly in time and resources and increase network latency. Importantly, it has been shown that no graph state can be created over a quantum network without classical communication <cit.>, which makes this latency unavoidable. The network latency not only increases the duration of quantum computations, but also, in realistic physical implementations, means that the time required to generate some particular multipartite entangled state can take longer than the coherence time of the individual qubits, making the computation impossible without the need of additional memory qubits. Although long-lived qubits have been demonstrated across different platforms, the qubits used for computation are not necessarily the most adequate to interface with the photonic network, adding error overheads when entanglement needs to be swapped between network and processing qubits (see e.g., <cit.>).

Hence, it is natural to look for quantum network resources beyond bipartite entanglement. In this article, we examine different schemes experimentally considered so far to create bipartite entanglement mediated by photons in different experimental platforms, and extend them to multiple parties. We identify the types of multipartite entanglement produced, and discuss their main limitations and strengths. All the schemes that we consider create multipartite entanglement in a single shot, i.e., they are processes that require at most one round of classical communication and feed-forward, and do not rely on bipartite remote entanglement and local entangling operations to create more complex entangled states. 

The various schemes considered in this article are summarised in Fig. <ref>. Fig. <ref>a shows the “photon exchange” scheme <cit.>, Fig. <ref>b shows the “itinerant photon” scheme <cit.>, Fig. <ref>c shows the “photon-to-atom mapping” scheme <cit.>, Fig. <ref>d shows the “entanglement swapping” scheme <cit.>, and Fig. <ref>e shows the “which-path erasing” scheme <cit.>.
In each case, each node contains one or several quantum emitters. These quantum emitters can be, for example, single trapped ions <cit.>, single neutral atoms <cit.>, clouds of neutral atoms <cit.>, superconducting qubits <cit.>, crystal defects <cit.>, quantum dots <cit.>, rare-earth ions in a crystal <cit.>, silicon T-centres <cit.>, among others. For simplicity, we refer to all these as “atoms" throughout this article. These atoms can emit single photons, which can then be captured either by using a resonator <cit.> or free-space optics <cit.>. An alternative scenario includes the case where atoms do not emit photons, but can interact with a quantised input field <cit.>.

The different schemes considered here produce directly different classes of multipartite entangled states, such as GHZ and W states. GHZ states have many applications, including remote atomic clock comparisons <cit.>, quantum secret sharing <cit.>, fundamental tests of quantum mechanics <cit.>, and teleportation <cit.>. They have also been proposed as a resource for distributed quantum computing <cit.>. W states have applications in secret voting <cit.>, secure quantum communication <cit.>, as a resource for teleportation <cit.>, metrology <cit.>, and repeaters <cit.>, among others. This article focuses on entanglement between remote nodes, but the schemes discussed here can also be employed to entangle atoms within the same node (see for example <cit.>). In the following sections, we describe each of these schemes and how they can be extended to generate entanglement between N nodes in a single shot, and find expressions for the maximum attainable entanglement fidelities and rates.



§ SINGLE PHOTON EXCHANGE


 §.§ Two Nodes

Using the single photon exchange protocol, the excitation of an atom in a node can be mapped to the state of a photon. The single photon is then routed  to a different node where the inverse mapping process is applied <cit.>. If the atom in the first node is in a superposition state, then this process creates bipartite remote entanglement. This technique has been demonstrated using superconducting qubits in microwave resonators <cit.> and with single neutral atoms in optical cavities <cit.>.

Let us consider the deterministic scheme realised using transmon-type artificial atoms in a microwave resonator, and a single microwave photon presented with number encoding, from ref. <cit.>. The three levels of the transmon, in ladder configuration, are |g⟩_A_n, |e⟩_A_n and |f⟩_A_n, with A_n labeling transmon A_1 or A_2. To transfer an excitation from A_1 to A_2, first, transmon A_1 is prepared in the highest excited state |f⟩_A_1, and driven coherently using the time-dependent process ĝ(t) to the ground state |g⟩_A_1. This process creates a microwave photon, |1⟩_P which is then coupled, with efficiency η_C, to a directional quantum channel that routes the photon to the second resonator-coupled transmon. The transmon A_2, initially in the ground state |g⟩_A_2, is then driven with the time-reversed process ĝ (-t), which has the exact opposite effect of ĝ (t), i.e., A_2 absorbs a microwave photon and transfers the population from |g⟩_A_2 to |f⟩_A_2. Finally, the excitation transfer ends with a pulse mapping |f⟩_A_2 to |e⟩_A_2. 

To create entanglement between A_1 and A_2 using this approach, the following sequence is applied. First, we prepare transmon A_1 in a superposition state of |e⟩ and |f⟩, transmon A_2 in the ground state, and vacuum in the microwave resonators and transfer line, i.e.,

    1/√(2)(|e⟩_A_1 + |f⟩_A_1)⊗|g⟩_A_2|0⟩_P,

where P denotes the field occupation state. Then, the ĝ(t) drive is applied in transmon A_1, which produces a photon only for the amplitude of the transmon that is in the state |f⟩ and maps it to the ground state |g⟩_A_1, i.e., 

    1/√(2)(|e⟩_A_1|g⟩_A_2|0⟩_P+ |g⟩_A_1|g⟩_A_2|1⟩_P).

Then the absorption ĝ(-t) process is applied in transmon A_2, producing the state |g⟩_A_2, i.e., 

    1/√(2)(|e⟩_A_1|g⟩_A_2|0⟩_P+ |g⟩_A_1|f⟩_A_2|0⟩_P).

Finally, a pulse transfers the amplitude from |f⟩_A_2 to |e⟩_A_2. Tracing out the field state, the resulting state of the transmon pair is the Bell state

    1/√(2)(|e⟩_A_1|g⟩_A_2+ |g⟩_A_1|e⟩_A_2).

A weakness of this particular scheme, which uses the photon number to encode the transferred information, is that the photon state |0⟩_P can be produced either from the desired process or from photon loss, and hence, the fidelity is drastically affected by photon loss. Therefore, to achieve high fidelity, a photon transfer probability close to unity is required. In ref. <cit.>, the observed state fidelity with respect to the ideal Bell state is 78.9%, with photon loss accounting for 10.5% infidelity, and limited coherence times contributing 9%. 
If the photon transmission efficiency is η_A_1A_2, then the  fidelity achieved by this protocol is limited to <cit.>
    F^ST_2 = 1/4(1+√(η_A_1A_2))^2.


As this is a deterministic scheme, the entanglement rate is only limited by the repetition rate of the experiment, and the time required by the photon to travel from one node to the other.


The implementation between two neutral atoms of ref. <cit.> uses the polarisation of the emitted photon instead of the number state. For an appropriate choice of atomic structure, such a scheme can be made robust against photon loss by heralding on the successful absorption of a photon (see also ref. <cit.>). The achieved entanglement fidelity is 85%, mostly limited by state preparation and readout, excitation of undesired electronic transitions, and multi-photon processes. By reducing the acceptance window of the herald photon, up to 98.7% fidelity has been reached. The probability for remote entanglement creation is proportional to

    R^ST_2∝η_Pη_    OUTη_NETη_ENTη_DET,

where η_P is the probability of emitting the desired photon in the cavity mode, η_NET is the probability of coupling the cavity mode into an optical fiber,  η_ENT is the probability of the second atom to absorb the photon from the fiber mode, and η_DET is the probability of detecting an emitted heralded photon. In ref. <cit.>, entanglement is created with an overall probability of 2. A detailed analysis of the performance of this process is presented in ref. <cit.>.
A similar approach has been implemented with transmon qubits and time-bin encoded photons <cit.>, where photon loss was detected using a non-destructive measurement of the transmon qubit state, reducing the transfer error by a factor of ∼ 2.



 §.§ N Nodes

If we now consider the transmon scheme, but such that the generated single photon is split and routed to different nodes, it is possible to generate entanglement between N different nodes. First, we prepare N+1 atoms in the initial state

    |f⟩_A_0(⊗_n=1^N|g⟩_A_n).

Then we drive the excitation process ĝ(t) in node 0, producing a photon P_0
    |g⟩_A_0|1⟩_P_0(⊗_n=1^N|g⟩_A_n).

Then, the photon is split, by e.g., using beam splitters <cit.>, and routed to each of the nodes. If the photon is split into different paths with equal probability, the state now is

    1/√(N)|g⟩_A_0∑_m=1^N|1⟩_P_m(⊗_n=1^N|g⟩_A_n)

where |1⟩_P_m represents the presence of a photon in resonator m, and no photons in all the rest, e.g., |1⟩_P_2 =|0⟩_P_1|1⟩_P_2|0⟩_P_3,...,|0⟩_P_N.
If we apply the time-reversed process in all the nodes, the rotation at the end, and trace out the state of the field and atom A_0, then the state is the multipartite entangled W state

    |W_1,N-1⟩=    1/√(N)( |e,g,g,...,g⟩_A_1,A_2,...,A_N.
       + .|g,e,g,...,g⟩_A_1,A_2,...,A_N. 
       ⋮
       + .|g,g,g,...,e⟩_A_1,A_2,...,A_N).


If no heralded absorption is performed, i.e, the deterministic scheme is followed, then the fidelity of the generated state depends strongly on the photon losses. If the probability of transmission of the photon from atom A_0 to A_N is the same for all the nodes and given by η_A_0,A_N, then the maximum attainable fidelity is proportional to

    F^ST_N ∝η_A_0,A_N.

If heralded absorption is used, then the fidelity is not limited by photon loses, instead, the entanglement generation rate is limited by

    R^ST_N∝η_Pη_OUTη_NETη_A_0A_NP_ENTη_DET,

where we have assumed that the different efficiencies are the same for all of the nodes. The heralding mechanism could be implemented by detecting a single photon emitted collectively by the receiver atoms, using a which-path eraser as described in section <ref>.




§ ITINERANT-PHOTON CONTROLLED ENTANGLING GATES


 §.§ Two nodes

Figure <ref>b) illustrates a scheme in which multi-node entanglement can be achieved using an itinerant photon interacting with multiple nodes. The photon is initially prepared in a balanced superposition, and at each node, the atom interacting with the cavity is prepared in the state |0⟩. The itinerant photon then interacts with each cavity through a “reflection gate" <cit.>. This implements a controlled-unitary interaction between the atoms and the photon. The initial state of the photon and the atoms in a two node network is given by

    1/√(2)(|0⟩+|1⟩)_P|0⟩_A_1|0⟩_A_2.

Here, P denotes the photon, and A_1 and A_2 represent atoms in the 1st and 2nd nodes, respectively. After subsequent interaction with the first and the second cavity, and considering that the controlled gate is applied in the X basis, the state of the system in the ideal and loss-less case is given by

    1/√(2)(|+⟩_P|Φ^+⟩_A_1,A_2 + |-⟩_P|Φ^-⟩_A_1,A_2)

where |±⟩=1/√(2)(|0⟩±|1⟩), and the Bell state |Φ^±⟩=1/√(2)(|00⟩±|11⟩). Subsequently, by measuring the state of the photon in the {|±⟩} basis, the states of the atoms can be projected into either Bell state |Φ^+⟩ or |Φ^-⟩. A similar proposal using continuous-variable encoding in light can be found in ref. <cit.>.

This scheme requires strong coupling between the atoms and the quantized modes of the field. It has been used to create two-node entanglement using atoms in optical cavities and photons with polarisation encoding <cit.>, and with SiV-centers in photonic crystal cavities and photons with time-bin encoding <cit.>. In both cases, the achieved entanglement fidelities are below 90%, mostly limited by the fidelity of the photon-atom gates and the use of weak coherent photon pulses instead of single photons. The photon-atom gate fidelities themselves are limited by spatial mismatch and the finite cooperativity of the atom-cavity systems. The entanglement rate is limited by the experiment repetition rate, optical losses, and the finite cooperativity. In both experimental platforms, a strong trade-off between the fidelity and the average number of photons in the coherent pulse was observed <cit.>, resulting, in practice, in a trade-off between fidelity and entanglement rate. The best observed entangling rate in the neutral atom experiment is 60<cit.>, while in the SiV-centre experiment, it is 0.2<cit.>, both corresponding to fidelities of ∼ 2/3.

A detailed study of the interplay between photon-atom gate fidelity versus cooperativity of the atom-cavity system is presented in ref. <cit.>. Details of how the mismatch between the input photon and the cavity affect the fidelities can be found in ref. <cit.>.

Assuming that both nodes can perform photon-controlled gates on the atoms, leading to the same atom-photon entangled fidelity F_A,P; if the error of the controlled gate can be fully described by a depolarising channel, the fidelity of the generated atom-atom entangled state will be limited by

    F^I_2  = 2F_P,A-1,

where F_A,P is the fidelity of the atom-photon system. 



 §.§ N Nodes



The extension of this protocol to an N-node network is depicted in Fig <ref>. The initial state of the photon and the N
nodes is 

    |+⟩_P⊗_n=1^N|0⟩_A_n ,

where A_n denotes the state of an atom in the n^th cavity. After the interaction with each cavity, and considering controlled gates in the X basis, the state of the system in the ideal, loss-less case is

    1/√(2)(|+⟩_P|G_0^+⟩_A_1,A_2,..,A_n + |-⟩_P|G_0^-⟩_A_1,A_2,..,A_n)

with the N-particle GHZ states

    |G_0^±⟩=1/√(2)(|00...0⟩±|11...1⟩) .

Thereafter, measuring the state of the photon in the {|±⟩} basis projects the states of the atoms either into the multipartite entangled state |G_0^+⟩ or into |G_0^-⟩.

If the dominant error mechanisms are the same as in the 2-node implementations, i.e., the fidelity of the photon-controlled atom gates,  the fidelity of the resulting multipartite entangled state decreases linearly with increasing number of nodes. : Fig. <ref> shows how the maximum attainable fidelity depends on the number of nodes and the atom-photon entangled state fidelity, where the controlled gates is performed with the same fidelity in all nodes. Importantly, as the success is heralded by the detection of a single photon, photon losses do not affect the atom state fidelity.
The rate of entanglement generation depends on the transmission efficiency of the photon along the different nodes. If the probability of transmission from one to another is η_T, the probability of the photon not being lost during the interaction with that atom-cavity system is  η_C, and η_DET is the efficiency of the photon detection at the end, then the success probability is proportional to

    P_N ∝η_T^N-1η_C^Nη_DET.


Using the fidelities and rates demonstrated in refs. <cit.> it would be possible to create GHZ states across several nodes, although with low fidelity and rates. New schemes for lower-loss and higher-fidelity atom-photon gates have been proposed <cit.>, which, together with higher cooperativity resonators and better single photon sources could achieve better performance for this networking scheme.



§ MAPPING ENTANGLEMENT FROM PHOTONS TO ATOMS


 §.§ Two nodes

Fig. <ref>c depicts a scheme where a multi-photon entangled source (MPES) is used to generate entanglement across different nodes. Let us first consider the case of two nodes. Following ref. <cit.>, a photon pair source produces polarisation-entangled pairs, for instance, photons in the state

    1/√(2)(|a_+⟩_P_1|a_-⟩_P_2-|a_-⟩_P_1|a_+⟩_P_2)

where P_1 and P_2 denote the two photons, and {|a_±⟩} are two orthogonal polarisation states. Let us also consider two separated network nodes, each with an atom with double-Λ level structure, see Fig. <ref>, initially prepared in a separable superposition state

    1/√(2)( |i_+⟩_A_1+|i_-⟩_A_1)⊗1/√(2)( |i_+⟩_A_2+|i_-⟩_A_2)..

The absorption of an a_+-polarised photon excites the atomic Raman transition |i_+⟩→|e_+⟩→|g_+⟩ emitting an a^'_+-polarised photon, whereas the absorption of an a_--polarised photon excites the atomic Raman transition |i_-⟩→|e_-⟩→|g_-⟩, emitting an a^'_--polarised photon. If each photon is sent to a different atom, upon absorption and emission of a photon, the state of the atoms and the emitted field is

    (|g_+⟩_A_1|g_-⟩_A_2|a^'_+⟩_1|a^'_-⟩_2 - |g_-⟩_A_1|g_+⟩_A_2|a^'_-⟩_1|a^'_+⟩_2)/√(2),

where |a^'_±⟩_n denotes the emission of a heralding photon with polarisation a^'_± from atom n=1 or 2. The detection of the emitted photons from each of the atoms in a basis that makes the orthogonal polarisations indistinguishable, e.g., {1/√(2)(|a^'_+⟩+|a^'_-⟩ ),1/√(2)(|a^'_+⟩-|a^'_-⟩ )}, projects the state of the atoms into a Bell state

    |Ψ^±⟩=1/√(2)(|g_+,g_-⟩_A_1A_2±|g_-,g_+⟩_A_1A_2),

where the sign depends on the parity of the photon measurement outcomes.
If we consider a pulsed entangled photon source which produces a polarisation-entangled pair with probability p_EPR, where the probability of a photon being absorbed by one of the atoms is η_ABS, and the probability of a heralding photon from one of the atoms being emitted and detected is η_DET, the total probability of creating a two-atom entangled state is

    P^EM_2 ∝ p_EPR(1/2η_ABSη_DET)^2

where the 1/2 factor comes from the fact that the |i_+⟩_A_1|i_-⟩_A_2 and |i_-⟩_A_1|i_+⟩_A_2 terms in the initial state of the atoms [Eq. (<ref>)] do not contribute to the absorption process.
An important issue of this scheme is that the photons from the entangled source need to match the linewidth, frequency, and wavepacket of the time-inverted atomic emission process. This can be done partially by using a filter cavity at the price of a considerable loss of photons <cit.>.
Using free-space optics to couple the single photons onto the atoms, Shuck et al. <cit.> achieved η_ABS∼e-4 and a collection efficiency of the heralded photons of η_DET∼5. A pulsed photon such as the one of ref. <cit.> can produce entangled pairs with probability p_EPR∼2.5e-3 per pulse. This adds up to a total probability of P_2∼e-13. Using an optical resonator to couple the atoms could increase the probabilities substantially, up to ∼e-5, using the cavity parameters discussed in ref. <cit.>. 



The achievable fidelities are limited by the fidelity and purity of the entangled photon source, F_2^ph, and given the extremely low rates of heralding photons, by false heralds due to dark counts in the detectors. The fidelity is therefore limited to

    F^EM_2 =  F_2^ph P_2^EM+1/4 P_2^false/P_2^EM+P_2^false,

assuming a maximally mixed state when a false herald occurs with probability P_2^false.
The effect of false heralds can be lowered by verifying that there is no population left in {|i_+⟩,|i_-⟩} by a subspace measurement on the atoms, to herald whether the atom absorbed the photon. Such a measurement would keep the coherence within the target qubit subspace {|g_+⟩,|g_-⟩} and otherwise flag unsuccessful attempts with high fidelity. The dark count probability then only competes with the local detection efficiency rather than with the overall efficiency P_2^EM.





 §.§ N Nodes

To scale-up this scheme and generate entanglement between N nodes simultaneously, we require a multi-photon entangled source (MPES). For instance, let us consider that the atoms are prepared in the separable superposition state

    ⊗_n=1^N1/√(2)(|i_+⟩ + |i_-⟩)_A_n.

If the source produces a polarisation-encoded N-photon GHZ state <cit.>
    |G_0^+⟩_P_1,P_2,...,P_N = 1/√(2)(|a_+,a_+,...,a_+⟩ +|a_-,a_-,...,a_-⟩),

then, upon absorption of the source photons by the atoms in the nodes, and subsequent emission and detection of N herald photons, the state of the atoms becomes

    |G_0^±⟩_A_1,A_2,,..,A_N = 1/√(2)(|g_+,g_+,...,g_+⟩±|g_-,g_-,...,g_-⟩),

where the sign depends on the parity of the photon measurement outcomes.
The successful completion of this process requires coincident detection of N photons, with probability proportional to

    P^EM_N ∝ p_GHZ,N(1/2η_ABSη_DET)^N.

Again, the main limitation for the fidelities here will be the initial entangled multi-photon fidelity, and given the extremely low process rates, the dark counts in the detectors. If each of the heralding detectors has a probability p_dark of measuring a dark count during the detection window, then the probability for a false herald is

    P_N^false = ∑_n=0^N Nnp_real^n p_dark^N-n ,

which represents the fact that the simultaneous detection of N heralding photons can be triggered by any combination of detections of n“real" heralding photons and N-n“dark" photons, with n∈[0,N]. Thus, the fidelity of the generated state is limited by

    F^EM_N = F_N^ph  p_N^EM + 1/2^N P_N^false/p_N^EM  + P_N^false .



Importantly, this N-node scheme is not restricted to mapping GHZ states from photons to atoms, but, in principle, any multi-photon state could be mapped.



§ REMOTE ENTANGLEMENT VIA ENTANGLEMENT SWAPPING


 §.§ Two Nodes

Fig. <ref>d illustrates the scheme for remote entanglement via entanglement swapping from atom-photon pairs to atom pairs by performing a Bell measurement on the states of the photons, as proposed in refs. <cit.>.

Let us start by considering the standard two-atom case. The four Bell states are defined as

    |Φ^±⟩ = 1/√(2)(|00⟩±|11⟩),   |Ψ^±⟩ = 1/√(2)(|01⟩±|10⟩).

One can then express the standard computational basis states in terms of the Bell states:

    |00⟩ = 1/√(2)(|Φ^+⟩ + |Φ^-⟩), 
      |11⟩ = 1/√(2)(|Φ^+⟩ - |Φ^-⟩), 
       |01⟩ = 1/√(2)(|Ψ^+⟩ + |Ψ^-⟩), 
      |10⟩ = 1/√(2)(|Ψ^+⟩ - |Ψ^-⟩).

The entangled state

    1/√(2)(|00⟩±|11⟩)_A_1P_1⊗1/√(2)(|00⟩±|11⟩)_A_2P_2

of two atoms (A_1 and A_2) and two photons (P_1 and P_2) can then be expressed as

    ±1/2|Φ^+⟩_A_1A_2|Φ^+⟩_P_1P_2 +
    	1/2|Φ^-⟩_A_1A_2|Φ^-⟩_P_1P_2
       ±1/2|Ψ^+⟩_A_1B_2|Ψ^+⟩_P_1P_2±1/2|Ψ^-⟩_A_1B_2|Ψ^-⟩_P_1P_2.


 For the photons, different degrees of freedom have been proposed and experimentally demonstrated, including polarisation, time-bin, and energy. For example, if the photon degree of freedom used is polarisation, with |H⟩ and |V⟩ representing horizontal and vertical polarisations respectively, we use the mapping

    |0⟩↔|H⟩,   |1⟩↔|V⟩.

A partial Bell state measurement on the photon polarisation state can be implemented using the setup in Fig. <ref>a. Upon the coincident detection of two clicks on different detectors, the outcome of the photons' Bell measurement is either |Ψ^+⟩ or |Ψ^-⟩. Therefore, these coincident clicks herald the creation of entanglement between the two atoms via entanglement swapping. Double detections in any single detector correspond to non-entangled states. Note that this limits the entanglement generation success probability to 1/2, which is the maximum efficiency allowed for a Bell state analyser (BSA) that uses only linear optics <cit.>. Importantly, unlike the cases shown below for more particles, the efficiency of 1/2 cannot be improved by using number-resolved detectors. A similar Bell-state analyser for time-bin encoded photons was proposed in <cit.>, which also achieves the theoretical maximum efficiency.



If the probability of detecting a photon emitted by two different nodes, A and B, is equal, and given by η_DET, then the probability of creating entanglement between the two atoms heralded by the detection of two photons in coincidence is

    R^SW_2 ∝1/2η_DET^2.


Importantly, in order to achieve high fidelities, the photons need to be completely indistinguishable. This includes the temporal wavepacket of the photons, necessitating the synchronisation of the arrival times of the photons at the BSA. 
So far, the fidelity of the created states has been limited by photon distinguishability, the atom-photon entanglement, and by imperfect optical systems. The atom-photon state fidelities are limited by the photon collection geometry, the electric field emission patterns of the involved transitions, and the purity of the emitted single photon (see refs. <cit.> for details).


This scheme has been implemented with neutral atom ensembles <cit.>, trapped ions, using photon-polarisation encoding and free-space optical collection <cit.>, trapped-ions using photon time-bin encoding and free space optical collection <cit.>, trapped ions coupled to optical cavities using photon polarisation encoding <cit.>, single neutral atoms and free-space optical collection <cit.>, and NV-centres and time-bin encoded photons <cit.>.
The best remote Bell-state fidelity achieved using this scheme was 96 at a rate of ∼ 100 entangled events per second <cit.>. For experiments based on photons emitted by spontaneous decay, the final atom-atom entanglement fidelity is independent of the generation rate <cit.>. On the other hand, experiments based on photons created via stimulated emission do suffer from a strong trade-off between maximum achievable fidelities and rates <cit.>.

 


 §.§ Three Nodes

Next, consider the same scenario with three identical atom-photon pairs. We denote the atoms as A_1, A_2, and A_3, and the photons as P_1, P_2, and P_3. The state can be described by

    ⊗_n=1^3 1/√(2)( |0⟩_A_n|0⟩_P_n±|1⟩_A_n|1⟩_P_n)=
       1/√(8)(|G_0^+⟩_A_1A_2A_3|G_0^+⟩_P_1P_2P_3 +
    	|G_0^-⟩_A_1A_2A_3|G_0^-⟩_P_1P_2P_3
       + 	|G_1^+⟩_A_1A_2A_3|G_1^+⟩_P_1P_2P_3  +
    	|G_1^-⟩_A_1A_2A_3|G_1^-⟩_P_1P_2P_3
       + 	|G_2^+⟩_A_1A_2A_3|G_2^+⟩_P_1P_2P_3  +
    	|G_2^-⟩_A_1A_2A_3|G_2^-⟩_P_1P_2P_3
       +	
    	|G_3^+⟩_A_1A_2A_3|G_3^+⟩_P_1P_2P_3  +
    	|G_3^-⟩_A_1A_2A_3|G_3^-⟩_P_1P_2P_3) .

The {|G_n^±⟩} states are GHZ states that form a complete orthonormal basis in the 3-qubit Hilbert space. Explicitly, they are given by

    |G_0^±⟩   = 1/√(2)(|000⟩±|111⟩) ,
    |G_1^±⟩   = 1/√(2)(|001⟩±|110⟩) ,
    |G_2^±⟩   = 1/√(2)(|010⟩±|101⟩) ,
    |G_3^±⟩   = 1/√(2)(|011⟩±|100⟩) .

The challenge is to implement a 3-photon generalized Bell state analyser (BSA) that can distinguish between different photonic |G⟩ states and project the corresponding 3-particle entangled state in the atoms, performing the entanglement swapping operation. Following the idea used for the 2-node BSA, such a generalised GHZ Bell state analyser requires the coincident detection of 3 photons and the capability of erasing any information about which atom emitted which photon. This problem has been considered before in photonic networks for the creation of multipartite photon entanglement using interferometers in post-selection experiments (see, e.g., refs. <cit.>. For the case of entangling three atoms, the requirement can be satisfied by a 3-to-3 symmetric multiport <cit.>, which is a generalisation of a 50:50 beam splitter to 3 inputs and 3 outputs. In a 3-to-3 symmetric multiport, a photon entering any port has the same probability of exiting any of the 3 output ports. An example of a 3-to-3 symmetric multiport is the tritter <cit.>. Fig. <ref>b shows a generalised 3-photon BSA implemented using a tritter, which can also be constructed using integrated photonics <cit.>. The tritter uses two 50:50 beam splitters and one 1:2 beam splitter (transmission 2/3 and reflection 1/3).
The creation of 3-particle entanglement is heralded by the detection of 3 photons in any combination of three different detectors.

Different implementations of the 3-photon generalised BSA will lead to different probabilities of generating entanglement. We label this probability as p_BSA_3. The success probability of this 3-photon heralded scheme to generate 3-particle entanglement is thereafter

    R^SW_3 ∝ p_BSA_3η_DET^3,

where it is assumed that  photons from different nodes have the same probability of being detected (η_DET).

For the particular implementation shown in Fig. <ref>b, using photon polarisation encoding, the probability of generating 3-particle entanglement by heralding in three different detectors and using single-photon detectors is p_BSA_3=1/4. If number-resolved photodetectors are available, the probability of generating a 3-particle entangled state heralded by the detection of three photons is 3/4. Notice that this scheme generates states of the GHZ class and of the W class, depending on the detection pattern (see appendix B).
A different proposed generalised BSA can be found in ref. <cit.>, achieving 3-particle entanglement with probability proportional to p_BSA_3= 1/8. 
In both cases, an interferometer is used to entangle atoms using photons with polarisation, time-bin <cit.>, or energy encoding <cit.>. Other proposed interferometers can be found in ref. <cit.>.



 §.§ N Nodes

Let us now consider the case of N pairs of atom-photon entangled states, i.e.,

    ⊗_n=1^N1/√(2)( |00⟩_A_n P_n + |11⟩_A_n P_n),

where A_n denotes an atom and P_n a photon. This state can be written as <cit.>
    1/√(2^N)∑_n=0^2^N-1|G_n^+⟩_A_1 … A_N|G_n^+⟩_P_1 … P_N + |G_n^-⟩_A_1 … A_N|G_n^-⟩_P_1 … P_N.

This equation shows that N entangled atom-photon pairs can be written as a superposition of states where the atom and photon part are separated, but each part is entangled.
The states |G_n^±⟩ are defined by

    |G_n^±⟩ = 1/√(2)( |B(n)⟩±|B(2^N-n-1 )⟩),

where n = 0,1,…,2^N-1-1, and B(n) = 0b_2b_3⋯ b_N is the
binary representation of n in an bit string of length N, and thereafter
n = ∑_k=2^N b_k·2^N-k.
The states |G_n^±⟩ are a set of 2^N elements, and form an orthonormal basis for the N qubits Hilbert space, i.e., ⟨ G_n^±|G_n^'^±⟩ = δ_n,n^'. The computation basis elements |B(n)⟩ are related to the |G_n^±⟩ states by

    i|B(n)⟩ = 1/√(2)( |G_n^+⟩ + |G_n^-⟩), 
       i|B(2^N-n-1 )⟩ = 1/√(2)( |G_n^+⟩ - |G_n^-⟩).

The |G_n^±⟩ states can also be written as

    |G_n^±⟩ = 1/√(2)( |0⟩⊗_k=2^N|b_k⟩±|1⟩⊗_k=2^N|b̅_k⟩).


By projecting the N-photon state to any of the |G_n^±⟩ states, or superpositions, we project the state of the atoms to the corresponding N-particle state. As before, if an N-photon generalised BSA can be constructed, and the entanglement can be heralded by the detection of N photons yielding to an entangled state with probability p_BSA,N, then the probability of generating N-particle entanglement is given by 

    P_BSA_Nη^N.



An explicit calculation for N=4 is shown in appendixes <ref> and <ref>.  
Thereafter, the challenge is to build a generalised BSA that can herald on the detection of N-photons and generate entanglement with some probability. For this purpose, it is important that the generalised BSA erases the which-input information. This can be achieved using an N-to-N symmetric multiport <cit.>. In an N-to-N symmetric multiport, a single photon entering any of the N inputs has the same probability of exiting any of the outputs, i.e., probability 1/N<cit.>. 
For N = 2^d, with integer d, a generalised BSA can be constructed by using a 2^d-to-2^d symmetric mutiport composed of multiple layers of 50:50 beams splitters, such that each input mode is mixed with every other input mode. Each of the 2^d input photons crosses d 50:50 beam splitters, and has a probability of 1/2^d of exiting via any of the outputs. Fig. <ref>b shows a particular implementation for the 4-node case, and in appendix <ref> we explicitly calculate the different states produced by each possible detection pattern. In appendix <ref> we show how to construct a general 2^d-to-2^d symmetric mutiport. 

Importantly, a 2^d-to-2^d symmetric mutiport allows for entanglement between any sub-network of m = 2, 3,...,  2^d-1 nodes. For example, we can entangle 3 nodes by emitting a photon from each of them using a 8-to-8 interferometer, heralded by the detection of 3 photons in any combination of output ports. This is beneficial, as it does not require the use of additional optical switches.

Alternative generalised BSAs have been considered in refs. <cit.>.





§ REMOTE ENTANGLEMENT VIA WHICH-PATH ERASING


 §.§ Two Nodes

The “which-path erasing” scheme, also known as the Cabrillo or photon-number scheme, is based on the detection of a single photon emitted indistinguishably by one of the two atoms <cit.>. An excitation scheme and appropriate levels are chosen such that the photon emission is the indicator of the atom being in a particular state. 
If we are able to detect a single photon, but the information of which atom emitted the photon has been erased, the measurement of that photon projects the atoms into a multipartite entangled state.

Let us consider that two atoms (n=1,2) are initially prepared in a superposition state

    √(1-p_n)|g_-⟩_A_n + √(p_n)|g_+⟩_A_n.

Then, a π pulse transfers the population from |g_+⟩ to a fast-decaying excited state |e⟩, which decays back to the |g_+⟩ state by emitting a photon. Provided that the excitation pulse is much faster than the decay rate of the excited state, the state of one atom is described by

    √(1-p_n)|g_-,0⟩_A_n,P_n + e^iϕ_D,n+iϕ_L,n√(p_n)|g_+,1⟩_A_n,P_n,

where {|0⟩, |1⟩} is the photon occupation number of the emitted field, ϕ_L,j is the phase of the laser driving the transition |g_+⟩→|e⟩ at the position of atom n, and ϕ_D,n is the phase acquired by the photon emitted by atom j on its path to the detector.

If two atoms simultaneously go through this process, the total state of the system is given by

    √(1-p_1)√(1-p_2)|g_-,g,0,0⟩
        + √(p_1)√(1-p_2) e^i(ϕ_L,1+ϕ_D,1)|g_+,g_-,1,0⟩
        + √(1-p_1)√(p_2) e^i(ϕ_L,2+ϕ_D,2)|g_-,g_+,0,1⟩
        + √(p_1)√(p_2) e^i(ϕ_L,1 + ϕ_L,2 + ϕ_D,1 + ϕ_D,2  )|g_+,g_+,1,1⟩.

Then, we direct the collected photons to a which-path eraser (WPE), as shown in Fig. <ref>a. For the case of two inputs, this can be implemented using a 50:50 beam splitter. If the detected modes of the emission from the atoms overlap perfectly, i.e., the photons emitted from either atom are indistinguishable, the detection of a single photon in either detector projects the state of the atoms into the (unnormalised) state 

    √(p_1)√(1-p_2) e^i(ϕ_L,1+ϕ_D,1)|g_+,g_-⟩
        + √(1-p_1)√(p_2) e^i(ϕ_L,2+ϕ_D,2)|g_-,g_+⟩.

 This will occur with probability η_1 p_1(1-p_2)+η_2 p_2(1-p_1), where η_1,2 is the overall detection efficiency of the emitted single photon from atom 1 and atom 2, respectively.

 If the probability of excitation is the same for both atoms, p_1 = p_2 = p, and the efficiency of the collection, transmission, and detection is the same for both, i.e., η_1 = η_2 = η, the generated state is approximately given by
 
    |Ψ^ϕ⟩ = 1/√(2)(|g_+,g_-⟩± e^iϕ|g_-,g_+⟩),

 with an overall probability of 2η p(1-p) and ϕ=(ϕ_L,2-ϕ_L,1) + (ϕ_D,B-ϕ_D,A). The sign of the superposition depends on what output of the beam splitter the photon was detected.

Notice that there is a non-vanishing probability,  proportional to p^2, of exciting both atoms and detecting just one photon. If p is kept small, then p^2 is negligible. Thereafter, upon a click in one of the detectors, the state is 

    1/√(2p(1-p)+p^2)×
       [√(2p(1-p))/√(2)(|g_+,g_-⟩+e^i(ϕ_L,2+ϕ_D,2-ϕ_L,1-ϕ_D,1)|g_-,g_+⟩)|1⟩. 
       + .  pe^i(ϕ_L,1+ϕ_D,1+ ϕ_L,2+ϕ_D,2)|g_+,g_+⟩|2⟩].

The fidelity of this state with respect to a two-particle Bell state is bounded by the process that scatters more than one photon, i.e., the maximum achievable fidelity with respect to a Bell state is

    F^WPE_1,2 = |⟨ψ^ϕ | ψ_1,2⟩|^2
    
        = 2p(1-p)/2p(1-p)+p^2.

For example, when using an excitation probability p=6, the maximum attainable fidelity is 96.9. 
The rate of entanglement heralding events is proportional to 
    R^WPE_1,2∝η_DET(2p(1-p)+p^2),

where η_DET includes the collection efficiency, fibre transmission, the transmission efficiency of the WPE and the detectors efficiency. It is worth noticing that this rate includes single photon heralding events, but also double photon detection. These are indistinguishable from single photon events and do not create entanglement, reducing the fidelity of the created state. 

This process has been used to demonstrate deterministic distribution of entanglement on NV-centre systems <cit.>,  and for several experiments demonstrating generation of entanglement in trapped-ion systems <cit.>,
in quantum-dot systems <cit.> in NV-centres <cit.> and rare-earth on qubits in crystalline structures <cit.>. It has also been used to create bipartite entanglement in a three-node network <cit.>.
In these implementations, the main sources of errors are the finite probability of exciting two atoms at the same time, residual which-path information in the detected photons, distinguishability of the atoms due to recoil, uncertainty in the phase due to residual thermal motion, and the limited fidelity of the coherent operations. Ref. <cit.> shows a detailed analysis of different sources of error for this implementation, and specifically discusses the ones that arise in NV-centre systems.

Notice also that in the  implementation with trapped ions, such as the one presented in <cit.>, the excitation is done via a non-deterministic Raman process. Since there are other decay channels in the atom's electronic structure, this results in additional errors.




 §.§ Three Nodes

The photon-number scheme can be generalised to any number of particles. Let us consider the example of three atoms. If three atoms, indexed by j, undergo the process described above, the total state of the system is now given by

    ×(√(1-p_1)|g_-,0⟩_1 + √(p_1)e^iϕ_D,1+iϕ_L,1|g_+,1⟩_1) 
       ⊗(√(1-p_2)|g_-,0⟩_2 + √(p_2)e^iϕ_D,2+iϕ_L,2|g_+,1⟩_2)
       ⊗(√(1-p_3)|g_-,0⟩_3 + √(p_3)e^iϕ_D,3+iϕ_L,3|g_+,1⟩_3)

expanding this, we get

    ×√(1-p_1)√(1-p_2)√(1-p_3)|g_-,g_-,g_,0,0,0⟩
    
       + √(p_1)√(1-p_2)√(1-p_3)e^iϕ_1|g_+,g_-,g_-,1,0,0⟩
       + √(1-p_1)√(p_2)√(1-p_3)e^iϕ_2|g_-,g_+,g_-,0,1,0⟩
       + √(1-p_1)√(1-p_2)√(p_3)e^iϕ_3|g_-,g_-,g_+,0,0,1⟩
    
       + √(p_1)√(p_2)√(1-p_3)e^iϕ_1e^iϕ_2|g_+,g_+,g_-,1,1,0⟩
       + √(1-p_1)√(p_2)√(p_3)e^iϕ_2e^iϕ_3|g_-,g_+,g_+,0,1,1⟩
       + √(p_1)√(1-p_2)√(p_3)e^iϕ_3e^iϕ_1|g_+,g_-,g_+,1,0,1⟩
    
       +√(p_1)√(p_2)√(p_3)e^iϕ_1e^iϕ_2e^iϕ_3|g_+,g_+,g_+,1,1,1⟩,

where ϕ_n = ϕ_L,n+ϕ_D,n. In this case, there are non-vanishing probabilities of emitting 0, 1, 2, or 3 photons. Let us consider that all the atoms are excited with the same probability p, so that the state simplifies to

    ± (1-p)^3/2|g_-,g_-,g_,0,0,0⟩
    
        + √(p)(1-p)e^iϕ_1|g_+,g_-,g_-,1,0,0⟩
       + √(p)(1-p)e^iϕ_2|g_-,g_+,g_-,0,1,0⟩
        + √(p)(1-p)e^iϕ_3|g_-,g_-,g_+,0,0,1⟩
    
       + p√(1-p)e^iϕ_1e^iϕ_2|g_+,g_+,g_-,1,1,0⟩
        + p√(1-p)e^iϕ_2e^iϕ_3|g_-,g_+,g_+,0,1,1⟩
       + p√(1-p)e^iϕ_3e^iϕ_1|g_+,g_-,g_+,1,0,1⟩
    
       + p^3/2 e^iϕ_1 e^iϕ_2e^iϕ_3|g_+,g_+,g_+,1,1,1⟩,


  §.§.§ Single-Excitation W-state

If we direct the collected photons to a WPE, as shown in Fig. <ref>b, the emission and detection of a single indistinguishable photon projects, approximately, the state to

    1/√(3)(|g_+,g_-,g_-⟩ + e^iϕ_2|g_-,g_+,g_-⟩. 
              + .e^iϕ_3|g_-,g_-,g_+⟩),

This multipartite entangled state is called a single excitation W-state. Notice that the optical setup shown in Fig. <ref>b is a symmetric 3-to-3 multiport, i.e, a tritter <cit.>, allowing for the erasure of the which-path information of the detected photon.  
If we use non-number-resolved photodetectors, the fidelity is limited by the non-negligible probability of not differentiating between one or more photons being detected in a single detector. Thereafter, the state of Eq. (<ref>) can be generated with a maximum fidelity of

    F^WPE_1,3 =3p(1-p)^2/3p(1-p)^2+3p^2(1-p)+p^3.

If we consider all the photon emissions that are detected as single clicks in the non-number-resolved photodetectors, the probability of detecting a heralding photon, and thereafter, the rate of entanglement generation is proportional to

    P^WPE_1,3∝η_DET(3p(1-p)^2+3p^2(1-p)+p^3).




A demonstration of this scheme for 3-atom entanglement, with two of them in the same node, has been shown in ref. <cit.>.



  §.§.§ Two-Excitation W-state

It is also possible to create an entangled state with a superposition of two atoms being excited, i.e., approximately 

    1/√(3)( e^iϕ_1e^iϕ_2|g_+,g_+,g_-⟩.
       + .e^iϕ_2e^iϕ_3|g_-,g_+,g_+⟩ +  e^iϕ_1e^iϕ_3|g_+,g_-,g_+⟩).

This process is heralded by the detection of two photons at the output of the WPE. The maximum fidelity with respect to the state of Eq. (<ref>) is limited by the non-negligible probability of emitting three photons and detecting just two, i.e., 

    F^WPE_2,3 = η_DET^23p^2(1 - p)/3p^2(1 - p) + p^3.

The probability of generating this state is proportional to the probability of detecting two photons at the outputs of the WPE. When using the tritter setup of Fig. <ref>b and using non-number-resolved photo-detectors, the probability of detecting two photons, and therefore the entanglement generation rate is given by

    P^WPE_2,3 = p_WPE,2,3·η_DET^2(3p^2(1 - p) + p^3),

where, for the case of the tritter,  p_wpe,2,3=1/3 is the probability of two photons exiting in two different outputs. The other 2/3 of the photons bunch together in any of the three detectors with equal probability 2/9, due to the Hong–Ou–Mandel effect. Thereafter, using number-resolved photodetectors can increase p_WPE,2,3 to reach unity.



If we choose p = 6%, then the probability of emitting one photon is ≈ 16%, two photons is ≈ 0.1%, and three photons is 2· 10^-4. With 83% probability, no photons are produced. The maximum fidelity of the one-excitation state using this value is F^WPE_1,3 = 93.88%, while for the 2-excitation entangled state is F^WPE_2,3 = 97.91%. The time required to create the one-excitation state scales linearly with η_DET, while the two excitation scales with η_DET^2, so there is a strong trade-off between fidelity and entanglement rate to be considered for practical implementations. 





 §.§ N Nodes

The photon-number scheme can be readily extended for any number of particles. If we consider N atoms, all prepared in the same initial superposition, the state after excitation can be written as <cit.>
    |Ψ⟩ = ∑_n=0^NNn^1/2 p^n/2 (1-p)^(N-n)/2|W_n,N-n⟩|n⟩,

where |n⟩ is the number of photons emitted collectively by the atoms, and |W_n,N⟩ are the so-called generalised W states, or generalised Dicke states. They can be defined as

    |W_n,N⟩ = Nn^-1/2∑_k 𝒫_k [|S_n,N-n⟩],

where 𝒫_k is the operator that produces all the possible different permutations with an equal number n of particles being in the |g_+⟩ state, and with |S_n,N-n⟩ defined by

    |S_n,N-n⟩ = ∏_α=1^n e^iϕ_α|g_+⟩_α∏_β=n+1^N |g_-⟩_β.

Thereafter, the probability of emitting only one photon is the prefactor of the |W_1,N-1⟩ state, and is given by 

    p_1,N = N p (1-p)^N-1.

The probability of emitting n photons is given by

    p_n,N = Nn p^n (1-p)^N-n.


As before, if we cannot distinguish the number of photons detected, the fidelity is limited by the multi-photon process. If we aim to generate the state |W_1,N⟩, the fidelity is limited to

    F^WPE_1,N = Np(1-p)^N-1/∑_n=1^NNn p^n (1-p)^N-n.

The probability of detecting a photon heralding this process is proportional to

    R^WPE_1,N∝η_DET∑_n=1^NNn p^n (1-p)^N-n.


If we aim to create a state with two excitations, i.e., |W_2,N⟩ by detecting two photons, the fidelity is limited by the multi-photon process, which is not distinguishable from the two-photon process due to optical losses, i.e.,

    F^WPE_2,N = Np^2(1-p)^N-2/∑_n=2^NNn p^n (1-p)^N-n.

The probability of detecting two photons heralding this process is proportional to

    R^WPE_2,N∝η^2∑_n=2^NNn p^n (1-p)^N-n.



In general, if we aim to create the m excitations state, i.e., |W_m,N-m⟩, the multi-photon processes limit the maximum attainable fidelity to

    F^WPE_m,N = Np^m(1-p)^N-m/∑_n=m^NNn p^n (1-p)^N-n.

The probability of detecting m photons heralding this process, and therefore the generation rate of the state |W⟩_m,N, is proportional to

    R^WPE_m,N∝η_DET^m∑_n=m^NNn p^m (1-p)^N-n.


The probability of detecting n photons will depend on the availability of number-resolved photodetectors, and the particular implementation of the n-to-n multiport, as in general, we will observe some outputs to be suppressed by the Hong-Ou-Mandel effect. As before, depending on the dimension, symmetric multiports can be built using combinations of beam splitters and tritters (see section <ref>). Other alternatives include the use of integrated multi-mode interferometers <cit.>.
Importantly, the entanglement rate using this scheme scales as η_DET^m, and since m<N, it is always better than the swapping scheme, which scales as η_DET^N. Fig. <ref> shows the maximum achievable fidelities and entanglement success rates in a 4-node network, for generalised states with 1, 2 and 3 excitations, using η_DET = 0.05.



§ OPTICAL PHASE CONTROL

The fidelity of entangled states is limited by the repeatability and stability of the optical phases of photons traveling through fibres and interferometers. Variations in these optical phases are imprinted on the generated entangled states, causing them to fluctuate over time. While the entanglement swapping scheme and the itinerant photon scheme are resilient to large phase drifts <cit.>, approaches such as the which-path erasing, single-photon exchange between multiple nodes, and entanglement mapping require sub-wavelength path length stabilisation.

Various methods can actively stabilise these phases. In <cit.>, the fluorescence light emitted by atoms during cooling is used to measure phase drift, and a movable mirror compensates for this drift using a sample-and-hold phase lock. In <cit.>, phase stabilisation is achieved by illuminating the sample substrate and using the reflected light to measure phase drift, with a fibre stretcher employed for compensation. A similar approach is described in <cit.>.

Even if the phase is unknown, it can be distilled out, provided that phase drifts are slower than the time required for multiple repetitions <cit.>. These conditions are typically met for short fibre lengths, as phase drift generally occurs on the scale of seconds, while the time required to generate two consecutive entangling events is less than a millisecond under realistic parameters.

Another source of phase uncertainty, particularly in atomic systems, is the intrinsic motion of atoms in trapping potentials. Uncertainty in the atom's position at the moment of absorbing or emitting a photon, causes path-length uncertainties. This uncertainty can be reduced by cooling the atoms to the ground state <cit.>. Additionally, residual motion-qubit entanglement due to recoil after the emission or absorption of single photons can be mitigated by selecting proper absorption and emission directions and, in some cases, by synchronising ion oscillation periods and absorption and emission times <cit.>.





§ DISCUSSION

We have presented and discussed different schemes for the generation of genuine multipartite remote entanglement over an N-node quantum network and considered its implementation on different experimental platforms used for quantum networking with matter qubits, including NV-centres, trapped ions, neutral atoms and superconducting qubits. The main advantage of using these schemes in comparison to schemes based solely on the use of bipartite entanglement (see, e.g., <cit.>) is that the single-shot schemes presented here do not require extensive use of local measurement, feed-forward and local entangling gates. Therefore, the desired entangled states can be generated without excessive network latency, and without the need of ancillary qubits or long coherence times. 


Typically, in a realistic network node, one could have many qubits, but a limited number of atom-photon interfaces per node. The qubit interacting with the interface will therefore need to be swapped, either physically or logically, during the execution of the scheme. Thereafter, using bipartite-entanglement-based schemes, bipartite entanglement will need to be created in a sequential way between different nodes. For this to work with high fidelity, the coherence time of the individual qubits needs to be much longer than the total time required to establish the desired multipartite state. Although this is feasible for some experimental platforms (see e.g., <cit.>), it is still a challenge for most experimental realisations of quantum networks. Furthermore, the overhead on the number of ancillary qubits and the number of local-entangling gates needed when using the bipartite-entanglement-based schemes grows with the number of nodes to entangle, and hence, the fidelity of the involved operations.


All the schemes presented here do not require any extra qubits to create the multi-node entangled state nor local entangling gates, and importantly, do not rely on long coherence times to generate multi-node entanglement because the entanglement is performed in one single step after the initial state preparation of the qubits. For all the heralded schemes presented here, if a try does not generate entanglement, the initial state is prepared again, and a new try is executed. A single try has been demonstrated to be as fast as 1 µs for atomic-based systems <cit.>, while the coherence time can be, even without the use of a dedicated memory ion, several orders of magnitude longer <cit.>.


Regarding efficiency, the best scaling with the number of nodes is expected to be achieved using the which-path erasing scheme. However, this scheme also shows the strongest trade-off between multipartite entanglement rate and achievable fidelity (Eqs. (<ref>) and (<ref>)), which may render it impractical for the most stringent applications. All other schemes discussed here show entangling rates that depend on η^N, where N is the number of nodes and η is the efficiency of the process. Generally, this will result in slow multi-node entangling rates compared to schemes based solely on bipartite entanglement <cit.>. However, this limitation diminishes as η approaches 1.




Consider, for example, a 4-node network of trapped ions using the parameters presented in <cit.> for Bell pair generation. In this case, the relevant efficiency η is η_DET, the probability of collecting and detecting a single photon emitted by the trapped ions. The generation rate of bipartite entanglement is given by (Eq. (<ref>))

    R_Bell = 1/2η_DET^2 r_T,

where r_T is the entanglement trial rate. To generate entanglement between four nodes, assuming that each node has at most one ion-photon interface (the Bell state generation cannot be parallelised), and neglecting the time required to perform the Bell measurements and local entangling gates necessary <cit.>, the rate of 4-node entanglement generation is ≤ (1/4) R_Bell, assuming that four rounds of bipartite entanglement is required to generate genuine entanglement between the four nodes (see e.g., <cit.>).

On the other hand, using the scheme for 4-node entanglement proposed in Appendix <ref>, the entanglement generation rate is given by

    R_Quad = 7/32η_DET^4 r_T,

assuming only non-number resolved detectors are available. Figure <ref> shows that for η_DET⪆ 0.76, the 4-photon scheme is more efficient than the bipartite-based scheme. This η_DET⪆ 0.76 is very close to efficiencies already achieved in trapped ion systems <cit.>. Conditions for the single-shot scheme being advantageous compared to bipartite entanglement-only schemes can be found for all the discussed schemes by varying the corresponding η parameters.



The diverse range of schemes presented here offers promising avenues for achieving multipartite remote entanglement in quantum networks, extending beyond the limitations of using only bipartite remote entanglement. Each scheme carries its unique advantages and challenges, tailored for specific experimental platforms. Further experiments and simulations are needed to understand the impact of various experimental errors on the quality of generated multipartite entangled states. Refs. such as <cit.> provide valuable insights into relevant calculations and simulations, paving the way for comprehensive exploration and realisation of these schemes in diverse quantum computing, metrology, and sensing architectures.



§ ACKNOWLEDGEMENTS

We thank David Lucas for valuable comments on the manuscript and Tommaso Faorlin for useful discussion. DM acknowledges support from the U.S. Army Research Office (ref. W911NF-18-1-0340).
DPN acknowledges support from Merton College, Oxford.
EMA acknowledges support from the U.K. EPSRC “Quantum Communications” Hub EP/T001011/1.
RS acknowledges funding from an EPSRC Fellowship EP/W028026/1 and Balliol College, Oxford.
GAM acknowledges support from Wolfson College, Oxford.
This work was supported by the U.K. EPSRC “Quantum Computing and Simulation” Hub EP/T001062/1.


 


§ ENTANGLEMENT 
SWAPPING FOR FOUR NODES

Using the entanglement swapping technique presented in section <ref>, let us now consider the case of four identical atom-photon entangled pairs, i.e., 

    ⊗_n=1^4 1/√(2)( |0⟩_A_n|0⟩_P_n±|1⟩_A_n|1⟩_P_n)=
       1/√(16)(  |G_0^+⟩_A_1A_2A_3A_4|G_0^+⟩_P_1P_2P_3P_4
       +|G_0^-⟩_A_1A_2A_3A_4|G_0^-⟩_P_1P_2P_3P_4
       + |G_1^+⟩_A_1A_2A_3A_4|G_1^+⟩_P_1P_2P_3P_4
       ±|G_1^-⟩_A_1A_2A_3A_4|G_1^-⟩_P_1P_2P_3P_4
       ±|G_2^+⟩_A_1A_2A_3A_4|G_2^+⟩_P_1P_2P_3P_4
       ±|G_2^-⟩_A_1A_2A_3A_4|G_2^-⟩_P_1P_2P_3P_4
       +|G_3^+⟩_A_1A_2A_3A_4|G_3^+⟩_P_1P_2P_3P_4
       +|G_3^-⟩_A_1A_2A_3A_4|G_3^-⟩_P_1P_2P_3P_4
       ±|G_4^+⟩_A_1A_2A_3A_4|G_4^+⟩_P_1P_2P_3P_4
       ±|G_4^-⟩_A_1A_2A_3A_4|G_4^-⟩_P_1P_2P_3P_4
       +|G_5^+⟩_A_1A_2A_3A_4|G_5^+⟩ _P_1P_2P_3P_4
       +|G_5^-⟩_A_1A_2A_3A_4|G_5^-⟩ _P_1P_2P_3P_4
       +|G_6^+⟩_A_1A_2A_3A_4|G_6^+⟩ _P_1P_2P_3P_4
       +|G_6^-⟩_A_1A_2A_3A_4|G_6^-⟩_P_1P_2P_3P_4
       ±|G_7^+⟩_A_1A_2A_3A_4|G_7^+⟩_P_1P_2P_3P_4
       ±|G_7^-⟩_A_1A_2A_3A_4|G_7^-⟩_P_1P_2P_3P_4),

with

    |G_0^±⟩ = 1/√(2)(|0000⟩±|1111⟩),
    |G_1^±⟩ = 1/√(2)(|0001⟩±|1110⟩),
    |G_2^±⟩ = 1/√(2)(|0010⟩±|1101⟩),
    |G_3^±⟩ = 1/√(2)(|0011⟩±|1100⟩),
    |G_4^±⟩ = 1/√(2)(|0100⟩±|1011⟩),
    |G_5^±⟩ = 1/√(2)(|0101⟩±|1010⟩)
    |G_6^±⟩ = 1/√(2)(|0110⟩±|1001⟩),
    |G_7^±⟩ = 1/√(2)(|0111⟩±|1000⟩),

where the {|G_n^±⟩} states correspond now to 4-particle entangled states of the GHZ class.
As before, by using a 4-photon generalized BSA it is possible to generate 4-particle entanglement. 

An example of such a 4-photon generalized BSA using photon-polarisation encoding, is shown in Fig. <ref>c. This BSA is a 4-to-4 symmetric multiport <cit.>: a single photon entering any of the input ports has the same probability of exiting at any of the output ports, erasing the information about which atom emitted that photon. Notice that this so-called quarter uses 50:50 beams splitters only <cit.>. The creation of entanglement of 4-particle entanglement is heralded by four coincidence clicks in some particular photon detection patterns. A detailed analysis for the different detection cases is shown in appendix <ref>.
If a particular implementation of the 4-photon BSA projects the atomic states into an entangled state with probability p_BSA_4, the probability of generating a 4-particle entangled state is then

    R^SW_4 ∝ p_BSA_4η_DET^4.




The implementation of the 4-photon generalised BSA shown in Fig. <ref>c, a called quarter, can generate 4-particle remote entanglement with probability p_BSA_4 = 7/32 upon coincident detection of single photons at four different detectors. When using number-resolved detectors, and heralding in any combination of four photons detected, the probability of generating entanglement is 7/8, see appendix <ref> for details. Note that, depending on the heralding detection patterns, not all the generated 4-particle entangled states are one of the {|G_n^±⟩} states, but more general 4-particle entangled states of the GHZ and W classes. Importantly, if we attempt to produce entanglement between only two nodes using the quarter shown in Fig. <ref>c, by sending two  photons and detecting two photons at the output, a Bell state 
 between the two nodes is produced also with the maximum probability allowed, i.e., 1/2η_DET^2. This enables the production of entanglement of any two in a 4-node without the need of an additional optical switch. The same applies for entangling 3 nodes. 

Using the alternative generalised BSA presented in ref. <cit.>, a 4-particle GHZ state can be generated with probability proportional to p_BSA_4 = 1/16. Other proposed interferometer structures can be found in  ref. <cit.>

§ ENTANGLEMENT SWAPPING USING A QUARTER

We consider now the case of entanglement swapping for a 4-node network using a quarter (see Fig. <ref>). The quarter is unitary transformation from the input modes a to the output modes b, such that the creation operators for each mode are related through

    (
    [ b^†_{h,v}1; b^†_{h,v}2; b^†_{h,v}3; b^†_{h,v}4;            ])= Q ·(
    [ a^†_{h,v}1; a^†_{h,v}2; a^†_{h,v}3; a^†_{h,v}4;            ]).

For example,  a^†_{h}1 is the creation operator for a h-polarized photon in the input one.
The unitary matrix representing the transformation exerted by the interferometer of Fig.<ref>c is the composition of the unitary transformations of a beam splitters acting on different sequentially on different modes

    Q =    (
    [      1      0      0      0;      0 1/√(2) i/√(2)      0;      0 i/√(2) 1/√(2)      0;      0      0      0      1;        ]) ·(
    [ 1/√(2)      0      0 i/√(2);      0      1      0      0;      0      0      1      0; i/√(2)      0      0 1/√(2);        ]) ·(
    [      1      0      0      0;      0      1      0      0;      0      0 1/√(2) i/√(2);      0      0 i/√(2) 1/√(2);        ])·(
    [ 1/√(2) i/√(2)      0      0; i/√(2) 1/√(2)      0      0;      0      0      1      0;      0      0      0      1;        ]).

Notice that this is a particular realisation of a quarter, and in principle, arbitrary phase shift could be placed in any path of the interferometer, changing this matrix and the obtained state that we will listen below. The phases could be optimised to achieve certain state with high probability.
We condition on the detection of some combination of clicks at the output states b, so we need to write the input states in terms of output states. This is done through the inverse matrix

    (
    [ a^†_{h,v}1; a^†_{h,v}2; a^†_{h,v}3; a^†_{h,v}4;            ])= Q^-1·(
    [ b^†_{h,v}1; b^†_{h,v}2; b^†_{h,v}3; b^†_{h,v}4;            ])

with

    Q^-1 = (
    [  1/2 -i/2 -1/2 -i/2; -i/2  1/2 -i/2 -1/2; -1/2 -i/2  1/2 -i/2; -i/2 -1/2 -i/2  1/2;      ]).


Any of the input states of the interferometer (Eq. <ref>) can be written using the specific combination of creation operators, for example

    1/√(2)(|0101⟩_P_1,P_2,P_3,P4+|1010⟩_P_1,P_2,P_3,P4)    = 1/√(4)(|hvhv⟩+|vhvh⟩)
       = 1/√(2)(a^†_h,1a^†_v,2a^†_h,3a^†_v,4+a^†_v,1a^†_h,2a^†_v,3a^†_h,4)|0⟩.

Then, each of the input field operators a^† operators can be written using Eq. <ref> in terms of the output operators of the GBSA. Then we can write the full extension of Eq. <ref>, and see how different output patterns, i.e., photon detection of four photons patterns in the eight detectors, correspond to the different projection of the atomic states. If we consider that the initial state of the atom-photon pairs is given by

    ⊗_n=1^4 1/√(2)( |0⟩_A_n|0⟩_P_n + |1⟩_A_n|1⟩_P_n),

And we consider detection patterns of four photons, with at most one click per detector, the possible patterns are:


    Detection Pattern   State   Probability
    
     b_h1 b_h2 b_h3 b_h4    |0000⟩   1/64
    
     b_h1 b_h2 b_h3 b_v1    -1/2 i
       (|0001⟩ +|0010⟩ -|0100⟩ -|1000⟩ )    1/256
    
     b_h1 b_h2 b_h3 b_v2   1/2
       (-|0001⟩ +|0010⟩ -|0100⟩ +|1000⟩ )    1/256
    
     b_h1 b_h2 b_h3 b_v3    -1/2 i
       (|0001⟩ -|0010⟩ -|0100⟩ +|1000⟩ )    1/256
    
     b_h1 b_h2 b_h3 b_v4   1/2
       (|0001⟩ +|0010⟩ +|0100⟩ +|1000⟩ )    1/256
    
     b_h1 b_h2 b_h4 b_v1   1/2
       (|0001⟩ -|0010⟩ +|0100⟩ -|1000⟩ )    1/256
    
     b_h1 b_h2 b_h4 b_v2    -1/2 i
       (|0001⟩ +|0010⟩ -|0100⟩ -|1000⟩ )    1/256
    
     b_h1 b_h2 b_h4 b_v3   1/2
       (|0001⟩ +|0010⟩ +|0100⟩ +|1000⟩ )    1/256
    
     b_h1 b_h2 b_h4 b_v4   1/2 i
       (|0001⟩ -|0010⟩ -|0100⟩ +|1000⟩ )    1/256
    
     b_h1 b_h2 b_v1 b_v2   |0110⟩
       +|1001⟩/√(2)   1/128
    
     b_h1 b_h2 b_v3 b_v4   |0110⟩
       +|1001⟩/√(2)   1/128
    
     b_h1 b_h3 b_h4 b_v1   1/2 i
       (|0001⟩ -|0010⟩ -|0100⟩ +|1000⟩ )    1/256
    
     b_h1 b_h3 b_h4 b_v2   1/2
       (|0001⟩ +|0010⟩ +|0100⟩ +|1000⟩ )    1/256
    
     b_h1 b_h3 b_h4 b_v3   1/2 i
       (|0001⟩ +|0010⟩ -|0100⟩ -|1000⟩ )    1/256
    
     b_h1 b_h3 b_h4 b_v4   1/2
       (-|0001⟩ +|0010⟩ -|0100⟩ +|1000⟩ )    1/256
     
     b_h1 b_h3 b_v1 b_v3   -|0101⟩
       -|1010⟩/√(2)   1/128
    
     b_h1 b_h3 b_v2 b_v4   |0101⟩
       +|1010⟩/√(2)   1/128
    
     b_h1 b_h4 b_v1 b_v4   |0011⟩
       +|1100⟩/√(2)   1/128
    
     b_h1 b_h4 b_v2 b_v3   |0011⟩
       +|1100⟩/√(2)   1/128
    
     b_h1 b_v1 b_v2 b_v3   1/2 i
       (|0111⟩ +|1011⟩ -|1101⟩ -|1110⟩ )    1/256
    
     b_h1 b_v1 b_v2 b_v4   1/2
       (-|0111⟩ +|1011⟩ -|1101⟩ +|1110⟩ )    1/256
    
     b_h1 b_v1 b_v3 b_v4   1/2 i
       (|0111⟩ -|1011⟩ -|1101⟩ +|1110⟩ )    1/256
    
     b_h1 b_v2 b_v3 b_v4   1/2
       (|0111⟩ +|1011⟩ +|1101⟩ +|1110⟩ )    1/256
    
     b_h2 b_h3 b_h4 b_v1   1/2
       (|0001⟩ +|0010⟩ +|0100⟩ +|1000⟩ )    1/256
    
     b_h2 b_h3 b_h4 b_v2    -1/2 i
       (|0001⟩ -|0010⟩ -|0100⟩ +|1000⟩ )    1/256
    
     b_h2 b_h3 b_h4 b_v3   1/2
       (|0001⟩ -|0010⟩ +|0100⟩ -|1000⟩ )    1/256
    
     b_h2 b_h3 b_h4 b_v4   1/2 i
       (|0001⟩ +|0010⟩ -|0100⟩ -|1000⟩ )    1/256
    
     b_h2 b_h3 b_v1 b_v4   |0011⟩
       +|1100⟩/√(2)   1/128
    
     b_h2 b_h3 b_v2 b_v3   |0011⟩
       +|1100⟩/√(2)   1/128
    
     b_h2 b_h4 b_v1 b_v3   |0101⟩
       +|1010⟩/√(2)   1/128
    
     b_h2 b_h4 b_v2 b_v4   -|0101⟩
       -|1010⟩/√(2)   1/128
    
     b_h2 b_v1 b_v2 b_v3   1/2
       (|0111⟩ -|1011⟩ +|1101⟩ -|1110⟩ )    1/256
    
     b_h2 b_v1 b_v2 b_v4   1/2 i
       (|0111⟩ +|1011⟩ -|1101⟩ -|1110⟩ )    1/256
    
     b_h2 b_v1 b_v3 b_v4   1/2
       (|0111⟩ +|1011⟩ +|1101⟩ +|1110⟩ )    1/256
    
     b_h2 b_v2 b_v3 b_v4    -1/2 i
       (|0111⟩ -|1011⟩ -|1101⟩ +|1110⟩ )    1/256
    
     b_h3 b_h4 b_v1 b_v2   |0110⟩
       +|1001⟩/√(2)   1/128
    
     b_h3 b_h4 b_v3 b_v4   |0110⟩
       +|1001⟩/√(2)   1/128
    
     b_h3 b_v1 b_v2 b_v3    -1/2 i
       (|0111⟩ -|1011⟩ -|1101⟩ +|1110⟩ )    1/256
    
     b_h3 b_v1 b_v2 b_v4   1/2
       (|0111⟩ +|1011⟩ +|1101⟩ +|1110⟩ )    1/256
    
     b_h3 b_v1 b_v3 b_v4   1/2 i
       (-|0111⟩ -|1011⟩ +|1101⟩ +|1110⟩ )    1/256
    
     b_h3 b_v2 b_v3 b_v4   1/2
       (-|0111⟩ +|1011⟩ -|1101⟩ +|1110⟩ )    1/256
    
     b_h4 b_v1 b_v2 b_v3   1/2
       (|0111⟩ +|1011⟩ +|1101⟩ +|1110⟩ )    1/256
    
     b_h4 b_v1 b_v2 b_v4   1/2 i
       (|0111⟩ -|1011⟩ -|1101⟩ +|1110⟩ )    1/256
    
     b_h4 b_v1 b_v3 b_v4   1/2
       (|0111⟩ -|1011⟩ +|1101⟩ -|1110⟩ )    1/256
    
     b_h4 b_v2 b_v3 b_v4   1/2 i
       (-|0111⟩ -|1011⟩ +|1101⟩ +|1110⟩ )    1/256
    
     b_v1 b_v2 b_v3 b_v4    |1111⟩   1/64

If we consider at most two clicks per detector, the possible patterns are:

    Detection Pattern   State   Probability
    
     b_h1^2 b_h2^2     |0000⟩   1/256
    
     b_h1^2 b_h2 b_v1   1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_h2 b_v2   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_h2 b_v3   1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1^2 b_h2 b_v4   1/2 (-|0001⟩
       +|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_h3^2     -|0000⟩   1/256
    
     b_h1^2 b_h3 b_v1   1/2 (-|0001⟩
       +|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_h3 b_v2   1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1^2 b_h3 b_v3   1/2 (-|0001⟩
       -|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1^2 b_h3 b_v4    -1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_h4^2     |0000⟩   1/256
    
     b_h1^2 b_h4 b_v1    -1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1^2 b_h4 b_v2   1/2 (-|0001⟩
       +|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_h4 b_v3    -1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_h4 b_v4   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h1^2 b_v1^2    |0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h1^2 b_v1 b_v2   i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h1^2 b_v1 b_v3   |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h1^2 b_v1 b_v4   i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h1^2 b_v2^2    |0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h1^2 b_v2 b_v3   i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h1^2 b_v2 b_v4   |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h1^2 b_v3^2    -|0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h1^2 b_v3 b_v4   i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h1^2 b_v4^2    -|0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h1 b_h2^2 b_v1   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h1 b_h2^2 b_v2    -1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1 b_h2^2 b_v3   1/2 (|0001⟩
       -|0010⟩ +|0100⟩ -|1000⟩ )    1/512
    
     b_h1 b_h2^2 b_v4   1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1 b_h2 b_v1^2     -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h1 b_h2 b_v2^2    i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h1 b_h2 b_v3^2    i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h1 b_h2 b_v4^2     -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h1 b_h3^2 b_v1   1/2 (-|0001⟩
       -|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1 b_h3^2 b_v2   1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1 b_h3^2 b_v3   1/2 (-|0001⟩
       +|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1 b_h3^2 b_v4    -1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1 b_h3 b_v1^2    |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h1 b_h3 b_v2^2    |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h1 b_h3 b_v3^2    |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h1 b_h3 b_v4^2    |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h1 b_h4^2 b_v1   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h1 b_h4^2 b_v2    -1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h1 b_h4^2 b_v3   1/2 (|0001⟩
       -|0010⟩ +|0100⟩ -|1000⟩ )    1/512
    
     b_h1 b_h4^2 b_v4   1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h1 b_h4 b_v1^2     -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h1 b_h4 b_v2^2     -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h1 b_h4 b_v3^2    i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h1 b_h4 b_v4^2    i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h1 b_v1^2 b_v2   1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h1 b_v1^2 b_v3   1/2 (|0111⟩
       -|1011⟩ +|1101⟩ -|1110⟩ )    1/512
    
     b_h1 b_v1^2 b_v4   1/2 i (|0111⟩
       +|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h1 b_v1 b_v2^2    1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h1 b_v1 b_v3^2    1/2 (-|0111⟩
       -|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h1 b_v1 b_v4^2    1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h1 b_v2^2 b_v3   1/2 (-|0111⟩
       +|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h1 b_v2^2 b_v4   1/2 i (-|0111⟩
       -|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h1 b_v2 b_v3^2    1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h1 b_v2 b_v4^2     -1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h1 b_v3^2 b_v4   1/2 i (|0111⟩
       +|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h1 b_v3 b_v4^2    1/2 (-|0111⟩
       +|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h2^2 b_h3^2     |0000⟩   1/256
    
     b_h2^2 b_h3 b_v1   1/2 (|0001⟩
       -|0010⟩ +|0100⟩ -|1000⟩ )    1/512
    
     b_h2^2 b_h3 b_v2    -1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h2^2 b_h3 b_v3   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h2^2 b_h3 b_v4   1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h2^2 b_h4^2     -|0000⟩   1/256
    
     b_h2^2 b_h4 b_v1   1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h2^2 b_h4 b_v2   1/2 (|0001⟩
       -|0010⟩ +|0100⟩ -|1000⟩ )    1/512
    
     b_h2^2 b_h4 b_v3   1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h2^2 b_h4 b_v4   1/2 (-|0001⟩
       -|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h2^2 b_v1^2    |0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h2^2 b_v1 b_v2    -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h2^2 b_v1 b_v3   |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h2^2 b_v1 b_v4   i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h2^2 b_v2^2    |0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h2^2 b_v2 b_v3   i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h2^2 b_v2 b_v4   |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h2^2 b_v3^2    -|0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h2^2 b_v3 b_v4    -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h2^2 b_v4^2    -|0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h2 b_h3^2 b_v1   1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h2 b_h3^2 b_v2   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h2 b_h3^2 b_v3   1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h2 b_h3^2 b_v4   1/2 (-|0001⟩
       +|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h2 b_h3 b_v1^2     -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h2 b_h3 b_v2^2     -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h2 b_h3 b_v3^2    i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h2 b_h3 b_v4^2    i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h2 b_h4^2 b_v1    -1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h2 b_h4^2 b_v2   1/2 (-|0001⟩
       -|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h2 b_h4^2 b_v3    -1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h2 b_h4^2 b_v4   1/2 (|0001⟩
       -|0010⟩ +|0100⟩ -|1000⟩ )    1/512
    
     b_h2 b_h4 b_v1^2    |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h2 b_h4 b_v2^2    |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h2 b_h4 b_v3^2    |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h2 b_h4 b_v4^2    |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h2 b_v1^2 b_v2   1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h2 b_v1^2 b_v3   1/2 i (-|0111⟩
       -|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h2 b_v1^2 b_v4   1/2 (|0111⟩
       -|1011⟩ +|1101⟩ -|1110⟩ )    1/512
    
     b_h2 b_v1 b_v2^2     -1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h2 b_v1 b_v3^2    1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h2 b_v1 b_v4^2     -1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h2 b_v2^2 b_v3   1/2 i (|0111⟩
       +|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h2 b_v2^2 b_v4   1/2 (-|0111⟩
       +|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h2 b_v2 b_v3^2    1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h2 b_v2 b_v4^2    1/2 (-|0111⟩
       -|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h2 b_v3^2 b_v4   1/2 (|0111⟩
       -|1011⟩ +|1101⟩ -|1110⟩ )    1/512
    
     b_h2 b_v3 b_v4^2    1/2 i (|0111⟩
       +|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h3^2 b_h4^2     |0000⟩   1/256
    
     b_h3^2 b_h4 b_v1    -1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h3^2 b_h4 b_v2   1/2 (-|0001⟩
       +|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h3^2 b_h4 b_v3    -1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h3^2 b_h4 b_v4   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h3^2 b_v1^2    -|0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h3^2 b_v1 b_v2    -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h3^2 b_v1 b_v3   |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h3^2 b_v1 b_v4    -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h3^2 b_v2^2    -|0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h3^2 b_v2 b_v3    -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h3^2 b_v2 b_v4   |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h3^2 b_v3^2    |0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h3^2 b_v3 b_v4    -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h3^2 b_v4^2    |0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h3 b_h4^2 b_v1   1/2 (|0001⟩
       -|0010⟩ +|0100⟩ -|1000⟩ )    1/512
    
     b_h3 b_h4^2 b_v2    -1/2 i (|0001⟩
       +|0010⟩ -|0100⟩ -|1000⟩ )    1/512
    
     b_h3 b_h4^2 b_v3   1/2 (|0001⟩
       +|0010⟩ +|0100⟩ +|1000⟩ )    1/512
    
     b_h3 b_h4^2 b_v4   1/2 i (|0001⟩
       -|0010⟩ -|0100⟩ +|1000⟩ )    1/512
    
     b_h3 b_h4 b_v1^2     -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h3 b_h4 b_v2^2    i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h3 b_h4 b_v3^2    i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h3 b_h4 b_v4^2     -i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h3 b_v1^2 b_v2   1/2 i (-|0111⟩
       -|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v1^2 b_v3   1/2 (-|0111⟩
       -|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h3 b_v1^2 b_v4    -1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v1 b_v2^2    1/2 (-|0111⟩
       +|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v1 b_v3^2    1/2 (|0111⟩
       -|1011⟩ +|1101⟩ -|1110⟩ )    1/512
    
     b_h3 b_v1 b_v4^2    1/2 (-|0111⟩
       +|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v2^2 b_v3   1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v2^2 b_v4   1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v2 b_v3^2    1/2 i (-|0111⟩
       -|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v2 b_v4^2    1/2 i (|0111⟩
       +|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h3 b_v3^2 b_v4    -1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h3 b_v3 b_v4^2    1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h4^2 b_v1^2    -|0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h4^2 b_v1 b_v2   i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h4^2 b_v1 b_v3   |0101⟩ -|1010⟩/√(2)   1/256
    
     b_h4^2 b_v1 b_v4    -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h4^2 b_v2^2    -|0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ -|1100⟩/√(6)   3/512
    
     b_h4^2 b_v2 b_v3    -i (|0011⟩
       -|1100⟩ )/√(2)   1/256
    
     b_h4^2 b_v2 b_v4   |1010⟩ -|0101⟩/√(2)   1/256
    
     b_h4^2 b_v3^2    |0011⟩ +|0101⟩
       -|0110⟩ -|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h4^2 b_v3 b_v4   i (|0110⟩
       -|1001⟩ )/√(2)   1/256
    
     b_h4^2 b_v4^2    |0011⟩ +|0101⟩
       +|0110⟩ +|1001⟩ +|1010⟩ +|1100⟩/√(6)   3/512
    
     b_h4 b_v1^2 b_v2   1/2 (|0111⟩
       -|1011⟩ +|1101⟩ -|1110⟩ )    1/512
    
     b_h4 b_v1^2 b_v3    -1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h4 b_v1^2 b_v4   1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h4 b_v1 b_v2^2    1/2 i (-|0111⟩
       -|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h4 b_v1 b_v3^2    1/2 i (|0111⟩
       +|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h4 b_v1 b_v4^2    1/2 i (-|0111⟩
       -|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h4 b_v2^2 b_v3   1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h4 b_v2^2 b_v4   1/2 (-|0111⟩
       -|1011⟩ -|1101⟩ -|1110⟩ )    1/512
    
     b_h4 b_v2 b_v3^2    1/2 (|0111⟩
       -|1011⟩ +|1101⟩ -|1110⟩ )    1/512
    
     b_h4 b_v2 b_v4^2    1/2 (-|0111⟩
       +|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_h4 b_v3^2 b_v4   1/2 (|0111⟩
       +|1011⟩ +|1101⟩ +|1110⟩ )    1/512
    
     b_h4 b_v3 b_v4^2    1/2 i (|0111⟩
       -|1011⟩ -|1101⟩ +|1110⟩ )    1/512
    
     b_v1^2 b_v2^2     |1111⟩   1/256
    
     b_v1^2 b_v3^2     -|1111⟩   1/256
    
     b_v1^2 b_v4^2     |1111⟩   1/256
    
     b_v2^2 b_v3^2     |1111⟩   1/256
    
     b_v2^2 b_v4^2     -|1111⟩   1/256
    
     b_v3^2 b_v4^2     |1111⟩   1/256



If we consider at most three clicks per detector, the possible patterns are:

    Detection Pattern   State   Probability
    
     b_h1^3 b_v1   1/2 (|0001⟩ +|0010⟩
       +|0100⟩ +|1000⟩ )    3/512
    
     b_h1^3 b_v2    -1/2 i (|0001⟩ -|0010⟩
       -|0100⟩ +|1000⟩ )    3/512
    
     b_h1^3 b_v3   1/2 (|0001⟩ -|0010⟩
       +|0100⟩ -|1000⟩ )    3/512
    
     b_h1^3 b_v4   1/2 i (|0001⟩ +|0010⟩
       -|0100⟩ -|1000⟩ )    3/512
    
     b_h1 b_v1^3    1/2 (|0111⟩ +|1011⟩
       +|1101⟩ +|1110⟩ )    3/512
    
     b_h1 b_v2^3    1/2 i (|0111⟩ -|1011⟩
       -|1101⟩ +|1110⟩ )    3/512
    
     b_h1 b_v3^3    1/2 (-|0111⟩ +|1011⟩
       -|1101⟩ +|1110⟩ )    3/512
    
     b_h1 b_v4^3    1/2 i (|0111⟩ +|1011⟩
       -|1101⟩ -|1110⟩ )    3/512
    
     b_h2^3 b_v1   1/2 i (|0001⟩ -|0010⟩
       -|0100⟩ +|1000⟩ )    3/512
    
     b_h2^3 b_v2   1/2 (|0001⟩ +|0010⟩
       +|0100⟩ +|1000⟩ )    3/512
    
     b_h2^3 b_v3   1/2 i (|0001⟩ +|0010⟩
       -|0100⟩ -|1000⟩ )    3/512
    
     b_h2^3 b_v4   1/2 (-|0001⟩ +|0010⟩
       -|0100⟩ +|1000⟩ )    3/512
    
     b_h2 b_v1^3     -1/2 i (|0111⟩ -|1011⟩
       -|1101⟩ +|1110⟩ )    3/512
    
     b_h2 b_v2^3    1/2 (|0111⟩ +|1011⟩
       +|1101⟩ +|1110⟩ )    3/512
    
     b_h2 b_v3^3    1/2 i (|0111⟩ +|1011⟩
       -|1101⟩ -|1110⟩ )    3/512
    
     b_h2 b_v4^3    1/2 (|0111⟩ -|1011⟩
       +|1101⟩ -|1110⟩ )    3/512
    
     b_h3^3 b_v1   1/2 (|0001⟩ -|0010⟩
       +|0100⟩ -|1000⟩ )    3/512
    
     b_h3^3 b_v2    -1/2 i (|0001⟩ +|0010⟩
       -|0100⟩ -|1000⟩ )    3/512
    
     b_h3^3 b_v3   1/2 (|0001⟩ +|0010⟩
       +|0100⟩ +|1000⟩ )    3/512
    
     b_h3^3 b_v4   1/2 i (|0001⟩ -|0010⟩
       -|0100⟩ +|1000⟩ )    3/512
    
     b_h3 b_v1^3    1/2 (-|0111⟩ +|1011⟩
       -|1101⟩ +|1110⟩ )    3/512
    
     b_h3 b_v2^3    1/2 i (-|0111⟩ -|1011⟩
       +|1101⟩ +|1110⟩ )    3/512
    
     b_h3 b_v3^3    1/2 (|0111⟩ +|1011⟩
       +|1101⟩ +|1110⟩ )    3/512
    
     b_h3 b_v4^3     -1/2 i (|0111⟩ -|1011⟩
       -|1101⟩ +|1110⟩ )    3/512
    
     b_h4^3 b_v1    -1/2 i (|0001⟩ +|0010⟩
       -|0100⟩ -|1000⟩ )    3/512
    
     b_h4^3 b_v2   1/2 (-|0001⟩ +|0010⟩
       -|0100⟩ +|1000⟩ )    3/512
    
     b_h4^3 b_v3    -1/2 i (|0001⟩ -|0010⟩
       -|0100⟩ +|1000⟩ )    3/512
    
     b_h4^3 b_v4   1/2 (|0001⟩ +|0010⟩
       +|0100⟩ +|1000⟩ )    3/512
    
     b_h4 b_v1^3    1/2 i (-|0111⟩ -|1011⟩
       +|1101⟩ +|1110⟩ )    3/512
    
     b_h4 b_v2^3    1/2 (|0111⟩ -|1011⟩
       +|1101⟩ -|1110⟩ )    3/512
    
     b_h4 b_v3^3    1/2 i (|0111⟩ -|1011⟩
       -|1101⟩ +|1110⟩ )    3/512
    
     b_h4 b_v4^3    1/2 (|0111⟩ +|1011⟩
       +|1101⟩ +|1110⟩ )    3/512



If we consider four clicks per detector, the possible patterns are:

    Detection Pattern   State   Probability
    
     b_h1^4     |0000⟩   3/512
    
     b_h2^4     |0000⟩   3/512
    
     b_h3^4     |0000⟩   3/512
    
     b_h4^4     |0000⟩   3/512
    
     b_v1^4     |1111⟩   3/512
    
     b_v2^4     |1111⟩   3/512
    
     b_v3^4     |1111⟩   3/512
    
     b_v4^4     |1111⟩   3/512.


Notice that not all detections lead to entangled atomic states. From this table we can see that, if we have non-number-resolved photodetectors, and we condition on the detection of four photons in 4 different detectors, the probability of generating entanglement between four atoms is proportional to (7/32)η^4. If number-resolved detectors are available, and we admit any combination of detections, the probability increases to to (7/8)η^4.

This particular implementation of quarter exhibit Hong-Ou-Mandel suppression of the following detection combinations:


    b_h1^3 b_h2
    
     b_h1^3 b_h3
    
     b_h1^3 b_h4
    
     b_h1^2 b_h2 b_h3
    
     b_h1^2 b_h2 b_h4
    
     b_h1^2 b_h3 b_h4
    
     b_h1 b_h2^3 
    
     b_h1 b_h2^2 b_h3
    
     b_h1 b_h2^2 b_h4
    
     b_h1 b_h2 b_h3^2 
    
     b_h1 b_h2 b_h4^2 
    
     b_h1 b_h2 b_v1 b_v3
    
     b_h1 b_h2 b_v1 b_v4
    
     b_h1 b_h2 b_v2 b_v3
    
     b_h1 b_h2 b_v2 b_v4
    
     b_h1 b_h3^3 
    
     b_h1 b_h3^2 b_h4
    
     b_h1 b_h3 b_h4^2 
    
     b_h1 b_h3 b_v1 b_v2
    
     b_h1 b_h3 b_v1 b_v4
    
     b_h1 b_h3 b_v2 b_v3
    
     b_h1 b_h3 b_v3 b_v4
    
     b_h1 b_h4^3 
    
     b_h1 b_h4 b_v1 b_v2
    
     b_h1 b_h4 b_v1 b_v3
    
     b_h1 b_h4 b_v2 b_v4
    
     b_h1 b_h4 b_v3 b_v4
    
     b_h2^3 b_h3
    
     b_h2^3 b_h4
    
     b_h2^2 b_h3 b_h4
    
     b_h2 b_h3^3 
    
     b_h2 b_h3^2 b_h4
    
     b_h2 b_h3 b_h4^2 
    
     b_h2 b_h3 b_v1 b_v2
    
     b_h2 b_h3 b_v1 b_v3
    
     b_h2 b_h3 b_v2 b_v4
    
     b_h2 b_h3 b_v3 b_v4
    
     b_h2 b_h4^3 
    
     b_h2 b_h4 b_v1 b_v2
    
     b_h2 b_h4 b_v1 b_v4
    
     b_h2 b_h4 b_v2 b_v3
    
     b_h2 b_h4 b_v3 b_v4
    
     b_h3^3 b_h4
    
     b_h3 b_h4^3 
    
     b_h3 b_h4 b_v1 b_v3
    
     b_h3 b_h4 b_v1 b_v4
    
     b_h3 b_h4 b_v2 b_v3
    
     b_h3 b_h4 b_v2 b_v4
    
     b_v1^3 b_v2
    
     b_v1^3 b_v3
    
     b_v1^3 b_v4
    
     b_v1^2 b_v2 b_v3
    
     b_v1^2 b_v2 b_v4
    
     b_v1^2 b_v3 b_v4
    
     b_v1 b_v2^3 
    
     b_v1 b_v2^2 b_v3
    
     b_v1 b_v2^2 b_v4
    
     b_v1 b_v2 b_v3^2 
    
     b_v1 b_v2 b_v4^2 
    
     b_v1 b_v3^3 
    
     b_v1 b_v3^2 b_v4
    
     b_v1 b_v3 b_v4^2 
    
     b_v1 b_v4^3 
    
     b_v2^3 b_v3
    
     b_v2^3 b_v4
    
     b_v2^2 b_v3 b_v4
    
     b_v2 b_v3^3 
    
     b_v2 b_v3^2 b_v4
    
     b_v2 b_v3 b_v4^2 
    
     b_v2 b_v4^3 
    
     b_v3^3 b_v4
    
     b_v3 b_v4^3



§ ENTANGLEMENT SWAPPING USING A TRITTER

The same treatment can be applied for the case of 3-atom entanglement swapping  using polarisation encoded photons. In this case the transformation between input modes a and output modes b of the tritter is given by

    (
    [ b^†_{h,v}1; b^†_{h,v}2; b^†_{h,v}3;            ])= T ·(
    [ a^†_{h,v}1; a^†_{h,v}2; a^†_{h,v}3;            ])

The unitary transformation of the titter (see Fig. <ref>) is given by

    T =  (
    [      1      0      0;      0 1/√(2) i/√(2);      0 i/√(2) 1/√(2);        ]) ·(
    [ √(2/3)      0 i/√(3);      0      1      0; i/√(3)      0 √(2/3);        ])·(
    [ 1/√(2) i/√(2)      0; i/√(2) 1/√(2)      0;      0      0      1;        ])

with the inverse matrix given by

    T^-1=(
    [               1/√(3)       1/6(-√(3)-3 i) -1/6 i
       (√(3)-3 i);              -i/√(3)        1/6(3+i √(3))       1/6(-√(3)-3 i);              -i/√(3)              -i/√(3)               1/√(3);                      ]).


Following the same procedure as before, if we just consider the initial state to be

    ⊗_n=1^3 1/√(2)( |0⟩_A_n|0⟩_P_n + |1⟩_A_n|1⟩_P_n),

and allow for, at most, one click per detector, the possible detection patterns are

    Detection Pattern   State   Probability
    
     b_h1 b_h2 b_h3    i |000⟩   1/24
    
     b_h1 b_h2 b_v1   2 √(3) |001⟩ -(√(3)-3
       i) |010⟩/2 √(6)   1/108
    
     b_h1 b_h2 b_v2   2 √(3) |001⟩ -(√(3)+3
       i) |010⟩/2 √(6)   1/108
    
     b_h1 b_h2 b_v3   i (|001⟩ +|010⟩ )/√(2)   1/108
    
     b_h1 b_h3 b_v1   2 i √(3) |001⟩ +(3-i
       √(3)) |010⟩/2 √(6)   1/108
    
     b_h1 b_h3 b_v2   i (|001⟩ +|010⟩ )/√(2)   1/108
    
     b_h1 b_h3 b_v3   -2 √(3) |001⟩ +(√(3)-3
       i) |010⟩/2 √(6)   1/108
    
     b_h1 b_v1 b_v2   -(√(3)+3 i) |011⟩
       -(√(3)-3 i) |101⟩ +4 √(3) |110⟩/4 √(3)   1/54
    
     b_h1 b_v1 b_v3   (-3-i √(3)) |011⟩
       +(3-i √(3)) |101⟩ +4 i √(3) |110⟩/4 √(3)   1/54
    
     b_h1 b_v2 b_v3   1/2 i (|011⟩ +|101⟩ +2
       |110⟩ )    1/54
    
     b_h2 b_h3 b_v1   i (|001⟩ +|010⟩ )/√(2)   1/108
    
     b_h2 b_h3 b_v2   2 i |001⟩ -(√(3)+i)
       |010⟩/2 √(2)   1/108
    
     b_h2 b_h3 b_v3   -2 |001⟩ +(1+i √(3))
       |010⟩/2 √(2)   1/108
    
     b_h2 b_v1 b_v2   -(√(3)-3 i) |011⟩
       -(√(3)+3 i) |101⟩ +4 √(3) |110⟩/4 √(3)   1/54
    
     b_h2 b_v1 b_v3   1/2 i (|011⟩ +|101⟩ +2
       |110⟩ )    1/54
    
     b_h2 b_v2 b_v3   1/4((√(3)-i)
       |011⟩ -(√(3)+i) |101⟩ +4 i |110⟩)    1/54
    
     b_h3 b_v1 b_v2   1/2 i (|011⟩ +|101⟩ +2
       |110⟩ )    1/54
    
     b_h3 b_v1 b_v3   (√(3)+3 i) |011⟩
       +(√(3)-3 i) |101⟩ -4 √(3) |110⟩/4 √(3)   1/54
    
     b_h3 b_v2 b_v3   1/4((1-i √(3))
       |011⟩ +(1+i √(3)) |101⟩ -4 |110⟩)    1/54
    
     b_v1 b_v2 b_v3    i |111⟩   1/24


If two clicks per detector are allowed, then the possible detection combinations are

    Detection Pattern   State   Probability
    
     b_h1^2 b_v1    -|001⟩ +|010⟩/√(2)   1/54
    
     b_h1^2 b_v2   -2 √(3) |001⟩ +(√(3)-3 i)
       |010⟩/2 √(6)   1/54
    
     b_h1^2 b_v3   i (√(3)+3 i) |010⟩ -2 i √(3)
       |001⟩/2 √(6)   1/54
    
     b_h1 b_v1^2    1/2 (-|011⟩ -|101⟩ -2 |110⟩ )    1/27
    
     b_h1 b_v2^2    (√(3)-3 i) |011⟩ +(√(3)+3
       i) |101⟩ -4 √(3) |110⟩/4 √(3)   1/27
    
     b_h1 b_v3^2    -(√(3)+3 i) |011⟩ -(√(3)-3
       i) |101⟩ +4 √(3) |110⟩/4 √(3)   1/27
    
     b_h2^2 b_v1   -2 √(3) |001⟩ +(√(3)+3 i)
       |010⟩/2 √(6)   1/54
    
     b_h2^2 b_v2    -|001⟩ +|010⟩/√(2)   1/54
    
     b_h2^2 b_v3   (√(3)+i) |010⟩ -2 i |001⟩/2
       √(2)   1/54
    
     b_h2 b_v1^2    (√(3)+3 i) |011⟩ +(√(3)-3
       i) |101⟩ -4 √(3) |110⟩/4 √(3)   1/27
    
     b_h2 b_v2^2    1/2 (-|011⟩ -|101⟩ -2 |110⟩ )    1/27
    
     b_h2 b_v3^2    1/4(i (√(3)+i) |011⟩
       +(-1-i √(3)) |101⟩ +4 |110⟩)    1/27
    
     b_h3^2 b_v1   4 √(3) |001⟩ -2 (√(3)-3 i)
       |010⟩/4 √(6)   1/54
    
     b_h3^2 b_v2   2 |001⟩ +(-1-i √(3)) |010⟩/2
       √(2)   1/54
    
     b_h3^2 b_v3   i (|001⟩ +|010⟩ )/√(2)   1/54
    
     b_h3 b_v1^2    (3+i √(3)) |011⟩ -i (4 √(3)
       |110⟩ -(√(3)+3 i) |101⟩)/4 √(3)   1/27
    
     b_h3 b_v2^2    1/4(-(√(3)-i) |011⟩
       +(√(3)+i) |101⟩ -4 i |110⟩)    1/27
    
     b_h3 b_v3^2    1/2 i (|011⟩ +|101⟩ +2 |110⟩ )    1/27



And finally, if we allow for three clicks in the same detector

    Detection Pattern   State   Probability
    
     b_h1^3     -|000⟩   1/36
    
     b_h2^3     -|000⟩   1/36
    
     b_h3^3     i |000⟩   1/36
    
     b_v1^3     -|111⟩   1/36
    
     b_v2^3     -|111⟩   1/36
    
     b_v3^3     i |111⟩   1/36

The probability of generating entanglement between three atoms when only non-number-resolved detectors are available is proportional to (1/4)η^3. When number-resolved detectors are available, this probability goes up to (3/4)η^3.


The following detection patterns are Hong-Ou-Mandel suppressed

    b_h1^2 b_h2
    
     b_h1^2 b_h3
    
     b_h1 b_h2^2 
    
     b_h1 b_h3^2 
    
     b_h2^2 b_h3
    
     b_h2 b_h3^2 
    
     b_v1^2 b_v2
    
     b_v1^2 b_v3
    
     b_v1 b_v2^2 
    
     b_v1 b_v3^2 
    
     b_v2^2 b_v3
    
     b_v2 b_v3^2


§ N-TO-N SYMMETRIC MULTIPORT


Figure <ref> shows how to construct an n-to-n symmetric multiport for n=2^d. Such a multiport can be used to entangle up to n=2^d nodes, including any subnetwork composed of m = 2, 3, 4, ..., n-1 nodes, using either the entanglement-swapping scheme or the WPE scheme. The number of 50:50 beam splitters that each photon crosses is d. Although the complexity of the interferometer increases with the number of input and output ports, when using the entanglement-swapping approach, a photon never interferes with itself, thereby maintaining the resilience of the scheme to path-length fluctuations. These kinds of interferometers can be built using bulk optics, although alignment and robustness are challenging. Fibre-based networks and photonic integrated circuits (PICs) are therefore more suitable.

When using PICs and photon-polarisation encoding, a technical challenge encountered is the difficulty in building polarisation-preserving waveguides for two orthogonal polarisations, as well as polarisation-agnostic beam splitters. This can be overcome by separating the two incoming photon polarisations into two separate interferometers, one for H-polarisation and another for V-polarisation, as illustrated in Fig. <ref> for the case of a 2-to-2 interferometer. This scheme still works because H and V photons never interfere anyway. Entanglement is heralded by the detection, in coincidence, of one H-photon and one V-photon in any of the outputs. The probability of entanglement generation is the same as using the standard setup (Fig. <ref>a), i.e., ∝1/2η^2. The main difference is that now the phase of the created state is sensitive to the H and V path differences. Depending on which detectors click, we create either

    1/2(|01⟩± e^iϕ|10⟩),

with ϕ = 2π[(β_H - β_V) - (α_H - α_V)]/λ, where α_H,V and β_H,V are the path lengths of the interferometers as illustrated in Fig. <ref>, and λ is the wavelength of the photons. Using temperature-stabilised PICs, achieving the required phase stability is not an issue, making this a viable option. The concept can be extended to any Bell state analyser, for any number of nodes, where the two polarisation modes can be separated into two different interferometers.

 Paper Title: Small correlation is sufficient for optimal noisy quantum metrology.

 


chao.yin@colorado.edu
Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder CO 80309, USA


vva@umd.edu
Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland, USA

sisi.zhou26@gmail.com
Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
Department of Physics and Astronomy and Institute for Quantum Computing, University of Waterloo, Ontario N2L 2Y5, Canada




§ ABSTRACT 








We propose a class of metrological resource states whose quantum Fisher information scales optimally in both system size and noise rate.


In these states, qubits are partitioned into sensing groups with relatively large correlations within a group but small correlations between groups.
The states are obtainable from local Hamiltonian evolution,


and we design a metrologically optimal and efficient measurement protocol utilizing time-reversed dynamics and single-qubit local measurements. 
Using quantum domino dynamics, we also present a protocol free of the time-reversal step that has an estimation error roughly twice the best possible value.





Small correlation is sufficient for optimal noisy quantum metrology
    Sisi Zhou
    August 5, 2024
===================================================================





Introduction.—
Quantum metrology <cit.> is the study of utilizing entanglement and other quantum effects to 

measure unknown data, such as gravitational waves <cit.>, images <cit.>, magnetic fields <cit.>, and time <cit.>. 
A typical metrological protocol involves initializing N probes in a particular (possibly entangled) state, letting the value of an unknown quantity θ be imparted on the state, and estimating θ via a measurement.
The Heisenberg limit (HL) <cit.>

restricts the best-case estimation precision to scale inversely with the number N of entangled probes, 
δθ∼ 1/N.

The prototypical example of a “metrologically optimal” state is the Greenberger-Horne-Zeilinger (GHZ) state <cit.>, which is used to attain the HL in estimating a Z-axis magnetic field in a noiseless N-spin system.
Unfortunately, the useful but fragile quantum correlations of this state quickly break down in the presence of noise.


Instead of attaining the HL, the optimal scaling 


in a generically noisy setting is the standard quantum limit (SQL), δθ∼ 1/√(N). 


For quantum metrology to continue to bear fruit, it is important to quantify the performance of noise-robust metrological protocols not only in terms of the number of probes N <cit.>, but also in terms of the noise rate p of each probe.



A more precise calculation for several models with non-negligible noise yields a precision of δθ≳√(p/N)  <cit.>, meaning that reducing the noise rate p can lead to a constant-factor improvement on top of the naive scaling with 1/√(N).

In this Letter, we continue this direction and describe both criteria on and examples of protocols whose variance scales optimally [We say a family of quantum states ρ as functions of N and p is optimal, if and only if for any N and p, there exists a universal constant c such that the QFI (ρ_θ) ≥ c min{N/p,N^2}.] with both N and p (see fig:summary for a summary).












We show that achieving optimal scaling requires a careful tuning of the quantum correlation length.
The extensive entanglement of the GHZ state is optimal only in the relatively noiseless setting of pN∼ constant <cit.>, while unentangled states do not yield any p-dependent precision improvement.
Interpolating between the two edge cases, we show that a “Goldilocks” amount of quantum correlation — quantified by a correlation length M ∼ 1/p — is required for optimality.

This general theory corroborates previous numerical results <cit.> that matrix product states with low bond dimension are optimal for strong noise. 







We construct states that satisfy our criteria for optimality using local-Hamiltonian evolution, and present a provably efficient measurement protocol utilizing time-reversed evolution (similar to the Loschmidt echo <cit.>) and single-qubit measurements. 
We showcase our results by an example of optimal states generated by analytically solvable quantum domino dynamics <cit.>, 
where local measurements without time reversal suffice for near-optimal performance.

















































Simple example—Our optimality conditions are stated for sensing a Z-axis magnetic field (with straightforward extension to any single-qudit Hamiltonian via local rotations) under arbitrary noise, but dephasing noise yields an illuminating warm-up example of the correlation that is required for optimality.

Let N qubits, labeled by i∈Λ with Pauli matrices X_i,Y_i,Z_i, be prepared in a state ρ, with each qubit i undergoing the channel 
    _θ, p(ρ_i) = (1-p)^-θ Z_iρ_i ^θ Z_i + p Z_i^-θ Z_iρ_i ^θ Z_iZ_i

for 0 < p < 1/2. 
For simplicity, we assume p > 1/N (otherwise the noise is asymptotically negligible and the HL is achievable).
One then measures the final state ρ_θ = _θ, p^⊗ N(ρ)
to estimate θ. 

The estimation precision can be bounded by the state's quantum Fisher information (QFI), δθ≥ 1/√(N_ e(ρ_θ)), via the quantum Cramér-Rao bound <cit.>, saturable as the number of repeated experiments N_ e→∞.
The QFI for the above example is bounded by <cit.> 

    (ρ_θ)≤ 4N/[1-(1-2p)^2] ∼ N/p,

yielding the aforementioned scaling of δθ∼√(p/N) [We use f∼ g,f≳ g and f≲ g to denote c_1 g ≤ f ≤ c_2 g, f ≥ c_3 g and f ≤ c_4 g, respectively, where c_1,2,3,4 are positive constants.]. 

This QFI bound cannot be surpassed with arbitrary state preparation and measurement protocols, including quantum error correction <cit.>. 


This QFI bound can be saturated (up to a constant factor) by partitioning the N-qubit system into N/M groups of M qubits each (with N/ assumed to be an integer for simplicity), tuning the groups' size inversely with the noise rate, and repeating the GHZ protocol within each group.
The corresponding quantum parity state, ρ = ⊗_I|⟩_I ⟨|_I with |⟩_I = 1/√(2)(|0⟩^⊗ + |1⟩^⊗), an instance of the [quantum parity code]quantum_parity <cit.>, is optimal as follows.





In the noiseless limit p=0, each group I gains a phase θ independently that corresponds to a QFI ∼^2, so the QFI of the entire state, equal to the sum of individual QFIs, scales as (N/)^2=N. 
This scaling is robust against noise if p is upper bounded by a constant, because then dephasing noise will only cause the individual QFI to reduce by a constant factor <cit.>. Therefore, (<ref>) is saturated by tuning ∼ 1/p. 






























Sufficient condition for optimality.—
The quantum parity state is not the only state that achieves the optimal QFI scaling ∼ N/p, and such states need not be tensor-product states w.r.t. the partititoning into groups I of size M. 
Any state with sufficiently large intra-group and sufficiently small inter-group correlations will do.
This partitioning idea also extends to general noise channels.

We generalize the final state to be ρ_θ=𝒩_θ, p^⊗ N(ρ), where 𝒩_θ, p is the noisy channel depending on θ, and the noise parameter p quantifies 


the diamond distance between noiseless (𝒩_θ, 0) and noisy (𝒩_θ, p) evolutions <cit.>.






Before stating our general sufficient condition (see <cit.> for the proof), we first introduce some notation.




For any state σ and operators A,B, define the expectation A_σ:=σ A, the correlation function 
    C_σ(A,B):=σ A B - σ Aσ B,

and the variance Δ A^2_σ=C_σ(A,A). 

Let ρ_I be the reduced density matrix (RDM) of group I before sensing, and let ρ_I,θ (ρ^(0)_I,θ) be the noisy (noiseless) RDM after sensing.


We use A_I to indicate that operator A acts inside group I, and denote the operator norm by ·.





For any given p, choose any <N with 
    ≤ c_ no/p,

where c_ no is a positive constant, and partition the qubits as above. Given any state ρ, suppose for any group I, its QFI in the noiseless limit is large, 
    ρ^(0)_I,θ∼^2,

    and its correlations with other groups are small, i.e.,
    
    ∑_J: J∉𝒮_IC_ρ(A_I, A_J)≲^-2max_J A_J^2,

where 𝒮_I is some set of groups containing a finite (independent of N,p,M) number of elements (including I) that may have large correlation with I.
Then
there exists an observable =∑_I _I such that 
    (ρ_θ)≥1/Δ^2_ρ_θ∂_θ_ρ_θ^2 ∼  N.




(<ref>) implies the optimal QFI scaling ((<ref>)) is saturated by choosing ∼ 1/p. Moreover,  here represents the optimal observable to measure, and Δ^2_ρ_θ and _ρ_θ represent its variance and expectation value w.r.t. ρ_θ, respectively (see, e.g., Ref. <cit.> for the signal-to-noise-ratio lower bound on QFI that we use in (<ref>)). 
The optimal observable is a sum of individual observables _I on each group, and individual measurements on each group suffice to achieve the optimal QFI. 
The two conditions generalize the quantum parity state example: (i) (<ref>) requires that each group has the asymptotically optimal QFI ∼ M^2 [In <cit.>, we show that if the global state ρ is pure, the mixed-state QFI for each group is equivalent (up to a constant factor) to certain correlation functions in ρ, which are much easier to work with.], and
(ii) (<ref>) requires each group to have a large correlation with 


a finite number of other groups, which guarantees the linear scaling of the QFI with respect to M.






Locally generated optimal states.— 

Our sufficient condition is naturally satisfied by states generated by local Hamiltonian evolution. 
Suppose the qubits lie on vertices of a d-dimensional square lattice, start from |0⟩:=|0⟩^⊗ N, and evolve to a state |ψ⟩ = U |0⟩ by a unitary U generated by a (possibly time-dependent) local Hamiltonian for time T.
Such evolution is quasi-causal because it satisfies the following Lieb–Robinson bound <cit.>: if an operator A acts within set S⊆Λ, then, in the Heisenberg picture,  
    A(T):= U^† A U ≈A_S,

i.e., A(T) can be approximated (up to small error) 

by an operator A  that is strictly supported in the “light cone” region 
    S:={i∈Λ: 𝖽(i,S)≤ T},
 
which contains all vertices within distance T to S. 
Here, T is rescaled such that the Lieb–Robinson velocity <cit.> 

is set to 1, and 𝖽(·, ·) is the distance function with 𝖽(i,S)=min_j∈ S𝖽(i,j). For illustrative purpose, we assume (<ref>) is exact from now on, with the rigorous treatment given elsewhere <cit.>.

The state ρ=|ψ⟩⟨ψ| =U |0⟩⟨0|U^† has correlation length at most 2T <cit.> because faraway light cones do not intersect. 
Therefore, it naturally satisfies (<ref>) if each group I is chosen to be an L×⋯× L hypercube with L=^1/d≳ T, as sketched in fig:ZI. 
We choose T ∼ L 

so that qubits within each group are sufficiently correlated so as to satisfy (<ref>) <cit.>. 

Based on Theorem <ref>, the class of optimal metrological states contains any state ρ 

defined above with large intra-group ZZ correlation: 
    C_ρ(Z_I, Z_I) ≥ c_Z L^d T^d,

corresponding to 

(<ref>) [3],
where Z_I=∑_i∈ IZ_i and c_Z ∼ 1. 
Such optimal states are efficiently preparable by definition, but it is still demanding to develop efficient measurement schemes because the measurement observable used in Theorem <ref>, although separate for each group, can be difficult to implement when the group size is large. 



















 







Local measurement scheme with time-reversal.— 
For any such state ρ, we propose a local measurement scheme by only assuming the ability to time-reverse the evolution U (e.g., by changing the sign of the generating Hamiltonian). Namely, after sensing, one applies a unitary U_ rev:=U^†^θ_ pr Z (Z:=∑_i Z_i) to the state ρ_θ, and then measures in the computational basis before post-processing the data. 
Here, θ_ pr represents prior knowledge of θ, such that θ̃:=θ-θ_ pr satisfies |θ̃| ∼ L^-d [The requirement |θ̃| ≲ L^-d is a common assumption in metrology because one can always gain the prior knowledge on the “digits” of θ one by one with negligible overhead until the requirement is satisfied <cit.>. We also assume |θ̃| ≳ L^-d, achievable by shifting θ_ pr if necessary, so that the slope (<ref>) is large.].
The idea comes from the Loschmidt echo protocol working in the noiseless case <cit.>, where one processes the measurement data by simply computing the final probability of returning to |0⟩: 
    𝖯_ echo:=⟨0|U^†^-θ̃ Z U|0⟩^2 = 1 - θ̃^2/4ℱ(ρ_θ) + (θ̃^4).
 
The measurement is optimal, i.e., the corresponding classical Fisher information 1/Δ𝖯_ echo^2∂_θ𝖯_ echo^2 ≈ℱ(ρ_θ) at small θ̃, where Δ𝖯_ echo^2=𝖯_ echo(1-𝖯_ echo) is the variance of a Bernoulli random variable with probability 𝖯_ echo. 

However, the global projective measurement |0⟩⟨0| in (<ref>) is not robust against noise. 
To guarantee robustness, we post-process the data in the following way. After the computational-basis measurement, one instead computes projections 

P_I := ⊗_i∈I|0⟩_i⟨0| onto small regions I 
(defined via (<ref>)), so that the overall observable is _ rev_ρ_θ with _ rev := ∑_I P_I := ∑_I U_ rev^† P_I U_ rev. (We often omit the identity operator on the rest of the system for simplicity, e.g., here P_I means P_I⊗𝕀.) This time-reversal local measurement scheme is asymptotically optimal. 





    For any p, choose L=M^1/d such that (<ref>) holds for some constant c_ no (which is not necessarily the same constant in Theorem <ref>). For any locally generated state ρ satisfying (<ref>), there exist constants C>c>1 determined by c_Z and d, such that
    for any C≥ L/T≥ c, 
    1/Δ^2_ rev_ρ_θ∂_θ_ rev_ρ_θ^2 ∼  N L^d.



Note that the observable _ rev here is independent of p, in contrast to  in Theorem <ref> that is a function of p. We sketch the idea of Theorem <ref> and leave the full proof to <cit.>. 
Note that all approximations “≈” below stem from the light-cone approximation in (<ref>).

Focusing on d=2 (generalization is straightforward) as shown in fig:ZI and recalling that |θ̃|∼ L^-2 and T∼ L, we first expand


in the noiseless case,

    ⟨P_I⟩_ρ_θ^(0)   = ⟨0|^θ̃ Z(T) P_I^-θ̃ Z(T)|0⟩
       = 1-θ̃^2/2⟨0|[Z(T), [Z(T), P_I] ]|0⟩ + ⋯.

The first-order term vanishes similarly as (<ref>), and we show that the second-order term is ∼θ̃^2 L^4.



(Note that we have required that |θ̃| is sufficiently large to guarantee that the second-order derivative is dominant.)



We expand Z=∑_J Z_J and observe that only J=I or neighbors of I contribute, because otherwise [Z_J(T), P_I]=0 since they do not overlap in space from (<ref>). Furthermore, when L/T is a sufficiently large constant, 

the coefficient is dominated by the term

    ⟨0|[Z_I(T), [Z_I(T), P_I] ]|0⟩
    ≈ 2  C_ρ(Z_I, Z_I)≥ 2c_Z L^2T^2,

which can be verified from direct computation and (<ref>). The reason is that the sum of all other terms is ≲ LT^3, which becomes subdominant when L/T is sufficiently large. To show this, we note that for i∈ I,j∈ J≠ I, 
    ⟨0|[Z_i(T), [Z_j(T), P_I] ]|0⟩≈ 0,

unless the two sites are close (d(i,j)≤ 2T), as shown in fig:ZI. 
Then the sum of (<ref>) over i,j is bounded by ∑_j  close to  I∑_i  close to  j≲ LT · T^2.

Above we showed, at sufficiently large L/T, the second-order coefficient in (<ref>) is ∼ L^4. This guarantees -∂_θ⟨P_I⟩_ρ_θ^(0)∼θ̃×[the second-order coefficient]∼ L^2, and thus 
    ∂_θ_ rev_ρ_θ^(0)∼ N.

(Note that the higher orders of θ̃ in (<ref>) are safely ignored by similar locality analysis.) Since faraway groups are hardly correlated, 

    Δ^2_ rev_ρ_θ^(0)∼∑_I ⟨ΔP_I^2⟩_ρ_θ^(0)≲ N/L^2,

leading to (<ref>) for the noiseless case.


Finally, we consider the noisy case p>0. (<ref>) still holds (where ρ_θ^(0) is replaced with ρ_θ) because adding noise does not create long-range correlation <cit.>. 
It then remains to show (<ref>) is also robust. For illustrative purposes, we focus on dephasing noise here, i.e., 𝒩_θ,p=_θ,p, yielding

    ∂_θ( ⟨P_I⟩_ρ_θ- ⟨P_I⟩_ρ_θ^(0)) = ⟨[ Z,_0, p^⊗ N(P_I)-P_I ]  ⟩_ρ_θ^(0)
       ≲ L^2   _0, p^⊗ c_0L^2(P_I)-P_I≲ L^4 p,


where we recall that the dual map of the dephasing noise _0, p 


is itself, and use the locality of P_I to restrict the channel to a region of size c_0 L^2 with c_0 ∼ 1. 
Taking p L^2 ≤ c_ no and c_ no 

sufficiently small guarantees the contribution from noise to (<ref>) is subdominant relative to the noiseless result.





Quantum domino dynamics with local measurement.—
We have shown that any state generated by local dynamics with correlations of the form (<ref>) is metrologically optimal when L^d∼ 1/p. Here we present a simple application of this framework using quantum domino dynamics <cit.>. 

Consider the 3-local spin-chain Hamiltonian that flips a spin only if its two neighbors are anti-aligned, 
    H_ do = 1/2∑_i=2^N-1 X_i(1-Z_i-1Z_i+1),

which can be engineered as the effective Hamiltonian for the transerve-field Ising model at weak X field <cit.>.

The vacuum |0⟩ is an eigenstate of H_ do, and, more generally, the number of domain walls (DWs)—transitions from |0⟩ to |1⟩ or vice versa—is conserved. If the system starts from the initial state |1_1⟩:=|10⋯ 0⟩, it only traverses the state space with one DW ^- T H_ do|1_1⟩ = ∑_i=1^N a_i |1_i⟩, where |1_i⟩ is the state with 1 on the first i qubits and 0 on the rest, and where the coefficients a_i are determined by T. This single-particle hopping process of the DW justifies the name “domino”. Before T ≳ N (i.e., when the DW propagates to the right), a_i is centered around i=vT with a velocity v. Then, an initial state |+⟩_1⊗ |0⟩ would evolve to 
    |ψ_ do⟩:=^- TH_ do|+⟩_1⊗ |0⟩ = 1/√(2)|0⟩+ ∑_i=1^N a_i |1_i⟩,
 
which is roughly a GHZ-like state on the left ∼ vT qubits (with other qubits in state |0⟩), useful for metrology <cit.>.







Adapting the domino dynamics to our partition framework, we propose to evolve the initial state containing one |+⟩ in every L qubits: 
    |Ψ_ do⟩ := ^- TH_ do|+⟩_L⊗|+⟩_2L⊗⋯⊗ |0⟩_ rest.

This final state, which we call the partitioned domino state, satisfies (<ref>) for T≲ L, because the two DWs of each |+⟩ roughly propagate in opposite directions with constant velocity. We numerically study noisy metrology using |Ψ_ do⟩ on a chain of 200 qubits, and fig:domino shows the results. Remarkably, this protocol is not only optimal asymptotically, but also quantitatively. By adjusting L for different p, the precision (δθ)^2 is only roughly twice larger than the ultimate limit (<ref>).





Besides having nearly the same metrological power as the quantum parity state,


the domino-state protocol requires only local measurements without any time-reversed evolution.
To show this, we focus first on the simplest one-DW case with state (<ref>). In <cit.>, we prove the following for various combinations 𝒩_θ, p of θ-imparting unitary evolution and a general noisy channel.




In a rotated local basis (i.e., redefining states |1⟩_i with extra phases) such that Arg(a_i) =-π/2(1+i) -2θ_ pr i, define the measurement observable 
    _ do := ∑_vT/2≤ i≤ 2vT |a_i| _i,
 
where _i := ⊗_j=1^i Y_j.
If 
    ∑_vT/2≤ i≤ 2vT |a_i|^2 ≥ c_ do∼ 1,

then 
    1/Δ^2_ do_ρ_θ∂_θ_ do_ρ_θ^2 ∼ T^2,

where ρ_θ=𝒩_θ, p^⊗ N(|ψ_ do⟩⟨ψ_ do|) with noise p≤ c_ do/(vT) for a constant c_ do.




(<ref>) can be realized by Y-basis 
local measurements with efficient classical post-processing.
The condition (<ref>) means that |a_i|^2, as a probability distribution, is concentrated around vT, which can be verified from the exact solution (Bessel function) of a_i <cit.>.
The idea is as follows. |ψ_ do⟩ is roughly a superposition of GHZ states |⟩_1⋯ i⊗|0⟩ of different size i. For each GHZ state, the canonical observable is _i. The problem is that each such observable does not have large sensitivity, i.e., ∂_θ_i∼ |a_i|≪ 1, because the DW wave packet spreads. To overcome this, we use a suitable linear combination of them, namely, _ do, to accumulate the small derivatives into ∂_θ_ do∼∑_i |a_i|^2 ∼ 1. It turns out this linear combination does not lead to large variance of _ do due to the lack of interference among the GHZ states in |ψ_ do⟩, making _ do a suitable observable. 



Generalizing the idea, the observable we use in fig:domino for state (<ref>) is of the form 
    (∑_i,jb_i,j_i,j) ⊗(∑_i,jb_i,j_L+i,L+j)⊗⋯,

where the real coefficients b_i,j come from simulating the two-DW subspace, which only requires polynomial (in L) resource of classical computation. See <cit.> for details.


Spin-squeezing.—
We show in <cit.> that certain types of spin-squeezed states (SSS) <cit.>, though demanding to prepare <cit.>, are another example of metrological optimality. 
Although spin squeezing is beyond the regime of Theorem <ref>, certain boundedness of correlation functions is also sufficient,

suggesting the general usefulness of correlation bounds for noisy metrology. 


Discussion.—
In this work, we prove sufficient conditions for optimal noisy metrology and illustrate them with several examples, from simple quantum parity states to quantum domino dynamics and spin-squeezed states.





Beyond state optimality, we also address the problem of efficient measurement schemes for metrology.
In particular, the domino example suggests that local measurements apply to wider sensing scenarios beyond GHZ: Although a local measurement scheme that exactly saturates the QFI <cit.> may not exist, the QFI can still be saturated up to a moderate prefactor. 


The example also raises an interesting connection between quantum metrology and many-body dynamics, hinting at the possibility of other natural (e.g. time-independent) Hamiltonian dynamics generating high-QFI states in higher spatial dimensions.






Acknowledgements.— 
C.Y. was supported by the Department of Energy
under Quantum Pathfinder Grant DE-SC0024324. S.Z. acknowledges funding provided by Perimeter Institute for Theoretical Physics, a research institute supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.
VVA thanks Ryhor Kandratsenia and Olga Albert for providing daycare support throughout this work.








in 1,...,
    
        [pages=,]
    







 Paper Title: Quantum thermalization and Floquet engineering in a spin ensemble with a clock transition.


  
 


	
	Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
	Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
	
	Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
	Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
 	
	Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
	Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA


	Department of Physics, Virginia Technology, Blacksburg, VA 24061, USA

	Department of Physics, Virginia Technology, Blacksburg, VA 24061, USA

        []joonhee.choi@stanford.edu
	Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA

 	[]faraon@caltech.edu
	Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
	Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

§ ABSTRACT 
Studying and controlling quantum many-body interactions is fundamentally important for quantum science and related emerging technologies. Optically addressable solid-state spins offer a promising platform for exploring various quantum many-body phenomena due to their scalability to a large Hilbert space. However, it is often challenging to probe many-body dynamics in solid-state spin systems due to large on-site disorder and undesired coupling to the environment. Here, we investigate an optically addressable solid-state spin system comprising a strongly interacting ensemble of millions of ytterbium-171 ions in a crystal. Notably, this platform features a clock transition that gives rise to pure long-range spin-exchange interactions, termed the dipolar XY model. Leveraging this unique feature, we investigate quantum thermalization by varying the relative ratio of interaction strength to disorder, dynamically engineering the XY model into other many-body Hamiltonian models, and realizing a time-crystalline phase of matter through periodic driving. Our findings indicate that an ensemble of rare-earth ions serves as a versatile testbed for many-body physics and offers valuable insights for advancing quantum technologies.


Quantum thermalization and Floquet engineering in a spin ensemble with a clock transition
    Andrei Faraon
    
=========================================================================================



Many-body quantum phenomena arise when multiple particles interact, playing a crucial role in contemporary physics and engineering <cit.>. Among the systems exhibiting many-body physics, optically addressable solid-state spins have become a particularly rich platform to study due to their scalability to very large numbers of spins <cit.> and their potential applications in quantum simulations <cit.>, sensing <cit.>, and information processing <cit.>.

To achieve versatility in quantum applications, it is essential to manipulate the time evolution of a many-body system in a programmable manner <cit.>. However, achieving local individual control of spins in solid-state systems is often challenging due to their nanometric proximity. To address this issue, a global control sequence–composed of either pulsed or continuous driving–can be applied periodically to the entire system, effectively manipulating many-body dynamics<cit.>. This periodic driving method, known as Floquet Hamiltonian engineering, enables the efficient realization of various many-body Hamiltonians with different interaction types and strengths, transformed from the original system Hamiltonian <cit.>. Notably, periodically driven quantum systems can also exhibit a variety of exotic non-equilibrium phases of matter, such as Discrete Time Crystals (DTCs)<cit.>, stabilized by many-body interactions.

In this context, a strongly interacting ensemble of spins with robust global control protocols can be employed to explore many-body physics and out-of-equilibrium dynamics. Rare-earth ions (REIs) doped in solids emerge as promising candidates for a many-body testbed due to their highly coherent optical and spin transitions at cryogenic temperatures, scalability, and ease of integration into photonic devices <cit.>. Moreover, the diversity in REI species, hosts, and concentrations, combined with naturally arising dipole-dipole interactions <cit.> or engineered cavity-mediated interactions <cit.>, introduces novel experimental control parameters and offers theoretical insights for exploring large-scale quantum many-body systems and their dynamics. However, despite numerous spectroscopic studies on large ensembles of REIs that have focused on identifying materials with high coherence times <cit.> for quantum technologies such as quantum transducers <cit.> and memories <cit.>, less effort has been devoted to the microscopic understanding, control, and engineering of these systems.



In this work, we report on the characterization and control of quantum many-body dynamics in a dense ensemble of approximately 10^6 ytterbium-171 ions in a nanophotonic cavity with a yttrium orthovanadate host crystal (^171Yb^3+:YVO_4). These high-density REIs are randomly positioned with an average distance of ≈9 nm (or equivalently, ≈86 ppm in concentration) in an effective three-dimensional volume defined by the cavity mode (Fig. <ref>a). Each Yb ion provides both microwave and optical transitions for spin state manipulation and state-selective readout, respectively. The nanophotonic optical cavity enables high-fidelity initialization and fast readout of Yb ions, while coherent microwave control is achieved via a coplanar waveguide. Specifically, the ground and optically excited states of Yb comprise {|0⟩, |1⟩, |Aux⟩} and {|0⟩_e, |1⟩_e, |Aux⟩_e}, respectively, resulting from the hybridization of its electron and nuclear spins (Fig. <ref>b) <cit.>. In this study, the spin transitions are defined within the ground state manifold using {|0⟩, |1⟩}, serving as an effective “qubit” with a microwave transition frequency of 675 MHz. For spin state readout, we utilize the cavity-enhanced, resonant optical transition between |1⟩ and |0⟩_e at a wavelength of 984 nm, enabling the optical detection of state-selective photoluminescence signals (Supplementary Information).

In essence, the goal of studying many-body physics in an experiment is to observe coherent quantum phenomena over extended durations governed by the target unitary dynamics of the system, while minimizing undesired incoherent coupling to external environments. In this regard, our REI platform offers distinct advantages over other solid-state spin systems for exploring the dynamics of a closed many-body quantum system, as outlined below.

First, the chosen spin states, {|0⟩, |1⟩}, are first-order insensitive to external electromagnetic fluctuations at zero magnetic fields, known as the “clock” transition <cit.>. Consequently, the spins are less susceptible to both decoherence and inhomogeneity induced by the external environment. Second, the electronic spin g factor, which determines the strength of the dipole moment, is approximately three times higher than that of a single electron <cit.>. This results in stronger dipole-dipole interactions between REIs (Fig. <ref>c). Third, we can control the overall interaction strength of the system, J, defined using the average nearest-neighbor distance (Supplementary Information), by adjusting the effective density of REIs within the qubit manifold {|0⟩, |1⟩} (Fig. <ref>d). This manipulation is achieved by varying the population distribution between the qubit manifold {|0⟩, |1⟩} and the nonparticipating auxiliary ground states |Aux⟩ through optical initialization (Supplementary Information). Lastly, unlike conventional dipole-dipole interactions, which include both spin-exchange and energy-shifting Ising interactions, our spin system realizes a pure spin-exchange interaction without the Ising component (Fig. <ref>e). This Hamiltonian is known as the dipolar XY model <cit.>, which has recently been experimentally investigated in various platforms to explore fundamental many-body phenomena and their applications, such as continuous symmetry breaking <cit.> and spin squeezing <cit.>. The naturally occurring XY model in three-dimensional dipolar systems offers a novel and complementary configuration for studying similar physics in the solid-state setting. 

The combination of strong spin-spin interactions, weak coupling to the external bath, cavity-enhanced fast readout, and effective spin density control positions our system as a promising platform for studying many-body spin dynamics. Notably, our spin system features the highest ratio of interaction strength to transition inhomogeneity among solid-state electronic spin systems, underscoring its significance in probing many-body dynamics where collective interaction effects are minimally hindered by on-site disorder (Fig. <ref>f). 




§ CHARACTERIZATION OF A STRONGLY INTERACTING SPIN SYSTEM

The very first step before we utilize our REI spin system as a many-body testbed is to quantitatively characterize the interaction and disorder strengths in the system. To this end, we describe the dynamics of our interacting spin system using the following Hamiltonian (in a rotating frame), Ĥ, defined within the qubit manifold {|0⟩,|1⟩}:

    Ĥ = Ĥ_dis+Ĥ_int

where Ĥ_dis = ∑_i^NΔ_i Ŝ_z^i is the on-site disorder Hamiltonian with spin detuning Δ_i for ion i, and Ĥ_int = ∑_ij,   i>j^N J_ij(Ŝ_x^i Ŝ_x^j+ Ŝ_y^i Ŝ_y^j) = ∑_ij,   i>j^N J_ij/2(Ŝ_+^i Ŝ_-^j+ Ŝ_-^i Ŝ_+^j) is the long-range, dipolar spin-exchange Hamiltonian with position- and orientation-dependent pairwise interaction strength J_ij between two ions i and j (see Supplementary Information for details). Here, Ŝ_μ^i is the spin-1/2 operator of ion i along the μ-axis (μ=x,y,z), and Ŝ_±^i = Ŝ_x^i ± iŜ_y^i are the creation and annihilation operators for spin excitation of ion i. We define the strength of on-site disorder, W, as the full width at half maximum of the probability distribution function of Δ_i.

In strongly interacting spin systems, the decoherence of a spin ensemble is influenced not only by the random, inhomogeneous on-site fields but also by interactions with the rest of the spin system, which act as an intrinsic bath <cit.>. To isolate the decoherence effect arising from spin-spin interactions, we employ the celebrated spin-echo sequence <cit.>, which effectively cancels out the on-site disorder in the Hamiltonian at short times (Fig. <ref>a). Here, we observe rapid decay of ensemble coherence within ≈1 μ s, which is much shorter than that of an isolated single spin independently measured from a reference sample. We confirm that the decay time is highly dependent on the effective spin density, where the decoherence rate increases as J increases (orange/blue markers, Fig. <ref>a; see Supplementary Information for additional experimental data). This implies that intrinsic spin-spin interactions dominate the decoherence mechanisms.

To extract the effective spin density and the corresponding average interaction strength, J, we conduct numerical simulations based on a closed many-body system (Supplementary Information). We find that the simulations show good agreement with the experiment, revealing spin concentrations of ≈46 ppm and ≈25 ppm for the cases with large J ≈ 2π× 0.35 MHz and small J ≈ 2π× 0.19 MHz, respectively (orange/blue lines, Fig. <ref>a). These spin densities within the qubit manifold are reasonable given the total spin densities of all ground states of ≈86 ppm, as independently measured by mass spectrometry <cit.>.

Having characterized the interaction strengths, we now focus on identifying the on-site disorder strength, W, using the disorder-sensitive Ramsey sequence (Fig. <ref>b). Experimental data show that the Ramsey signal decays faster than the spin-echo signal in both the small and large J regimes, due to the additional contribution of disorder-induced decoherence. By comparing the experimental data to the corresponding Ramsey sequence simulation, we estimate an on-site disorder strength of W ≈ 2π× 0.65 MHz, which is independent of interaction strengths (orange/blue lines, Fig. <ref>b).

To further corroborate the significance of spin-spin interactions in our system, we employ the so-called ϵ-CPMG sequence <cit.>, a variant of the conventional CPMG sequence (Fig. <ref>c). Specifically, the ϵ-CPMG sequence purposely uses an imperfect echo pulse with a rotation angle of π + ϵ with nonzero ϵ. For either an isolated spin or a non-interacting spin ensemble affected only by disorder, such imperfect spin rotation with a sizable ϵ results in non-ideal dynamical decoupling, leading to rapid decoherence compared to ϵ = 0 (black markers, Fig. <ref>c). In contrast, we observe drastically different behavior in our high-density spin system, where nonzero ϵ pulses better preserve ensemble coherence (orange/blue markers, Fig. <ref>c).

This feature originates from the different sensitivities of the ϵ-CPMG sequence to on-site disorder and many-body interactions (Supplementary Information). Specifically, when ϵ = 0, spin-spin interactions still lead to rapid decoherence of the spin ensemble because the π pulses do not alter their spin-exchange interaction Hamiltonian,  Ĥ_int (whereas the effects from the on-site disorder Hamiltonian, Ĥ_dis, can be decoupled). However, when ϵ = ±π/2, effectively corresponding to a π/2 pulse, Ĥ_int transforms into a Hamiltonian that is the sum of the Ising interaction along the y-axis and the Heisenberg interaction (Supplementary Information). The initially prepared spins along the y-axis then become an eigenstate under this Hamiltonian. Consequently, using π/2 pulses instead of π pulses in the ϵ-CPMG sequence significantly extends the coherence of the interacting spin ensemble (Supplementary Information). Using the Hamiltonian parameters extracted from the Ramsey and echo sequences, we reproduce the coherence dependence on ϵ that matches experimental data (orange/blue lines, Fig. <ref>c). 



§ MICROSCOPIC UNDERSTANDING OF DECOHERENCE MECHANISMS
 



When probing the spin-echo dynamics over longer timescales, we observe that the decoherence profile starts deviating from a simple exponential decay, displaying much slower relaxation at later times (orange markers, Fig. <ref>b). Remarkably, a numerical simulation with calibrated disorder and interaction strengths (Model III) shows excellent agreement with the experimental data even at longer timescales (orange line, Fig. <ref>b). To comprehend these late-time observations, we further consider two simpler theoretical models, I and II, which consider two-body and many-body systems with no disorder, respectively (Fig. <ref>a; see Supplementary Information for details). We find that both fail to capture the late-time slowdown in decoherence (purple/red lines, Fig. <ref>b).



Notably, the cross-over from fast to slow decoherence occurs after a time approximately given by the inverse disorder strength, τ∼ 1/W (Fig. <ref>b). We attribute this to the breakdown of perfect decoupling of on-site disorder when the free evolution time becomes comparable to the inverse disorder strength, implying the critical role of on-site disorder in late-time many-body dynamics. Specifically, high-order disorder effects, arising from the non-zero commutator between the disorder and interaction Hamiltonians ([Ĥ_dis,Ĥ_int] ≠ 0), become pronounced at late times, imposing energetic penalties on spin-exchange processes and thereby slowing down the decoherence rate (Supplementary Information).

Our microscopic analysis allows us to dissect the underlying decoherence mechanisms of the interacting spin system influenced by both interactions and disorder. Moreover, the analysis strongly supports an exceptionally high degree of coherence in a many-body regime, as evidenced by its excellent agreement with simulations based on a closed many-body system.




§ CONTROL OF SYSTEM EVOLUTION VIA DYNAMIC HAMILTONIAN ENGINEERING
 
Having established that the decoherence dynamics of an individual spin are governed by interactions within the system, we use the decoherence profile as a proxy to explore dynamically engineered many-body Hamiltonians with control pulse sequences (Fig. <ref>a-c). Intuitively, each spin can be viewed as a quantum sensor sensitive to interactions with surrounding spins, experiencing interaction-induced dephasing, which is often termed quantum thermalization <cit.>. 

Specifically, we prepare an initial state in a globally polarized state along the y-axis, let the spin system evolve under a control pulse sequence over an interrogation time T, and apply a π/2 pulse with a variable rotation axis θ at the end of the sequence to measure the mean coherence of all individual spins. Through this experiment, we can probe the decoherence dynamics of a spin ensemble as a function of T, defined as the average spin polarization along the y-axis, P(T) = 1/N⟨Ψ(T)|Ŝ^tot_y|Ψ(T)⟩. Here, Ŝ^tot_y=∑_i^NŜ_y^i is  the total spin operator along the y-axis with N being the total number of spins, and  |Ψ(T)⟩ = e^-i Ĥ_eff T|Ψ(0)⟩ is the quantum state at time T obtained through time evolution under the dynamically-engineered effective Hamiltonian Ĥ_eff.

Crucially, with dynamic Hamiltonian engineering of the original dipolar XY Hamiltonian, we can realize a wide class of different effective Hamiltonians, Ĥ_eff, parameterized as

    Ĥ_eff   = w_HeisĤ_Heis + ∑_μ=x,y,z(w_on-site^μĤ_on-site^μ + w_Ising^μĤ_Ising^μ).

Here, w_Heis, w_on-site, and w_Ising are relative weights between the different Hamiltonians: the Heisenberg Hamiltonian Ĥ_Heis=∑_ij,   i>j^N J_ijS⃗^i·S⃗^j, the on-site Hamiltonian Ĥ_on-site^μ = ∑_i^N h_μ^i Ŝ_μ^i, and the Ising Hamiltonian Ĥ_Ising^μ = ∑_ij,   i>j^N J_ijŜ_μ^i Ŝ_μ^j, where S⃗^i = (Ŝ_x^i, Ŝ_y^i, Ŝ_z^i) is a vectorized spin operator, and h⃗^i = (h_x^i, h_y^i, h_z^i) is an effective on-site field for ion i. Note that the Heisenberg Hamiltonian Ĥ_Heis is isotropic, whereas the other two Hamiltonians exhibit directionality along the μ-axis. 

As depicted in Figs. <ref>b and <ref>c, we implement and compare four different effective Hamiltonians using Ramsey, spin-echo, Waugh-Huber-Haberlen (WAHUHA)-echo <cit.>, and spin-locking sequences. The on-site Hamiltonian is determined by the inhomogeneous disordered field along the z-axis (h⃗^i = (0, 0, Δ_i)) for the Ramsey and spin-echo sequences, and the homogeneous control field along the y-axis with strength Ω_y (h⃗^i = (0, Ω_y, 0)) for the spin-locking sequence. Meanwhile, the WAHUHA-echo sequence is designed to realize the pure Heisenberg Hamiltonian (to leading order), thereby protecting initially polarized spins from interaction-induced decoherence. This protection arises because a globally polarized spin state is an eigenstate of the Heisenberg Hamiltonian <cit.>. Additionally, the spin-locking sequence prevents spin dephasing by employing a strong pinning field along the y-axis with strength Ω_y ≫ J after the initial state preparation, effectively arresting spin dynamics.

In line with these theoretical expectations, both the WAHUHA-echo and spin-locking sequences demonstrate significantly prolonged coherence times compared to the Ramsey and spin-echo sequences, confirming the effective engineering of the underlying many-body Hamiltonian (Fig. <ref>d). However, we observe that the coherence of the WAHUHA-echo and spin-locking sequences still decays over time, with 1/e time constants of ≈20 μ s and ≈73 μ s, respectively. We attribute these decays to imperfections in spin ensemble control caused by finite pulse duration and rotation angle errors, as well as fast time-dependent fluctuations of on-site fields originating from the external spin bath (Supplementary Information). These issues could be addressed by implementing a more robust control sequence and improving control pulse fidelity <cit.>.



§ SIGNATURES OF DISCRETE TIME-CRYSTALLINE PHASE
 

Studying non-equilibrium dynamics under periodic driving unveils the microscopic mechanisms of driven many-body phenomena, such as DTCs <cit.>, revealing an interplay among disorder, dimensionality, interactions, and robustness to both intrinsic and extrinsic perturbations. The canonical model for observing DTCs relies on many-body localization (MBL) and disordered Ising interactions, which lead to the emergence of symmetry-broken spin-glass ordering <cit.>. Since our system Hamiltonian consists of pure spin-exchange interactions, we employ a spin-locking sequence to effectively realize an Ising interaction along the y-axis (Fig. <ref>a). In particular, the spin-locking sequence enables long-lived many-body coherence, providing an ideal setting to study the emergent dynamics at late times under periodic driving (Supplementary Information). While we do not expect our three-dimensional spin system to exhibit MBL, we investigate the possibility of inducing time-crystalline behavior in a disordered system, akin to the critical DTCs where the relaxation of spin ordering occurs exponentially slowly <cit.>. 


Specifically, we apply a Floquet pulse sequence for probing DTCs as follows: we first prepare the globally polarized spin state along the y-axis, immediately drive the polarized spins along the same direction for a variable spin-locking duration τ, rotate every spin by an angle π + ϵ around the x-axis, and repeat this “spin-lock-and-rotate” operation k times (Fig. <ref>a). Subsequently, using the last π/2 pulse with two opposite phases of x and -x, we measure the normalized spin polarization, P(k), at stroboscopic times t=kτ, where k is an integer. The repeated base sequence imposes a discrete time-translation symmetry with a period of τ, during which spins are allowed to interact (ignoring the finite spin-rotation duration), providing the desired setting to test whether our many-body system can break this discrete time-translation symmetry.

Experimentally, we find that when ϵ≈ 0, corresponding to near-perfect π rotations, with a short interaction period τ≈ 0, the spin polarization, P(k), oscillates up and down over time, exhibiting a 2τ-periodic oscillation (Fig. <ref>b). The decay envelope of the oscillation is caused by imperfections in the π pulse, as well as the dephasing of the spin-locking signal itself. The discrete Fourier transform (DFT) of the oscillating P(k) reveals a subharmonic peak at a frequency of ν = 1/2 in units of the base period τ. However, this subharmonic peak arises trivially from the finetuned condition of ϵ = 0; as soon as we introduce a non-zero systematic rotation angle error, ϵ≠ 0, as a perturbation to the system, the subharmonic response is disrupted, and instead, the system exhibits a beat note where the corresponding DFT spectrum shows ϵ-dependent frequency splitting (Fig. <ref>c). Crucially, however, the subharmonic oscillation of the polarization can be restored by allowing for a longer interaction time of τ≈ 0.145 ×2π/J≈ 425 ns at a nonzero ϵ, which is indicative of a DTC phase exhibiting robustness against perturbations (Fig. <ref>d). 

We proceed to investigate the stability of this subharmonic behavior more systematically for various values of interaction time τ and perturbation strength ϵ by constructing a phase diagram using the subharmonic peak intensity at ν=0.5 (Fig. <ref>e). The resulting DTC phase diagram shows a characteristic linear phase boundary (red dashed lines, Fig. <ref>e), consistent with observations from other many-body platforms <cit.>. We further substantiate the robustness of the observed DTC phases by confirming the persistent subharmonic oscillations when varying the initial spin states through global rotations away from the y-axis (Supplementary Information).



§ CONCLUSION AND OUTLOOK
 
Our experimental demonstrations showcase that a REI system provides a versatile and flexible testbed for many-body physics. We envision that REI platforms hold greater potential to serve as large-scale analog quantum simulators in the solid state, offering unique features compared to other solid-state spin systems. First, REI platforms offer flexible engineering options, including a variety of fabrication-friendly choices for ion species, host crystals, and doping concentrations spanning a wide dynamic range from ppb to a few percent. Second, co-doping different REI species allows for the simultaneous engineering of two distinct groups of many-body systems, facilitating the study of heterogeneous spin-spin interactions <cit.>, analogous to the dual-species experiments using neutral atom arrays  <cit.>. Third, proximal nuclear spins within the crystal can serve as an additional quantum register. The nuclear spin-spin interactions can be mediated through engineered hyperfine interactions with a Yb ion, offering additional functionality such as a quantum memory <cit.>. Lastly, all of these control knobs, combined with on-chip integration using nanotechnology <cit.>, enable a scalable REI system, opening up a range of practical applications from quantum simulation to networking and sensing.


§ ACKNOWLEDGMENTS

We thank S. L. N. Hermans and T. Xie for reading the paper and providing useful feedback, and A. Ruskuc and A. Beckert for useful discussions. Funding: This work was supported by U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (contract number DE-SC0012704), Institute of Quantum Information and Matter, an NSF Physics Frontiers Center (PHY-1733907) with support from the Moore foundation EPI program, NSF UCLA, NSF QUIC-TAQS (Award Number: 2137984). The initial device and experimental setup development was supported by the Office of Naval Research award no. N00014-19-1-2182 and N00014-22-1-2422. The device nanofabrication was performed in the Kavli Nanoscience Institute at the California Institute of Technology. R.F. acknowledges the support from the Quad fellowship. E.B. acknowledges support from the National Science Foundation (grant no. 1847078). J.C. acknowledges support from the Terman Faculty Fellowship at Stanford. Author Contributions: M.L., J.C., and A.F. conceived the idea and experiment. M.L. built the experimental setup, performed the measurements, and analyzed the data. R.F. performed the numerical simulations. C.J.W. took the data for a reference sample. M.L., R.F., E.B., S.E.E., J.C., and A.F. interpreted the results. M.L., J.C., and A.F. wrote the manuscript with inputs from all authors. All work was supervised by J.C. and A.F. Competing interests: The authors declare no competing interests. Data and materials availability: All data are available in the manuscript or the supplementary information.

 










 Paper Title: Analyzing the Effectiveness of Quantum Annealing with Meta-Learning.





[1]Riccardo Pelliniriccardo.pellini@polimi.it

1]Maurizio Ferrari Dacremamaurizio.ferrari@polimi.it


*[1]
Politecnico di Milano,

Milan, 
Italy



The field of Quantum Computing has gathered significant popularity in recent years and a large number of papers have studied its effectiveness in tackling many tasks.
We focus in particular on Quantum Annealing (QA), a meta-heuristic solver for QUBO problems.
It is known that the effectiveness of QA is dependent on the task itself, as is the case for classical solvers, but there is not yet a clear understanding of which are the characteristics of a problem that makes it difficult to solve with QA.
In this work, we propose a new methodology to study the effectiveness of QA based on meta-learning models. 
To do so, we first build a dataset composed of more than five thousand instances of ten different optimization problems. We define a set of more than a hundred features to describe their characteristics, and solve them with both QA and three classical solvers. We publish this dataset online for future research.
Then, we train multiple meta-models to predict whether QA would solve that instance effectively and use them to probe which are the features with the strongest impact on the effectiveness of QA. 
Our results indicate that it is possible to accurately predict the effectiveness of QA, validating our methodology. Furthermore, we observe that the distribution of the problem coefficients representing the bias and coupling terms is very informative to identify the probability of finding good solutions, while the density of these coefficients alone is not enough. The methodology we propose allows to open new research directions to further our understanding of the effectiveness of QA, by probing specific dimensions or by developing new QUBO formulations that are better suited for the particular nature of QA. Furthermore, the proposed methodology is flexible and can be extended or used to study other quantum or classical solvers.





Analyzing the Effectiveness of Quantum Annealing with Meta-Learning
    [
    August 5, 2024
===================================================================





§ INTRODUCTION


In recent years the field of Quantum Computing has gathered significant popularity, thanks to remarkable advancements that led to the development of several quantum computers of different architectures and technologies that can be used to tackle numerous problems. Although quantum computers are still limited both by their relatively small size and by the noise that limits the precision of the computation, the field is rapidly moving forward.
Among the existing Quantum Computing paradigms, Quantum Annealing (QA) is a meta-heuristic that can be used to solve Quadratic Unconstrained Binary Optimization (QUBO) problems, a family of NP-hard optimization problems. 
The key idea of QA is to represent a QUBO problem as an energy minimization problem of a real and configurable quantum device. To do so, the problem variables are mapped onto physical quantum bits, or qubits.
The quantum device is steered towards a state of minimal energy, called ground state, with a controlled evolution. The ground state corresponds to the optimal solution of the original QUBO problem. The devices that implement the QA process are called Quantum Annealers.

The ability of QA to tackle NP-hard optimization problems and its flexibility to heterogeneous domains is what makes it an interesting technology for industries and researchers. Many applications of QA have been proposed in the fields of machine learning <cit.>, chemistry <cit.> and logistics <cit.>, but the results are not always competitive against classical heuristics solvers.

An important issue is that the quality of the solutions found by QA is limited by multiple factors. 
First of all, Quantum Annealers are physical devices which have a limited number of qubits and connections between them. This limits the size of the problems that they can tackle and requires to process the QUBO problem adapting it to the physical structure of the Quantum Annealer. 
A second important aspect is that the quality of the solutions found by QA depends on the behaviour of the underlying physical quantum system, which is very difficult to study. It is known that some problems appear to be more difficult to solve with QA <cit.>, but understanding why is not a trivial task and still an open research question.

Most of the previous studies on QA compare its performance, in terms of required time for computation with respect to other heuristic solvers, rather than on the quality of the solutions it finds, its effectiveness. There are two ways in which one can study the effectiveness of QA, one is by analytically describing the underlying quantum behaviour and the other is to perform empirical experiments.
A theoretical analysis has been performed for very small QUBO instances <cit.>, which however are too simple to assess the effectiveness of QA when compared to other classical solvers. Furthermore, analytically analysing such a quantum system becomes rapidly very expensive and is generally impossible for problems of interesting size. 
On the other hand, the existing empirical studies on the effectiveness of QA have explored much larger problems but focus mainly on specific tasks such as feature selection <cit.>, clustering <cit.> and classification <cit.>, and therefore lack generality.

To the best of our knowledge, there is no published research which has investigated extensively how the characteristics of the problem impact the effectiveness of QA. 
For this reason, in this study we propose a novel empirical methodology for the analysis of the effectiveness of QA, based on the study of the characteristics of QUBO problems with a meta-learning approach. The general idea consists in generating many QUBO instances, defining a set of features which can describe them and train meta-models to predict whether QA would solve that problem or not. 
Our key contributions are as follows:

    
  * The design of an experimental methodology which can be applied to study the effectiveness of QA. This methodology can be used also for other quantum algorithms, such as QAOA <cit.> or VQE <cit.>;
    
  * The selection of ten classes of optimization problems, each one with specific characteristics, from which we generate approximately five thousand QUBO instances; 
    
  * The design and the generation of a meta-learning dataset, which contains for each of the five thousand instances a selection of a hundred features based on probability theory, statistics and graph theory. We show that using them it is possible to effectively predict whether QA would solve a problem instance effectively or not. We share the meta-learning dataset online for further research;
    
  * The analysis of the features of a QUBO problem with the strongest impact on the effectiveness of QA;




§ BACKGROUND
 


 §.§ QUBO and Ising Models

In order to use QA to tackle optimization problems, these should be represented with one of two equivalent formulations called QUBO and Ising, suitable for NP-Complete and some NP-Hard optimization problems <cit.>. While the two are equivalent, the QUBO formulation is closer to traditional Operations Research, the Ising formulation is instead closer to Physics.

The objective function in the QUBO model is given by Equation <ref>, where x ∈{0,1}^n is a column vector representing the assignment of the binary variables x_1, x_2, ..., x_n, n is the number of problem variables, y the cost, and Q ∈ℝ^n × n is a real square matrix, either symmetric or upper triangular.

    min_x y = x^TQx

We will refer to combinatorial optimization problems written in the QUBO formulation as QUBO problems. 
Note that the QUBO formulation does not allow for hard constraints. 
An optimization problem with constraints can be transformed into a QUBO problem by introducing a quadratic penalty term multiplied by a penalty coefficient p. The idea is that the hard constraints are transformed in soft constraints, such that if they are violated a positive penalty p is added to the cost function making the cost of that variable assignment worse. Note that by using soft constraints we do not have the guarantee that the optimal solution will satisfy the constraints, which may happen frequently if the penalty coefficient p has a value that is too low. 
In general, a quadratic binary optimization problem with equality constraints formulated as Ax - d = 0, where d ∈ℝ^m and A ∈ℝ^m × n, can be transformed into the following QUBO problem:

    min_x y    = x^TQx + p · x^TCx 
    
        C    = (Ax - d)^T(Ax - d)

If the quadratic binary optimization problem also has inequality constraints, those need to be transformed first into equality constraints using binary slack variables.
For example, if we have the following constraint:

    x_1 + 2x_2 + 4x_3 ≤ 3

we can transform it into an equality constraint by introducing the binary slack variables x_4 and x_5:

    x_1 + 2x_2 + 4x_3 + x_4 + 2x_5 = 3

There exist no general rule to choose the best number of slack variables, so multiple strategies can be followed.

A second useful formulation is the Ising model, which was developed to describe an energy minimization problem for a system of particles <cit.>.
The objective function of the Ising model is given by Equation <ref>, where s ∈{-1,1}^n is the column vector representing the assignment of the n problem variables s_1, s_2, ..., s_n also called spin variables, J ∈ℝ^n × n is the coupling matrix that describes the quadratic terms of the objective function and has zero diagonal, b ∈ℝ^n is the bias vector, which contains the linear terms of the objective function. The constant term c ∈ℝ is called offset.

    min_s y = s^TJs + b^Ts + c

A QUBO problem can be transformed into an Ising problem through a linear mapping of the variables. In particular, a binary variable x_i is transformed into a spin variable s_i according to the following conversion[This mapping is equivalent to the more commonly used x_i = 1+s_i/2 with the difference that in our case a binary 0 is mapped onto spin 1 and binary 1 is mapped onto spin -1.]:

    x_i = 1-s_i/2




 §.§ Quantum Annealing and Quantum Annealers

Quantum Annealing (QA) is a meta-heuristic solver for QUBO problems.
It is based on the Adiabatic Quantum Computation (AQC) paradigm, with some relaxations <cit.>.
The idea is to represent the optimization problem as an energy minimization one, and then use a configurable device that exhibits the needed quantum behaviour to minimize it. Such a device, the Quantum Annealer, is composed by multiple qubits connected between each other. QA works based on a time evolution of the quantum system. The initial state of the system is a default one, easy to prepare, so that the qubits are in a state of minimal energy, the ground state. Then, the physical system is evolved slowly over a short amount of time by introducing a dependency on the Ising coefficients of the problem one wishes to solve. This means, for example, slowly changing the magnetic fields the qubits are subject to. At the end of the evolution, the physical system will depend only on the problem and, if the evolution was careful enough, it will still be in the ground state. Since the state of minimal energy is also the solution of the optimization problem, measuring the state of the qubits will yield the values that the problem variables should have. 

The evolution of the system in QA occurs in a noisy environment and is subject to quantum fluctuations, quantum tunneling, which helps it explore the solution space. 
The noise of the system and the duration of the evolution influence the results of QA, if the evolution is too fast the system will likely escape its ground state and find a worse solution, while if the evolution is too slow noise may build up and push the system again out of the ground state. 
Due to its stochastic nature, QA acts as a device sampling low-cost solutions in a similar way as other classical solvers do, such as Simulated Annealing.
For this reason, QA is repeated multiple times in order to obtain samples of the final state of the quantum system. 

The physical devices that implement QA are called Quantum Annealers.
Currently, D-Wave Systems Inc. is the company that provides the Quantum Annealers with the largest number of qubits[The documentation of the D-Wave Systems' services: <https://docs.ocean.dwavesys.com/en/stable/index.html>]. 
For example, the D-Wave Advantage has more than 5000 qubits with a topology called Pegasus, where each qubit is connected to other 15 ones.

Solving a QUBO problem with a Quantum Annealer requires the following steps:

    
  * Formulate the problem as a QUBO or an Ising problem: the coefficients that are needed to configure the Quantum Annealers are those of the Ising formulation, as such the problem needs to be in this form. If the problem has a simpler formulation as a QUBO, the transformation is straightforward. Note that some problems can be formulated as QUBO or Ising easily, while others require more expensive processing.
    
    
  * Embed the problem on the topology of the device: since the Quantum Annealer is a physical object, we must fit the problem we want to solve on it, accounting for the limited number of qubits and of the connections between them. 
    This procedure is called minor-embedding <cit.> and maps each problem variable to one or more qubits. If multiple qubits are needed to represent a single problem variable, that is called a qubit chain. If the problem has a large number of quadratic terms, a substantial number of qubits may be needed to create all the physical connections.  
    Figure <ref> shows an example of how a simple problem can be mapped on a Quantum Annealer.
    
    Minor-embedding is an NP-Hard problem but polynomial-time heuristic algorithms are available <cit.>[We use the library  offered by D-Wave: <https://docs.ocean.dwavesys.com/en/stable/docs_minorminer/source/sdk_index.html>].

    
  * Evolution of the system and sampling of the solutions:
    once the minor-embedding is done, the problem is transferred to the Quantum Annealer. First, the device is programmed with the problem coefficients, then we can perform a sequence of multiple evolutions to obtain the desired number of samples n_s. Each sample requires three steps: (i) the evolution is run for the desired duration, called annealing time t_a, (ii) the final state of the system is measured, which requires a read-out time t_r dependent on the number of qubits used, and (iii) the device pauses shortly for cooling.




More formally, the energy of a system can be modeled with an Hamiltonian, H ∈ℝ^2^n × 2^n, and the evolution that occurs in QA is described by the time-dependent Hamiltonian H(t) that models the transition from the initial default Hamiltonian H_i[
The initial Hamiltonian H_i is also called driver Hamiltonian.] and the Hamiltonian describing the problem H_p:

    H(t) = A(t)H_i + B(t)H_p

The coefficient A(t) decreases as the evolution progresses, while B(t) increases introducing the dependency on the characteristics of the problem, but their exact values depend on the hardware. At the beginning of the evolution B(t) is zero, while at the end A(t) is zero. Note that this is just a description of the underlying physical system and there is no need to compute this representation to use QA.

In the ideal Adiabatic Quantum Computing setting, it is possible to compute the exact annealing time needed to ensure the system remains in the ground state and finds the global optimum, this result dates back from a century ago <cit.>. This optimal annealing time is inversely proportional to the smallest difference between the two smallest eigenvalues λ_1(t), λ_2(t) of H(t). Such difference is called minimum gap. Although this result may be useful to understand its behaviour, it is not applicable to QA because it is subject to noise. Furthermore, computing the eigenvalues of H(t) is prohibitive for all but the smallest problems.

To exemplify how this representation works, assume to have an Ising problem of n variables with coupling J and bias b, H_p is a 2^n × 2^n matrix computed as follows:    
    H_p = ∑_i=1^n ∑_j=i^nJ_ijσ_z^(i)σ_z^(j) + ∑_i^n h_i σ_z^(i)

The matrix σ_z^(i) is the Z-Pauli operator σ_z acting on qubit i:

    σ_z = [r]
            1     0 
    
            0     -1


    σ_z^(i) = ⊗_k=1^i-1I  ⊗σ_z ⊗_k=1^n-iI

with ⊗ being the tensor product and I the identity matrix. A useful property of H_p is that it is a diagonal matrix that contains all the cost values for all possible variable assignments of the problem. Since it is diagonal, these values are also its eigenvalues and the corresponding eigenvectors encode the variable assignment that has that cost. The minimum eigenvalue of H_p corresponds to the minimal cost and the corresponding eigenvector to the optimal variable assignment.

As an example, consider the following QUBO problem, which is minimized when x_1 = x_2:

    min_x_1, x_2 y = x_1 + x_2 - 2x_1 x_2

The equivalent Ising formulation is:

    min_s_1, s_2 y = 1/2 - 1/2s_1 s_2

For this small instance we can compute H_p easily.
The matrices σ_z^(1) and σ_z^(2) are:

    σ_z^(1) = σ_z ⊗ I = [  1  0  0  0;  0  1  0  0;  0  0 -1  0;  0  0  0 -1;    ]  σ_z^(2) = I ⊗σ_z = [  1  0  0  0;  0 -1  0  0;  0  0  1  0;  0  0  0 -1;    ]

H_p is then equal to:

    H_p = 1/2[ -1  0  0  0;  0  1  0  0;  0  0  1  0;  0  0  0 -1;    ]

The smallest eigenvalue of H_p is λ_min = -1/2 and has a multiplicity of two corresponding to the first and last eigenvalues. Indeed, both x_0 = 0, x_1=0 and x_0 = 1, x_1=1 are optimal solutions of the problem. If we sum λ_min with the offset of the Ising problem, 1/2, we obtain 0, that is the same value of the QUBO cost function when x_1 = x_2.



 §.§ Studies on the Effectiveness of QA

Most of the previous studies on QA focus on its performance, by measuring the time required to solve a problem and comparing it to that of classical solvers.
To the best of our knowledge there is no consensus on whether QA provides a general and consistent speedup compared to other traditional solvers for QUBO problems <cit.>, while a recent paper claims substantial speedup for a quantum simulation task <cit.>. Although some papers may claim a speedup, this is often based on measurements that only account for part of the process. Indeed, one should consider the time required by all phases: (i) formulating the optimization problem as QUBO or Ising, (ii) embedding the problem on the QA, (iii) sampling the solutions on the device and (iv) postprocessing the results if needed (for example by checking if the constraints are satisfied). Frequently, the efficiency of QA is measured by only accounting for the usage of the quantum device itself (programming time and the repeated annealing and read-out) while ignoring the time needed for minor embedding and for creating the QUBO formulation. This gives an incomplete picture of the technology that does not account for two significant bottlenecks. For example, it may be that in a certain situation QA is faster than other traditional methods in solving a specific QUBO problem, but that may not be the case anymore if one includes the minor embedding phase. Furthermore, if it is very computationally expensive to formulate the problem as QUBO, it may be more efficient to use other traditional methods that do not need a QUBO formulation at all.

When comparing the quality of the solutions found by QA and classical solvers, their effectiveness, the published literature usually focuses on problems related to specific fields or even to very specific instances of those problems. Due to this, there is still a limited understanding of how would QA compare in a more general setting. For example, the effectiveness of QA has been analyzed for feature selection <cit.>, classification <cit.> and  clustering <cit.>, which are typical machine learning tasks. In the field of chemistry, QA has been applied and analyzed to find the equilibria of polymer mixtures <cit.>, to find similarities between molecules <cit.> and to find their ground state <cit.>.
The effectiveness of QA in solving problems related to logistics has been analyzed too, for example in solving the Nurse Scheduling Problem <cit.> and in optimizing the assignments of the gates at the airport <cit.>.

Previous research also studied the effectiveness of QA from a theoretical perspective by representing analytically the evolution of the time-dependent Hamiltonian H(t) (see Equation <ref>) and computing the probability of escaping the ground state <cit.>. This approach is however limited by the fact that the size of the Hamiltonian grows exponentially on the number of QUBO problem variables n, and the analytical analysis of the Hamiltonian becomes rapidly impractical for all but the smallest problems.
An alternative way is to adopt an empirical approach, by using the outcome of multiple experiments to probe the underlying physical system <cit.>. The idea is to allow the evolution to progress up to a certain intermediate stage and then drastically accelerating it (a quench, according to D-Wave terminology), observing how the effectiveness changes based on when was the evolution accelerated. While this approach allows to tackle large problem instances, applying the acceleration at different stages of the evolution requires to repeat the experiment a large number of times and therefore this approach too is very resource intensive.

To overcome the limitations of the methods adopted in the literature, this paper proposes a new empirical approach to study how the quality of the solutions found by QA is impacted by the characteristics of the problem.
To achieve this, we first collect a dataset of problem instances belonging to 10 selected problem classes and solve them using both QA and three classical solvers. Then, for each problem instance, we compute a set of features describing various characteristics, from the distribution of the bias coefficients of its  QUBO formulation to the topology of the graph that describes the instance once it has been embedded on the Quantum Annealer. Using this dataset we train a machine learning classifier to identify whether QA was able to find a good solution for that instance and, finally, use it to assess which are the most important problem features.



§ META-LEARNING DATASET GENERATION

In this section, we present the methodology used to generate our meta-learning dataset, on which we train the meta-models to predict the effectiveness of QA. We publish this dataset online for future research.
First, we describe how we select the ten classes of problems we want to solve with QA and the strategies we use to generate the five thousand instances.
Then, we describe how we evaluate the effectiveness of QA, in terms of closeness to the optimal solution of the problem and by comparing QA with other classical methods (Simulated Annealing, Tabu Search and Steepest Descent).
Third, we describe the representations of the QUBO problem we used to compute the approximately one hundred features used to train the meta-models.
Finally, we describe how we solve the instances with QA and with the classical solvers, with a particular focus on the choice of the optimal hyperparameters of the solvers.



 §.§ Selection of Problems and Instances

We identify a selection of ten different optimization problems that exhibit different characteristics: some have constraints, others do not; some have linear terms, others do not; some have a large number of quadratic terms while others do not, etc. The details on their formulations are reported in Appendix <ref> and the details on the generation of the instances are in Appendix <ref>.

The first group contains five classes of optimization problems defined over a graph: Max-Cut, Minimum Vertex Cover, Maximum Independent Set, Max-Clique and Community Detection.
They were selected for the following reasons. Both the Max-Cut and Community Detection problems have a straightforward QUBO formulation that does not require penalties to represent constraints. The Max-Cut, Maximum Independent Set and Minimum Vertex Cover problems share the same quadratic terms in their QUBO matrix, but not the diagonal (the linear terms or bias). The Max-Clique problem is formulated as a Maximum Independent Set problem but defined on the complement graph. The Community Detection problem has a very dense QUBO matrix as there are quadratic terms between all variables and is a relevant problem in Machine Learning <cit.>.
Since these problems are formulated on a graph, we apply them on four different graph topologies: Erdös-Renyi, Cyclic, Star and 2d-grid. Note that in order to have a diversified set of instances we introduce small random perturbations to each topology, consisting in few edge insertions and deletions. The number of insertions and deletions depends on the number of nodes of the graph. More details are reported in Appendix <ref>.

The second group of five optimization problems contains: Number Partitioning, Quadratic Knapsack, Set Packing, Feature Selection and 4 × 4-Sudoku. These are a more heterogeneous set than the previous graph-based problems ad so require ad-hoc strategies to generate their instances which we detail in Appendix <ref>.
Similarly to the Max-Cut and Community Detection problems, the Number Partitioning problem has a straightforward QUBO formulation with no penalty terms to represent constraints.
Similarly to the Community Detection problem, the Feature Selection problem has a dense QUBO matrix with quadratic terms between all variables.
Finally, the Quadratic Knapsack, Set Packing and 4 × 4-Sudoku problems are all Constraint Satisfaction Problems, each with different types of constraints.
In particular, the Quadratic Knapsack problem has inequality constraints that need to be converted in equality constraints using slack variables.

We generate multiple instances of all the problem classes we selected. 
Concerning the size of the problem instances, measured in the number of problem variables, there are two constraints to take into account. First, the D-Wave Quantum Annealer that we use has more than 5000 qubits but, due to their limited connectivity, it is generally possible to tackle problem instances up to between 100-200 variables depending on the structure of the QUBO problem. This is due to the minor-embedding phase. Second, we want the instances to be representative of problems that are not trivial and with a Hamiltonian that could not be analyzed analytically. 
In order to provide a more complete picture, we are also interested to assess the impact of the distribution of the solution space of the problem. This, formally, corresponds to the set of eigenvalues and eigenvectors of the Hamiltonian of the problem H_p (see Section <ref>). Unfortunately, it is impractical to compute them for instances of more than 32 problem variables, which may be too small and easy to allow a comparison on the effectiveness of different solvers. For this reasons we decided to create two separate sets of instances:

    
  * One set of large instances, with 5114 instances of between 69 to 99 variables, the upper range of what can be tackled with the QA; 
    
  * One set of small instances, with 246 instances of between 27 to 32 variables. With this set of instances we can do a more complete analysis which includes also the distribution of the solution space.


The number of problem instances for each optimization problem is summarized in Table <ref>. 
Notice that the instances of the 4 × 4-Sudoku problem are included only in the small instances set, since the largest possible instance is unique and it has at most 64 variables.
All the generated instances are satisfiable and, when needed, the penalty term coefficient p used in the QUBO formulation is optimized with a Bayesian Search[The range of possible values of p, for a particular instance, depends on the cost function of that instance. More details are given in Appendix <ref>] <cit.>, in order to maximise the number of feasible solutions for Simulated Annealing.





 §.§ Evaluating the Effectiveness of a Solver

In this section, we describe how to evaluate the effectiveness of a solver and, in particular, of QA.
Both QA and the traditional solvers we compare it to are stochastic and are executed multiple times to obtain a set of variable assignments that aim to minimize the cost function, which we call a set of samples. 
A sample is represented by an assignment of the decision variables x and by the related cost value y.

We solve all instances with QA, Simulated Annealing (SA), Tabu Search (TS) and Steepest Descent (SD).
Our definition of how much a solver is effective is based on whether it finds samples that meet some quality constraints. While for the small instance set it is possible to compute the global optimum, for the large ones it is not feasible to do so and therefore we define the effectiveness in relative terms with respect to the other solvers.

In particular, we evaluate the effectiveness of QA on the large instances set by comparing its samples with those of the traditional heuristic solvers. We define the samples associated to the best cost value for a solver S as y^S_min.
An instance I is QA-over-all if the best solution found by QA is at least as good as the best one found by SA, SD, TS combined. More formally, if y^QA_min≤min{y^SA_min, y^TS_min, y^SD_min}.
Comparing QA with a pool of multiple solvers results in a stricter evaluation of its effectiveness, but the condition that QA has to be at least as good as all the other solvers combined may be too strict.
For this reason, we also compare QA with each individual solver.
An instance I is QA-over-S if the best solution found by QA is at least as good as the one found by solver S, hence y^QA_min≤ y^S_min.

For the small instances we can perform a deeper analysis of the effectiveness because we can explore the full solution space and find the global optimum. This is in practice done by computing the Hamiltonian of the problem, H_p, which is a diagonal matrix enumerating the eigenvalues λ, sometimes called energy, of all the variable assignments. The eigenvalue is equivalent to the cost function y but does not include possible constant offsets c from the Ising formulation, therefore y = λ + c. 
The variable assignment x associated to an eigenvalue  λ can be computed starting from the corresponding eigenvector of H_p.
The global optimum of an instance is the assignment x_min corresponding to the minimal eigenvalue of H_p, λ_min. We will refer to the maximum eigenvalue as λ_max.

We define a sample with energy λ as ϵ-Optimal if the following condition holds:
    
    λ≤λ_min + ϵ· (λ_max- λ_min)

The ϵ-Optimality condition describes how close is the eigenvalue of a sample to the solution of the instance.
The coefficient ϵ∈ [0,1] allows to restrict the interval under which λ is considered close enough to the optimal eigenvalue λ_min. Notice that if ϵ=0 only the global optimum of the instance meets the constraint in Equation <ref>. 

We also define a sample x as Hamming-Optimal (h-Optimal) if 
it differs from any solution x_min in at most one decision variable.
This corresponds to check the Hamming distance between a sample and a solution of the instance:
    
    ||x - x_min||_Hamming≤ 1




 §.§ Meta-Learning Features


In this section, we introduce the features we define to describe a problem instance.
We rely on a selection of metrics used in statistics and probability theory, such as the Gini coefficient <cit.>, the Herfindahl-Hirschman index <cit.> and the Shannon entropy <cit.>, as well as metrics used in graph theory, such as the spectral gap, the radius a graph, its diameter and its connectivity. In total we compute 107 features, which we describe in detail in  Appendices <ref> and <ref>.

The features we compute can be grouped in multiple domains of analysis.
Overall we identify seven domains, among which we describe the three most relevant ones:

    
  * Logical Ising Graph (LogIsing): This domain uses the Ising formulation of a QUBO problem. It is represented as a graph having one node per problem variable, associated to the corresponding bias b, and an adjacency matrix that corresponds to the coupling matrix J;
    
  * Embedded Ising Graph (EmbIsing): This domain uses the Ising formulation of a QUBO problem obtained after its minor embedding on the QA. The target architecture is D-Wave Advantage with the Pegasus topology. Therefore, this formulation represents the actual problem solved by the Quantum Annealer, in which multiple qubits may be used to represent one problem variable. This formulation is represented as a graph in the same way as LogIsing; 
    
  * Solution Space (SolSpace): This domain uses the eigenvalues, or energy values, of all possible variable assignments, which can be computed only for the small instances, and aims to describe how they are distributed.

Other domains we identify are: (i) Normalized Multiplicity (NorMul), whose features are related to the multiplicity of the eigenvalues of H_p; (ii) Matrix Structure (MatStruct), which contains features related to the distribution of the values of the matrix Q of a QUBO problem; (iii) 25%-SolSpace and 25%-NorMul, which contain the same features of SolSpace and NorMul, but computed by considering only the 25% lowest eigenvalues of H_p, the energies of the 25% best solutions.

For the LogIsing and EmbIsing domains we compute several features on different mathematical objects, such as the coupling matrix J, the Laplacian matrix of the corresponding graph and the bias vector. We call such objects components. 

We can also identify sets of features that refer to the same mathematical object, but are computed on different domains. For example, both LogIsing and EmbIsing domains include features computed on the bias. We refer to them as component sets and allow us to perform an analysis of the importance of those mathematical objects that is orthogonal to that of the domains. We identified the following component sets: Coupling, Bias,  Laplacian, Structural Adjacency (StructAdj), Structural Laplacian (StructLap), where
StructAdj and StructLap gather features related to the binarized versions of the coupling and Laplacian matrices. 



 §.§ Hyperparameter Optimization of the Solvers

Since the goal of this study is to compare the effectiveness of different solvers, it is essential to ensure that each solver is using the best hyperparameters. Indeed, it is well-known in many fields that comparing methods that are not consistently optimized leads to inconsistent results that cannot be used to draw reliable conclusions <cit.>. The same applies in our case.

We optimize the hyperparameters of each solver (QA, SA, TS, SD) 
on the instance with the largest minor-embedding on D-Wave Quantum Annealer for each optimization problem class. The goal is to identify the hyperparameters that will lead the solver to find the variable assignment with the lowest cost y. Once the optimal hyperparameters have been found, they are used to solve all instances of the corresponding problem class. We optimize separately the hyperparameters used for the large and small instances sets. 
To optimize the hyperparameters of the classical solvers we use the standard QUBO formulation while for QA we use the embedded QUBO formulation: we followed this strategy because the embedded QUBO formulation is required only for QA. In this way, we have a fair comparison between different solvers since, for each one of them, we take into account only the necessary steps to solve a QUBO instance.



  
Optimal Hyperparameters of Quantum Annealing
QA has several hyperparameters that can be optimized[For a detailed explanation of all the hyperparameters of D-Wave Quantum Annealers, refer to the documentation of the devices: <https://docs.dwavesys.com/docs/latest/c_solver_parameters.html>], some of which refer to the evolution process as a whole while others allow to fine-tune it at the level of each individual qubit.
The access to the D-Wave Quantum Annealers is limited and for such a large set of instances we have devised a methodology to optimize the hyperparameters we believe are the most important: 
the annealing time t_a and the number of samples n_s. In order to perform an efficient optimization within the available resources, we define a fixed computational budget T for each instance. 
Using the default annealing time, 20 μ s, and drawing 100 samples requires, in the worst case, at most  37 ms. In our experiments we allocated T=70 ms and T=300 ms per each problem instance. 

The optimization is performed by iterating over 10 values for t_a, approximately equidistant from each other, between 5 μ s and 200 μ s. Given t_a, the number of samples n_s is computed as the maximum value allowed within the computational budget T, according to Equation <ref>. For T =70 ms, n_s is between 145 and 537 while, for T =300 ms, n_s is between 766 and 2826.

    n_s = ⌊T - t_p/t_a + t_r + Δ⌋

The term t_p ≃ 15  ms is the time needed to program the instances on the Quantum Annealer, t_r is the read-out time, needed to read the results of the annealing process, and Δ≃ 20 μ s is the delay applied after each read-out operation. The read-out time t_r is unknown a-priori because it depends on the size of the embedded problem. Based on empirical observation we use t_r=75  μ s for small instances and t_r = 150 μ s for large instances. 
We choose t_a and the related n_s which provide the sample with the lowest energy. If for multiple pairs (t_a, n_s) QA finds samples with the lowest energy, we choose the pair with the smallest t_a.

Since the results we obtained when using both computational budgets are very similar, we report those for T=70 ms. The selected hyperparameters are reported in Table <ref>.
Notice that the annealing time t_a for the large instances is often smaller than for the small instances. This highlight that, for large instances, a larger t_a does not improve the effectiveness of QA, at least in the range of values we considered and for the number of samples it allows to draw. Such result suggests that QA may require an additional optimization, for example, of the annealing schedule, which is not straightforward and it goes beyond the scope of this study.





  
Optimal Hyperparameters of the Classical Solvers

The hyperparameters of Simulated Annealing (SA), Tabu Search (TS) and Steepest Descent (SD) are optimized with the following procedure.
For the optimization of these methods we do not use a fixed computational budget because the technology is fundamentally different and, due to the various stages required by QA, it is not trivial to define such a comparison in a way that is fair.
First, we fix the number of samples to n_s = 200 which is a value comparable to that used for QA. For half of the the large instances QA uses more samples than the classical solvers, while for the remaining half the opposite is true.
We optimize the hyperparameters with a Bayesian Search of 100 iterations <cit.>. The results are available in Appendix <ref>. 
For TS, we optimize the 
number of restarts of the algorithm and the initialization strategy.
For SD, there are no hyperparameters to optimize, except for the number of samples, which we have already set.
For what concerns SA, we optimized the number of [A sweep consists in flipping a randomly chosen decision variable.], the [The schedule determines how the temperature of the system decreases over time.] and the initial state generator. We noticed however that hyperparameters we found for SA produced worse results compared to the default ones in our following analysis, which may be due to the sensitivity of SA to some of them. For this reason, we retain the default hyperparameters of 1000 , a  and a random initial state generator. 



§ META-MODEL TRAINING AND OPTIMIZATION

In this study we aim to identify which are the characteristics of a problem that impact the effectiveness of QA. We do this by first training a classification model on the dataset we have created in order to predict whether QA would solve that instance well or not based on its features. Since the classifier is trained to predict the outcome on another experiment, it is called a meta-model. Once the meta-model is trained, we can use it to probe how important are the various features.

We train the meta-models with Random Forest, AdaBoost, XGBoost and Logistic Regression, using as input data either a specific domain (LogIsing, EmbIsing, SolSpace) or a specific component set (Bias, Coupling, Laplacian), which are described in Section <ref>.
The target labels are described in Section <ref> (Optimal, ϵ-Optimal, h-Optimal) 
and they are binary, according to whether the solver meets that effectiveness condition or not.

The first step is to train the meta-model and optimize its hyperparameters to ensure it is effective in predicting the label. 
In order to measure the effectiveness of the meta-models, we have to account for the significant class imbalance of the labels towards the negative class, instances that are not solved well by QA  (see Section <ref>).
We use Balanced Accuracy (BA) to evaluate the meta-models because it is robust to class imbalance.
Given the true positives as TP, the true negatives as TN, the number of positive labels in the data as P and the number of negative labels as N, the Balanced Accuracy BA is computed as: 

    BA = 1/2(TP/P + TN/N)


The training and optimization of the meta-models is performed with 5-fold Nested Cross-Validation. First, we create a 5-folds testing split with a training fold and a testing one which we will use to train and evaluate the meta-model. In order to find the optimal hyperparameters for the meta model, we split each training fold with a further 5-folds split, the optimization split. This results in 5 optimization splits for each of the 5 training folds of the testing split and is aimed at preventing the overfitting of the meta-models. 
The splits are all stratified with respect to the problem class of the instances, to ensure every split has an equal distribution of the problem classes. All meta-models are trained on the same data splits and we perform different splits for the large and small instances.
The hyperparameters of the meta-models are optimized according to a Bayesian Search <cit.> exploring 50 configurations, we select those that provide the best Balanced Accuracy on the optimization split.

Once the meta-models have been optimized, we use them to assess which problem characteristic, feature, is most important. We use Permutation Feature Importance (PFI), 
which evaluates how the accuracy of a model drops when the values of a certain feature are shuffled. The idea is that the more important a feature is the larger will be the drop when the values of that features are shuffled. For each feature the process is repeated multiple times and the corresponding importance is given by the mean of the drop in accuracy observed.



§ RESULTS AND ANALYSIS

In this section we provide the most relevant insights of our analysis regarding the effectiveness of QA.
We have three goals: (i) determine hitch classes of
problems are more difficult to solve with QA, (ii) understand whether it is possible to predict the effectiveness of QA based on the features we have identified; and (iii) discover the domains, the component sets and the features that impact the effectiveness of QA.
To do so, we describe the results obtained by solving the instances with QA and with the other classical solvers. Then, we describe the results of the validation of the meta-models and of the Permutation Feature Importance performed on their features. We publish online a dataset with all the instances we generated, the features we computed and the samples obtained for each solver.[The instances, the dataset with the features, the results of the solvers and an example script to train meta-models are available at this GitHub repository: <https://github.com/qcpolimi/QA-MetaLearning>.]



 §.§ Effectiveness of QA for Large Instances





In this section we discuss the effectiveness of QA compared to the other classical solvers (SA, TS and SD) on the large problem instances. Table <ref> reports the results on each problem class according to the labels we defined in Section <ref>, whether the best sample found by QA is at least as good as that found by a specific solver (QA-over-SA, QA-over-TS and QA-over-SD) or by all of them combined (QA-over-all).

As a general comment, we can observe that for less than half of the problem classes (4 out of 9) QA solves effectively more instances than at least one classical solver, while for most of the problem classes (7 out of 9) QA is more effective than at least one of the classical solvers for some particular instances.
However, if we combine all classical solvers QA is more effective only in three problem classes but mostly to a limited extent. Only for Max-Cut QA shows a consistently high effectiveness. 
These results confirm that the effectiveness of QA depends on the problem class, as is the case for classical solvers, which is consistent to what observed in previous studies <cit.>.
If we compare QA and classical solvers throughout the problem classes, we can see that QA is very frequently more effective than SA, while it is more effective than TS or SD only on some specific problem classes.
As a result, we conclude that comparing QA only with SA, without considering other solvers, is not the best practice to evaluate the effectiveness of QA.

Regarding the characteristics of the problem classes, a first observation we can make is that QA is more effective on problems that do not require penalties to represent constraints: Max-Cut, Community Detection and Number Partitioning. This suggests that the presence of constraints is a factor that makes a problem more difficult to solve with QA. The reason for this may be due to the type of quadratic terms introduced by the penalties which could open new research directions in whether one could use a different formulation for the same constraint that is more suitable for QA <cit.>.
Furthermore, remember that Max-Cut, Maximum Independent Set and Minimum Vertex Cover share the same Ising coupling matrix J, with the exception of a multiplicative factor, but have a different bias vector b. We can observe how Max-Cut is the only one among them that is solved effectively by QA, suggesting that the bias structure plays an important role as well.

Lastly, a high number of quadratic terms (a dense coupling matrix J) does not always negatively affect QA. In particular, both Community Detection and Number Partitioning have a dense coupling matrix but still 13% of Community Detection instances and the 30% of Number Partitioning instances are solved effectively with QA.



 §.§ Effectiveness of QA for Small Instances



In this section we discuss the effectiveness of both QA and the other classical solvers (SA, TS and SD) on the small problem instances. For these instances we can compute the cost associated to all variable assignments and the global optimum. We do this by computing the Hamiltonian of the problem H_p and use its eigenvalues (its diagonal). 
We also compute the maximum energy values needed to assess the ϵ-Optimality for the samples of QA.



Table <ref> compares the effectiveness of the solvers according to the labels defined in Section <ref>, if the solver finds the global optimum (Optimal), if the energy of the best sample is close to that of the global optimum (for ϵ-Optimal we use ϵ = 10^-5), if the variable assignment of the best sample has an Hamming Distance of at most 1 with any of the global optimum solutions (h-Optimal).

Consistently with what observed for the large instances, QA is less effective than the classical solvers on all the effectiveness conditions. 
If we compare 10^-5-Optimal and h-Optimal, we can see that QA finds more h-Optimal samples than 10^-5-Optimal samples, as opposed to the other solvers. This indicates that QA finds more easily samples which are close to the optimal ones in terms of Hamming distance rather than energy.
Notice that in this experiment SD is the most effective solver, this may be related to the small size of the instances which may make them relatively easy to solve with simple strategies.



As done for the large instances, we compare the effectiveness of the solvers on the problem classes. Table <ref> shows the fraction of problem instances in which the solver finds the global optimum (Optimal). Overall, as opposed to what we observed for the large instances, on the small ones QA is never more effective than the classical solvers. 
QA seems to be more effective for problems that are defined over graphs (Max-Cut, Maximum Clique, Community Detection) compared to the ones that are not. Note however that on two graph problems, Minimum Vertex Cover and Maximum Independent Set, QA performs significantly behind the classical solvers. 
Based on the analysis of the large instances (see Section <ref>) we observed that the bias of the problem seems to play a role in affecting the effectiveness of QA. This observation is confirmed here as well since QA is much more effective in solving Max-Cut problems than in solving Maximum Independent Set and Minimum Vertex Cover ones.
We can also observe that the effectiveness of QA is quite poor for Set Packing and Feature Selection, being significantly behind the classical solvers. 

Interestingly, on Max-Cut and 4 × 4-Sudoku problems almost all the solvers find the global optimum, while for the Quadratic Knapsack hardly any instance can be solved optimally at all, with the most effective solver being TS with 4% of the instances solved optimally.

As a second analysis we study the effectiveness of QA in sampling solutions with a variable assignment that is close to one of the optimal ones in terms of Hamming distance (h-Optimality). The results reported in Table <ref> are consistent with the previous ones in which we assessed the ability of the solvers to find the global optimum, see Table <ref>.
We should note QA exhibits much better effectiveness, when measured in this way, being able to sample solutions close to the optimal ones in the majority of cases. For example, the effectiveness on the Number Partitioning problem goes up from 33% to 96% while for Minimum Vertex Cover goes from 17% to 58%.
The results also confirm that QA is quite effective for problems that do not require penalties to model constraints (Max-Cut, Community Detection and Number Partitioning). On the other hand, QA is still ineffective for the Feature Selection problem. Quadratic Knapsack remains very challenging for all the solvers.






 §.§ Meta-Models and Feature Importance Analysis







The goal of our analysis is to identify the domains and the component sets whose features allow the meta-models to predict well the effectiveness of QA.
We limit our analysis to the meta-models trained to predict whether QA will be at least as effective compared to all the classical solvers combined (target label QA-over-all) for the large instances. We also analyze the meta-models predicting whether QA will find the global optimum on the small instances (QA-Optimal). The full results are available in the online appendix. 

The first important question is whether it is possible to train meta-models able to predict the effectiveness of QA. The results in terms of Balanced Accuracy are shown in Figure <ref> (target QA-over-all) and in Figure <ref> (target QA-Optimal) for the two most effective classifiers.
We can immediately see that for several domains or components of the large instances the Balanced Accuracy is approximately 85%, while for the small instances it is often exceeding 90 or even 95%. This, combined with the fact that we selected a heterogeneous set of problem classes and instances that are solved by QA with different degrees of success, allows us to conclude that it is indeed possible to build accurate meta-models to predict the effectiveness of QA. These meta-models can then be used for many purposes, among which, studying the behaviour of this technology.

We now analyse which are the domains or components that produce the best meta-models. Concerning the domains, the more informative ones are those related to the graph structure of an Ising problem (LogIsing and EmbIsing) which, based on the high accuracy of the meta-model, we conclude are very informative on the effectiveness of QA. 
Among the domains, the distribution of the values in the Q matrix of the QUBO problem (MatStruct) is less informative, this can be explained by the fact that the problem that is actually solved on the quantum device is represented as Ising and not as QUBO.

Secondly, if we consider the domains related to the distribution of the energies of an Ising problem which are only available for the small instances (SolSpace, NorMul, 25%-SolSpace, 25%-NorMul in Figure <ref>) it is possible to build meta-models which, again, predict well the effectiveness of QA with a Balanced Accuracy well above 90%. 
This result shows that the effectiveness of QA also depends on the distribution of the energies of the problem, on how the cost y of the QUBO problem is distributed. Notably, using features based on the solution space allows to achieve comparable Balanced Accuracy with other domains, indicating that both are equally very informative. This is a particularly good result because it is relatively easy to compute the features for the other domains once the problem has been formulated as QUBO.

If we look at the orthogonal grouping of the features, by component sets, we notice that with the Bias, Coupling the Laplacian component sets it is possible to train at least a meta-model with good Balanced Accuracy (higher than 80%). On the other hand, the Structural Adjacency (StructAdj) and Structural Laplacian (StructLap) are the least informative and, in particular for large instances, do not allow to build a meta-model better than random guess. Since both StructAdj and StructLap are computed on the binarized problem structure, they only account for how the problem variables are connected and not the coefficient values, this means that the structure of the problem alone is not informative at all.
The bias and the coupling of an Ising problem, together with the Laplacian matrix related to the graph of the Ising problem, are the most informative on the effectiveness of QA.

In general, we can confirm that the characteristics of the problem are important to determine the effectiveness of QA but one must account for the actual coefficient values of the problem and, preferably, use features derived from the Ising formulation. This could open new research questions on whether one can change the formulation of a problem so that its coefficients have a different distribution that is more adequate for the QA. Furthermore, since the coefficients are a function of the problem class and the data, it may be possible to identify which types of graph topologies may be more or less difficult to tackle based on the distributions of the coefficients that they would produce.

We now move to analyzing which specific features are the most important among the ones we identified.
We limit our analysis on feature importance to the XGBoost and AdaBoost meta-models trained on two domains (LogIsing and EmbIsing) and on two component sets (Bias and Coupling).
In the case of meta-models related to small-instances, we include in the analysis also the domain SolSpace.
The best five features of each of these domains and component sets are listed in Table <ref> (target QA-over-all, large instances) and in Table <ref> (target Optimal, small instances).








  
Domains Feature Importance
We consider in particular the domains LogIsing and EmbIsing.
The majority of the most important features are related to the bias and to the coupling of the problem, which is consistent with our previous analysis. Some features are related to the distribution of the values of the bias and the coupling (Gini index, Shannon entropy, Herfindahl-Hirschman index), while other features are related to precise values of these mathematical objects (minimum value, maximum value).

In particular, notice that  (related to the distribution of the bias),  (related to the values of the bias) and  (related to the eigenvalues of the coupling) are among the best features in the majority of the meta-models. We deduce that the distribution of the values and the values themselves of the bias are important to study the effectiveness of QA, together with the eigenvalues of the coupling. This is again an interesting observation because it would allow us to identify in advance whether a problem could be well-suited for QA.

Notice also that the number of qubits needed to embed the problem on the Quantum Annealer () is important, for one meta-model, to predict the effectiveness of QA for the large instances, but not for the small instances. 
This difference may be linked to the fact that, for the small instances, the number of qubits required after the minor-embedding process is limited and therefore has a lower impact.

We analyze, for these two domains, the least important features too. The majority of them is related to the structural adjacency and to the structural Laplacian matrix. This confirms that the sole structure of a problem is not sufficient to determine the effectiveness of QA. Thus, we must consider also the coefficient values between the variables. 

For what concerns the SolSpace domain (see small instances in Table <ref>), notice that both the meta-models have the same top three features, although in different order: such features are mostly related to the distribution of the eigenvalues of the problem ( and ), which plays therefore a role in determining the effectiveness of QA.



  
Component Sets Feature Importance
If we consider the Bias component set, the majority of the most important features are related to the distribution of the values of the bias, both considered in the LogIsing domain and in the EmbIsing domain.
In particular, observe that the Gini index and the condition number of the Bias, computed in both domains, are among the five most important features, as they were for the meta-models trained on the domains LogIsing and EmbIsing.
This corroborate our statement that the distribution and the values of the bias are important to determine the effectiveness of QA.

If we instead consider the Coupling component set, we observe that most of the important features we identify are related to the values and to the eigenvalues of the coupling. Features computed in both the LogIsing and EmbIsing domains are important and the most important features are related to the values of the eigenvalues of the coupling.
Notice that the spectral gap and the Gini index of the coupling, computed both in the LogIsing and EmbIsing domains, are shared with almost all meta-models as some of the most important features. We conclude that the eigenvalues of the coupling and their distribution are important to analyze the effectiveness of QA. 



  
Feature Correlation with Target Label
In the previous analysis we identified the features that are the most important for the meta-models.
Some of these features, furthermore, are important also if computed in different domains and for different meta-models. These features are, for the LogIsing and EmbIsing domains: , , ; while for the SolSpace domain:  , , and . 
We want now to give an intuition of which values of these features determine a low or an high effectiveness of QA.
For each feature we identified, we compute the Spearman rank with the targets of the meta-models (QA-over-all and QA-Optimal).

For most of these features, the Spearman rank does not highlight a strong correlation with the value of the target (in general, the Spearman rank in absolute value is below 0.40). This suggests that the complexity of the underlying behaviour might require more powerful tools.
Two exceptions are given by  and  in domain SolSpace, which have respectively a Spearman rank of -0.596 and 0.537.
This mean that as the Gini index of the eigenvalues increases, the Ising problem becomes less difficult to solve.
On the other hand, high values of  of the eigenvalues implies that the Ising problem is difficult to solve with QA.



§ CONCLUSIONS


In this paper, we have studied the effectiveness of QA with an empirical approach based on meta-learning models.

First, we select a pool of ten optimization problems which can be formulated as QUBO. Then, we generated more than five thousand instances, based on different problem sizes and structures. In particular, we created two sets, one containing large instances and another with small ones for which we can study also the properties of the whole solution space.

As a second step, we define a set of more than a hundred features to describe each problem instance. 
The features are heterogeneous, based on graph theory or on metrics largely used in statistics, probability theory and economics and account for the structure of the problem, its coefficients and its solution space. 
We gather all the features into a meta-learning datasets, which we share on GitHub for further research.

Third, we compare the effectiveness of QA and three classical solvers: Simulated Annealing (SA), Tabu Search (TS) and Steepest Descent (SD).
We observe that QA is frequently less effective than the classical solvers, for both the large and small instances, except for specific problems. 
In particular, we have observed that QA solves more effectively problems with no constraints in their formulation (Max-Cut, Number Partitioning and Community Detection).

Lastly, we train different classification algorithms to predict whether QA will solve an instance effectively or not and show that it is possible to do so accurately. We then use the meta-models to probe the behaviour of QA.
In particular, by analyzing the feature importance of the meta-models, we can observe how the distribution of the bias and the coupling of a problem play a key-role in determining whether QA will be effective in solving it. 

In conclusion, we successfully applied an empirical analysis of the effectiveness of QA based on meta-learning. 
Possible future directions include the analysis on how different distributions of the coupling and bias values relate to the effectiveness of Quantum Annealing. Such results could be correlated to specific kinds of problems. For example, problems characterized by graphs with a power-law distribution (e.g. problems involving social networks) may be more or less difficult to tackle than those characterized by regular graphs. This can be done, for example, by defining new features which describe how much the distribution of the bias and the coupling differs from another distribution, e. g. from a Gaussian or a uniform distribution.
Thanks to its generality, the methodology can be easily extended to other heuristic solvers of QUBO problems, such as the Variational Quantum Eigensolver (VQE) <cit.> and Quantum Approximate Optimization Algorithm (QAOA) <cit.>, providing a useful tool to further our understanding of how to use these quantum algorithms effectively.



Supplementary information
The meta-learning dataset with all the problem instances, the corresponding graphs and features, as well the samples obtained with each solver can be accessed here: <https://github.com/qcpolimi/QA-MetaLearning>.



§ DECLARATIONS


This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: <https://doi.org/10.1007/s42484-024-00179-8>.

Funding We acknowledge the financial support from ICSC - “National Research Centre in High Performance Computing, Big Data and Quantum Computing”, funded by European Union – NextGenerationEU. We acknowledge the CINECA award under the ISCRA initiative, for the availability of high-performance computing resources and support. We also acknowledge the support and computational resources provided by  E4 Computer Engineering S.p.A.

Competing interests The authors declare no competing interests.

Data availability The instances, the dataset with the features, the results of the solvers and an example script to train meta-models are available at this GitHub repository: <https://github.com/qcpolimi/QA-MetaLearning>.

Author contribution All authors contributed to the study conception and design.
M. Ferrari Dacrema conceived the methodology.
Material preparation, data collection and analysis were performed by R. Pellini.
The first draft of the manuscript was written by R. Pellini and all authors contributed to the final version. All authors read and approved the final manuscript.





§ QUBO FORMULATIONS OF OPTIMIZATION PROBLEMS


In this Appendix, we show the QUBO formulations of the ten optimization problems we selected for our study.
We divide the problems in two different groups: problems defined over a graph, explained in Section <ref>, and problems not defined over a graph, explained in Section <ref>.



 §.§ Graph Problems





  §.§.§ Max-Cut

Given the graph G = (V, E), a cut over graph G induced by the set of vertices S and V - S is the set of edges which connect vertices in S with vertices in V-S.
The Max-Cut problem aims at finding the largest cut which can be induced on graph G <cit.>.
Consider, for each  vertex i of graph G, a binary variable x_i, such that

    x_i = 
            1  if  i ∈ S 
    
            0  otherwise

Then, the Max-Cut problem is formulated as written in Equation <ref>

    min_x y = -∑_(i,j) ∈ E(x_i + x_j - 2x_i x_j)




  §.§.§ Maximum Independent Set


Given the graph G = (V, E), an independent set is a set S of vertices which are not adjacent to each other.
The Maximum Independent Set problem aims at finding the largest independent set in G.
Consider, for each vertex i of graph G, a binary variable x_i, such that

    x_i = 
            1  if  i ∈ S 
    
            0  otherwise

Then, the Maximum Independent Set problem can be formulated as a QUBO problem as it is written in Equation <ref> <cit.>.

    min_x y = -a∑_i ∈ Vx_i + b∑_(i,j) ∈ Ex_i x_j

The penalty term b must outweigh the term a, to penalize the choice of having two connected vertices in S.



  §.§.§ Minimum Vertex Cover

Given the graph G = (V, E), a Vertex Cover C is a set of vertices such that all the edges E are connected at least to a vertex in C.
The Minimum Vertex Cover is the Vertex Cover with the smallest number of vertices <cit.>.
Consider, for each  vertex i of graph G, a binary variable x_i, such that

    x_i = 
            1  if  i ∈ C 
    
            0  otherwise

The Minimum Vertex Cover can be formulated as a QUBO problem as written in Equation <ref>.

    min_x y =  ∑_i ∈ V x_i + p ·∑_(i,j) ∈ E(1 - x_i - x_j + x_i x_j)

As for the Maximum Independent Set problem, also in the Minimum Vertex Cover problem the penalty term p must be chosen large enough, to penalize the selection of a set of vertices which is not a vertex cover of G.



  §.§.§ Max-Clique

Given the graph G = (V, E), a clique C is a sub-graph of G such that C is fully connected.
The Max-Clique problem aims at finding the clique C of G with the highest number of nodes.
The QUBO formulation of the Max-Clique problem has the same QUBO formulation of the Maximum Independent Set, but it leverages the complement graph G of G <cit.>.
Please refer to the Section <ref>.



  §.§.§ Community Detection

Given the graph G = (V, E), the aim of the Community Detection problem is to partition G in two communities C_1, C_2 of similar nodes <cit.>.
Remember that the graph G is represented by the adjacency matrix A, where the  (i,j) entry is A_ij = 1 if nodes i and j are connected by an edge, otherwise it is A_ij.
The degree vector d keeps track of how edges are incident to every node of the graph.
The similarity between nodes is expressed by the modularity matrix B, which is computed according to the following formula:
    
    B = A - dd^T/2|E|

    The binary variable x_i is related to the node i ∈ V of graph G and it is defined as follows:
    
    x_i = 
                1     if  i ∈ C_1 
    
                0     if  i ∈ C_2

    The matrix Q is proportional to B. In particular, it is equal to:
    
    Q = -1/|V|B

    The negative sign is due to the fact that QUBO problems require to be minimization problems in order to be solved with a QA.
    The formulation of the problem is therefore given by:
    
    min_x y = -1/|V| x^T Bx


    


 §.§ No-Graph Problems





  §.§.§ Number Partitioning

Given a set of real numbers Z = {z_1, z_2, ..., z_n }, 
the Number Partition problem aims at finding two partitions Z_1, Z_2 of Z in order to minimize the following expression <cit.>:
    
    (∑_z' ∈ Z_1z' - ∑_z”∈ Z_2 z”)^2


The binary decision variable x_i is defined as follows:

    x_i = 
                1     if  z_i ∈ Z_1 
    
                0     if  z_i ∈ Z_2


The QUBO formulation of the Number Partitioning problem is then derived from the expression <ref>, and it can be written as follows:

    min_x y = ( ∑_i=1^|Z|x_i z_i - ∑_i=1^|Z|(1-x_i) z_i)^2

    


  §.§.§ Set Packing

Given a collection of n sets S_1, S_2, ..., S_n, each one with a given capacity c_1, c_2, ..., c_n, the Set Packing problem aims at finding a selection of sets providing the largest total capacity, while respecting m constraints for the selection of the sets.
The binary decision variable x_i is associated to the set S_i, in particular:

    x_i = 
                1     if  S_i  is selected 
    
                0     otherwise

The Set Packing problem can be formulated as it follows:

    max_x y =    ∑c_i x_i
    s.t.   ∑a_ji x_i ≤ 1,    j = 1,2,...,m,    a_ji∈{0,1}


In order to map the constraints of the Set Packing problem into a quadratic penalty term, we leverage the following conversion rule <cit.>:
    
    ∑_i=1^na_ji x_i≤ 1 → p ·∑_i=1∑_k>ia_ji a_jk x_i x_k

    where coefficients a_ji and a_jk are either 0 or 1.
    The objective function is therefore derived as
    
    min_x y = -∑c_i x_i + p ·∑_i=1^n∑_k>i^nx_i x_k ∑_j=1^m a_ji a_jk

    


  §.§.§ Quadratic Knapsack

Given a set of projects P_1, P_2, ..., P_n such that for each pair (P_i, P_j) it exists a joint revenue r_ij = r_ji, the Quadratic Knapsack problem aims at maximizing the total revenue of activating the projects under a budget constraint. 
The Quadratic Knapsack problem is formulated as it follows:
    
    max_x y =  ∑_i^n ∑_j^nr_ij x_i x_j
    s.t.∑_i^nc_i x_i ≤ b


    To formulate the Quadratic Knapsack problem in the QUBO formulation, we introduce m binary slack variables t_1, t_2, ..., t_m.
    Each slack variable has a budget coefficient c_t_j, which has to be chosen accordingly to the budget b, e. g. by stating that ∑_j^m c_t_j = b.
    With these type of contraints, we rewrite the problem using the formulation given in Equation <ref>, defined in Section <ref>.
    The QUBO formulation of the Quadratic Knapsack problem is therefore:
    
    min_x y = - ∑_i^n ∑_j^nr_ij x_i x_j + p ·(∑_i^nc_i x_i  + ∑_j^mc_t_j t_j  - b )^2

    


  §.§.§ Sudoku

A Sudoku can be considered as a constraint satisfaction problem <cit.>.
In our study, we analyze only small instances of Sudoku problems, so we consider Sudoku with a 4 × 4 grid and some fixed assignments. We call such problems 4 × 4-Sudoku
The binary variable we use is defined as
    
    x_(i,j),k = 
                1   if cell  (i,j)  has value  k 
    
                0  otherwise

There are four kinds of constraints in a Sudoku:

    
   Cell Constraints. 
 A cell can contain only one number;
    
    ∑_k = 1^4x_(i,j),k = 1   ∀ i,j ∈{1,2,3,4}

    
   Row Constraints. 
 Two cells in the same row must have distinct numbers;
    
    ∑_j = 1^4x_(i,j),k = 1   ∀ i,k ∈{1,2,3,4}

    
   Column Constraints. 
 Two cells in the same column must have distinct numbers;
    
    ∑_i = 1^4x_(i,j),k = 1   ∀ j,k ∈{1,2,3,4}

    
   Block Constraints. 
 Two cells in the same block must have distinct numbers;
    
    ∑_(i,j) ∈ Bx_(i,j),k = 1      ∀ k ∈{1,2,3,4}, 
       ∀ blocks  B

    There are totally 64 constraints in a 4 × 4-Sudoku problem.
    If some cells are already assigned, the number of constraints and the number of variables reduces.
    In particular:

    
  * If cell (i,j) is assigned, there exist no variable related to this cell;
    
  * If cell (i,j) is not assigned and the number of values it can have is m, then there exist m variables related to cell (i,j) and each variable refer to a possible value of the cell;


    We can map all the constraints into a single square matrix A, which has at most 64 rows and 64 columns.
    Each column is related to a variable x_(i,j), k, while each row refers to a constraint.
    Assume that x is a vector where each element is a variable x_(i,j), k.
    If 1⃗ is a vector of 64 elements, all equal to 1, the QUBO formulation of the 4 × 4-Sudoku problem is:
    
    min_x y = (Ax - 1⃗)^T(Ax - 1⃗)





  §.§.§ Feature Selection


Given a dataset 𝒟, the Feature Selection problem aims at finding the subset S ⊂ F = {f_1, f_2, ..., f_n}  of the best k features able to represent the dataset. This is done especially in machine learning, to reduce the number of features of a predictive model and therefore its complexity.
Assume to have n features. For each feature f_i, we have the binary decision variable x_i, defined as follows:

    x_i = 
            1     if  f_i  is selected
    
            0     otherwise

We model the Feature Selection problem using Pearson correlation Corr(·, ·) <cit.>. 
For the quadratic terms, we compute the correlation between two different features f_i, f_j of 𝒟.
For the linear terms, we compute the correlation between a feature f_i and the target t of 𝒟.
We can compute directly the QUBO matrix Q, where in particular its (i,j) element of the matrix Q is equal to:
    
    Q_ij = 
            
                Corr(f_i, f_j)     if  i ≠ j 
    
                - Corr(f_i, t)     otherwise

Suppose we want to select k features. To do so, we consider the following QUBO formulation of the Feature Selection problem:
    
    min_x y = x^TQx + (∑_i=1^nx_i - k)^2

Notice that, if less or more than k features are selected, the penalty term is larger than 0 and the objective function value gets worse.



§ INSTANCE GENERATION STRATEGIES


In this section, we explain the strategies we applied to generate the instances of the optimization problems we selected.
The strategies varies whether the problem is defined over a graph or not.
An instance of a problem is characterized by its structure, which is either the topology of a graph or a general title which refers to the strategy used to generate the constraints and the objective function, and the number of variables n.

In general, for all the problems we generate instances with a minimum of n_min variables to a maximum of n_max variables.
For every optimization problem, except for the 4 × 4-Sudoku and for the Feature Selection problem, there exist n_rep instances having a certain structure and with the same number of variables.
Such instances are however different between each other, thanks to graph tweaking (explained in Section <ref>) and to the random generation of the coefficient of the cost function and of the constraints.

For the small instances, we have chosen n_min = 27, n_max = 32 and n_rep = 1.
For the large instances, we have chosen n_min = 69, n_max = 99 and n_rep = 5.



 §.§ Graph Problems


The instance of a problem defined over a graph is totally determined by the topology of the graph.
The strategy we applied to generate instances of graph problems is essentially divided in two distinct phases: the first is the generation of the graph G of n nodes, according to a certain topology; the second is the generation of an instance, for every graph problem, defined over G.

We have selected four different graph topologies to generate the graph G: the Star topology, the Cycle topology, the 2d-grid topology and the Erdös-Rényi topology.
A key step in the generation of G is the random insertion and removal of a limited number of edges, resulting in the tweaking of graph G. In this way, we generate more graphs starting from a given topology and with n nodes, which however get tweaked in different ways. 
We ensure always that the tweaked graph is connected.
In particular, the graph tweaking process happens according to the following rules:

    
  * At most n_ins = ⌊n/6⌋ edges are inserted;
    
  * At most n_rem = ⌊n/8⌋ edges are removed;
    
  * To perform the tweaking, we randomly modify the adjacency matrix A of the original graph.
    We start the tweaking of A from the first element of the first row, A_11, and we proceed left-to-right, till the element in the last column and in the last row, A_nn.
    
  * If A_ij=0 and less than n_ins edges have been inserted, insert the edge (i,j) by setting A_ij = 1 with probability p_ins = 40%;
    
  * On the contrary, if A_ij=1 and less than n_rem edges have been removed, remove the edge (i,j) by setting A_ij = 0 with probability p_rem = 30%;
    
  * If the tweaked graph is not connected, repeat the procedure;


For the Minimum Vertex Cover and the Maximum Independent Set problems we have to set the value of the penalty term coefficient p.
For the Minimum Vertex Cover, we have set p = n, that is to the number of nodes of the graph G.
For the Maximum Independent Set, we have set p = 2n. 
In both cases, we have that an assignment which violates a constraint highly penalize the objective function and that it is not a solution of the instance.



 §.§ No-Graph Problems



For what concerns the problems not defined over a graph, to generate an instance we have to generate the matrices related to the objective function and to the constraints. In particular, we want to analyze satisfiable instances.
We discuss now the strategies we used to generate the instances of each problem we selected.



  §.§.§ Number Partitioning

To generate the instances of the Number Partitioning problem, we have to define the set Z to partition.
For simplicity, we generate only sets of integer numbers.
We selected three probability distribution to generate the set Z, in addition to a fourth strategy where Z contains all the numbers between 1 and the number of variables n.
The three probability distributions are: (i) an uniform distribution between 1 and 99; (ii) a geometric distribution with probability of success p = 0.02; (iii) a Poisson distribution of mean value μ = 50.



  §.§.§ Set Packing

An instance of the Set Packing problem is determined by the capacities of the n sets c_1, c_2, ..., c_n, the coefficients a_j,i of the constraints, and the coefficient of the penalty term p.

For what concerns the capacities c_i, we sample highest possible capacity coefficient, called c_max, from a uniform distribution between 10 and 39.
This number represents the highest possible capacity coefficient, called c_max.
Then, we generate the capacities c_1, c_2, ..., c_n of the sets by sampling them from to a uniform distribution between 1 and c_max.

To generate the coefficients a_j,i, we consider a matrix A where the element in the j-th row and in the i-th columns is equal to a_j,i. Notice that A has always n columns.
We have defined four strategies to generate matrix A:

    
  * A is rectangular, with n-1 rows and the elements a_i,i and a_i,i+1 are equal to 1, for i = 1,2,...,n-1. All the other elements are set to zero. We say that this matrix has a step structure.
    We provide below, as an example, a matrix of this kind with n = 5:
    
    A = [ 1 1 0 0 0; 0 1 1 0 0; 0 0 1 1 0; 0 0 0 1 1;   ]


    
  * A is rectangular, with a number of rows m randomly sampled from a uniform distribution between 2 and ⌊n/2⌋. All the columns of A have exactly one element equal to 1. We say that this matrix has disjoint rows structure.
    Below, we provide an example of a disjoint rows matrix, with n = 5 and m = 3:
    
    A = [ 1 0 0 1 0; 0 0 1 0 1; 0 1 0 0 0;   ]


    
  * A is a square matrix. The elements on the main diagonal are all equal to 1. Furthermore, for each row of A, a random off-diagonal element is set to 1 with probability p=60%. We say that this matrix has an almost diagonal structure;

    
  * A is rectangular, with a number of rows m sampled from a uniform distribution between 10 and ⌊n^2/2⌋.
    The elements of A are randomly generated between 0 and 1, with equal probability. We say that this matrix has a random structure;
    


Also the penalty term p must be chosen, in order to penalize the assignments which violate the constraints.
We use 100 steps Bayesian optimization to choose p, by maximizing the percentage of feasible solutions found by Simulated Annealing in 30 executions.
The range of values of p is determined by the sum c_all = ∑_i^n |c_i|. In particular:

    ⌊c_all/3⌋≤ p ≤ c_all




  §.§.§ Quadratic Knapsack

In the Quadratic Knapsack problem, to generate the instances we have to choose: (i) the values of the joint revenue r_ij; (ii) the values of the coefficients c_1, c_2, ..., c_n of the constraint; (iii) the budget b; (iv) the number m of slack variables t_i to introduce and their constraints coefficients c_t_i; (v) the value of the penalty term coefficient p.

We generate the joint revenue values by generating a matrix.
The general joint revenue value r_ij is the element in the i-th row and in the j-th column of a revenue matrix R. By definition of the Quadratic Knapsack problem, R is symmetric.
We generate R according to the following four strategies:

    
  * R is diagonal. The elements on the diagonal are sampled from a uniform distribution of integer numbers between 15 and 39. We say that the structure of R is diagonal;

    
  * R has the whole diagonal, with some other few random off-diagonal elements.
    The elements on the diagonal of R are sampled from a uniform distribution of integers numbers between 15 and 39. 
    Then, for every column of R, we set with probability 40% a random off-diagonal r_ij with an integer number sampled from a uniform distribution between 1 and 24; when the element is set, we make the matrix symmetric by dividing r_ij by 2 and setting r_ji = r_ij.
    In the case r_ji element is set again, when considering column j, the previous value of r_ji and r_ij is overwritten. We say that R has an almost diagonal structure.
    An example of almost diagonal 4 × 4 matrix is:
    
    [l]
            17.0     13.5     0.0     16.0 
    
            13.5     32.0     0.0     0.0 
    
            0.0     0.0     19.0     0.0 
    
            16.0     0.0    0.0     26.0


    
  * The on-diagonal elements of R are non-zero; also the elements immediately on the right and on the left of on-diagonal elements are non-zero; the on-diagonal elements are integer numbers sampled from a uniform distribution between 15 and 39; the off-diagonal elements immediately to the right of on-diagonal elements are sampled from the same distribution; then, the off-diagonal elements are divided by 2 and the matrix is made symmetric, by setting r_ij = r_ji, for every i, j ∈{1,2,..., n}. We say that R has an enlarged diagonal structure.
    An example of a 4 × 4 enlarged diagonal is:
    
    [l]
                15.0     11.0     0.0     0.0 
    
                11.0     25.0     9.5     0.0 
    
                0.0     9.5     31.0     17.0 
    
                0.0     0.0     17.0     37.0


    
  * R is generated randomly, every element is sampled from the uniform distribution of integer numbers between 15 and 39; then, all the elements below the diagonal are ignored and set to 0; finally, the off-diagonal elements are divided by 2 and the matrix is made symmetric, by setting r_ij = r_ji, for every i, j ∈{1,2,..., n}. We say that R has a random structure.


The coefficients c_1, c_2, ..., c_n which appear in the constraint are integer numbers sampled from a uniform distribution between 1 and 15.
The budget b is set as ϕ * ∑_i=1^n c_i, where ϕ∈ℝ is sampled from a uniform distribution between 0.20 and 0.70.

We have chosen to introduce four binary slack variables t_1, t_2, t_3, t_4 to transform the inequality constraints into equality constraints. The slack variables are respectively associated to the budget coefficients c_t_1, c_t_2, c_t_3, c_t_4. We choose also the values of these coefficients as it follows:
    
    c_t_1 = ⌊b/2⌋   
            c_t_2 = ⌊b/4⌋   c_t_3 = ⌊b/8⌋   c_t_4 = b - ∑_i=1^3 c_t_i


We use 100 steps Bayesian Search to choose p, by maximizing the percentage of feasible solutions found by Simulated Annealing in 30 executions.
The range of the values of p depends on the sum r_all = ∑_i=1^n∑_j=1^n|r_ij|.
In particular:

    1 ≤ p ≤ r_all




  §.§.§ Sudoku

To generate 4 × 4-Sudoku instances, we generated randomly 30 different solved 4 × 4-Sudoku games.
We then iteratively removed the value of a cell, chosen randomly, of the 4 × 4-Sudoku game.
Each time we remove a cell, we increase the number of variables of the related 4 × 4-Sudoku instance.
We ensure that the generated instance has a number of variables between 27 and 32.

Notice that it is not possible to build instances larger than 64 variables of the 4 × 4-Sudoku problem. For this reason, we generate only small instances for this kind of problem.



  §.§.§ Feature Selection

We generated the instances of the Feature Selection problem starting from the following public datasets[Datasets are available on the website OpenML: <https://www.openml.org>]: , , , , , , .
An instance of the Feature Selection problem depends on the dataset 𝒟, on the number of features to select k and on the target variable t.

Assume we generate m instances with a number of variables n, with n bounded between n_min and n_max.
To select the dataset D, we use an iterative procedure.
We start from the dataset  and we check if it has more than n = n_min non-target features: if this condition is true, we select this dataset and we reduce its number of features to n, by deleting randomly chosen features; otherwise, we go to the the next dataset, chosen according to the order we used to list them, and we repeat this check. After the last dataset, , we repeat staring from .
After that m features are selected, we continue the procedure by incrementing n.
The procedure goes on until we generate m instances with n = n_max variables. 

Assume that the dataset 𝒟 is the i-th dataset selected to generate an instance of n variables, in the iterative procedure, the number of features to select is computed as follows:

    k = ⌊n/5⌋ + i


For what concerns the target variable t, every dataset we selected has one or multiple target variables. 
Since this formulation can tackle only one target variable t, in case of multiple target variables we randomly select one of them and delete all the others.

For small instances, we considered n_min = 27, n_max = 32 and m = 4, for a total of 24 instances. The number of features k we select is bounded between 5 and 9.
For large instances, we considered n_min = 69, n_max = 99 and m = 5, for a total of 155 instances. The number of features we select is bounded between 13 and 23.



§ DEFINITION OF THE FEATURES


We introduce in this section the definitions of the features we used in our study.
A subset of these features is based on probability theory and statistics, another subset is based on graph theory, and other features are related to the study of the spectrum of matrices.



  
Gini index
Given a set X of positive numbers, the Gini index Gini(Y) is a real number which measures the degree of inequality between the values y ∈ Y <cit.>. It is comprised between 0 (all the values y ∈ Y are equal) and 1 (only one value of Y is different from 0).
Given an ordered collection of values Y = {y_1, y_2, ⋯, y_m}, such that 0 ≤ y_1 ≤ y_2 ≤⋯≤ y_m, the Gini index is computed as:
    
    g(Y) = 2∑_i=1^miy_i/m∑_i=1^my_i - n+1/n




  
Herdindahl-Hirschman index
Given a set Φ, called industry, composed of m firms F_1, F_2, ..., F_m, each one characterized by a market share S(F_i) ∈ [0,1], such that ∑_i=1^m F_i = 1.
The Herfindahl-Hirschman index (HHI) is a measure used in economics to quantify the competitiveness of an industry with respect to the market share of the firms which compose the industry <cit.>.
The HHI of inustry Φ is defined as:

    HHI(Φ) = ∑_F ∈ΦS(F)^2

Clearly, 0 < HHI(Φ) ≤ 1. In particular, HHI(Φ) = 1 when there exists only one firm inside industry Φ, that is Φ is a monopoly.
On the contrary, if we assume that all the firms have equal market share, the market is competitive and, if m → +∞, we have that HHI(Φ) → 0. 

If two firms F_i, F_j merge into a larger new firm F_z, we have that S(F_z) = S(F_i) + S(F_j). The HHI computed on the new industry, which include F_z, increases.



  
Shannon entropy
Given a random variable Y, distributed according to a certain distribution p_Y(y).
The Shannon entropy of Y measures the level of uncertainty in the distribution p_Y(y).
In the discrete and finite case, assuming that the possible values of Y are y_1, y_2, ..., y_m, Shannon entropy is defined as:

    Sh(Y) = -∑_i=1^m p_Y(y_i)log_2 p_Y(y_i)

In general, Shannon entropy can be greater than 1.
The maximum value of Shannon entropy occurs when Y is distributed according to a uniform distribution, where all the values of Y are equally probable and none of them can be predicted more easily than the others.



  
Condition Number
Given a matrix M, the condition number CN(M) measures how close is matrix M to be singular.
If M is symmetric and real, the CN(M) is computed according to the maximal and minimal eigenvalues of M, respectively λ_max and λ_min:

    CN(M) =  | λ_max/λ_min |




  
Radius of a Graph
Given a graph G(V, E), described by the adjacency matrix A.
The radius of graph G is the largest eigenvalue of A in absolute value.



  
Diameter of a Graph
Given a graph G(V, E), the shortest path between two nodes i, j is the sequence of edges having minimal cost which connects i and j.
The diameter of G is the length of the longest shortest path.
If G is not connected, the diameter is not defined.
In the case of negative weights on the edges that form a cycle, the diameter is not defined too.



  
Spectral Gap of a Graph
Given a graph G(V, E), described by the adjacency matrix A, the spectral gap of G the difference, in modulus, between the the two largest eigenvalues of A.
Another definition we use is related to the Laplacian L of the graph. In this case, the spectral gap is the smallest non-zero eigenvalue of the Laplacian L related to the graph.



  
Connectivity of Graph
Given a graph G(V, E), described by a positive semi-definite Laplacian matrix L, the connectivity is equal to the second smallest eigenvalue of L.



  
Connected Components of a Graph
Given a graph G(V, E), described by a positive semi-definite Laplacian matrix L, the number of connected components of the graph is equal to the multiplicity of the eigenvalue 0 of L.




§ DOMAINS AND COMPONENT SETS




 §.§ Matrices of an Ising Graph


We call Ising graph a graph having the coupling matrix J of an Ising problem as adjacency matrix. Each node of the graph corresponds to a variable of the Ising problem.
The nodes of an Ising graph has a weight too, determined by the bias vector b of the Ising problem.
In the case the Ising problem is mapped onto the topology of a Quantum Annealer, we call it embedded Ising graph; otherwise, we refer to it as logical Ising graph.
We call structural adjacency matrix A of an Ising graph the adjacency matrix related to the unweighted version of the Ising graph.

The degree matrix D_J of the Ising graph is computed starting from the coupling J, while the structural degree matrix D_A of the Ising graph is computed starting from A.

The Laplacian L of an Ising graph is computed according to the following formula:

    L = D - J

The structural Laplacian L_A is computed using A instead of J and D_A instead of D_J.

The normalized adjacency matrix A_N is computed according the following formula:

    A_N = D_A^-1/2 A D_A^-1/2

The normalized Laplacian L_N is instead equal to:

    L_N = D_A^-1/2 L_A D_A^-1/2




 §.§ Domains and Related Features

We list here all the features computed in every domain we consider.
We refer to the definitions of the features given in Appendix <ref>.



  §.§.§ Embedding and Logical Ising Graph (EmbIsing and LogIsing)

In these domain, we gather features related to the embedded and logical Ising graph.
We compute the features starting from the following mathematical objects: bias, coupling, Laplacian, degree, structural adjacency, structural degree, structural Laplacian, normalized adjacency and normalized Laplacian matrices.
In both the domains, we compute the same features, but the considered mathematical objects vary in size and values.
For both the domains, we use the notation  to refer to the feature , in lowercase letters, computed on the mathematical object .



  
Coupling
Given the coupling J, we compute:

    
  * : Gini index of the eigenvalues of J. Since eigenvalues may be negative, we shift all the eigenvalues by the subtracting to all of them the smallest eigenvalue of J;
    
  * : HHI of the eigenvalues of J. The share of the eigenvalues is equal to their multiplicity divided by n;
    
  * : Shannon entropy of the eigenvalues of J. The probability of an eigenvalue is equal to its multiplicity divided by n;
    
  * : condition number of J;
    
  * : radius of the graph described by adjacency matrix J;
    
  * : diameter of the graph described by the adjacency matrix J. It may be not defined;
    
  * : spectral gap of the graph described by the adjacency matrix J;
    
  * : smallest eigenvalue of J;
    
  * : largest eigenvalue of J.




  
Bias
Given the bias b, we compute:

    
  * : Gini index of the values of b. Since some values of b may be negative, the values are all shifted by subtracting the minimal value of b;
    
  * : HHI of the values of b. The share of the values corresponds to their multiplicity divided by n;
    
  * : Shannon entropy of the values of b. The probability of a value corresponds to its multiplicity divided by n;
    
  * : the minimal value of b;
    
  * : the maximal value of b;
    
  * : the fraction between the maximal and the minimal value of b, in absolute value.




  
Degree
Given the degree matrix D of an Ising graph, we compute:

    
  * : Gini index of the eigenvalues of D. Since eigenvalues may be negative, we shift all the eigenvalues by the subtracting to all of them the smallest eigenvalue of D;
    
  * : HHI of the eigenvalues of D. The share of the eigenvalues is equal to their multiplicity divided by n;
    
  * : Shannon entropy of the eigenvalues of D. The probability of an eigenvalue is equal to their multiplicity divided by n;
    
  * : minimal eigenvalue of D;
    
  * : maximal eigenvalue of D;
     
  * : condition number of D.




  
Laplacian
Given the Laplacian matrix L of an Ising graph, we compute:

    
  * : Gini index of the eigenvalues of L. Since eigenvalues may be negative, we shift all the eigenvalues by the subtracting to all of them the smallest eigenvalue of L;
    
  * : HHI of the eigenvalues of L. The share of the eigenvalues is equal to their multiplicity divided by n;
    
  * : Shannon entropy of the eigenvalues of L. The probability of an eigenvalue is equal to its multiplicity divided by n;
    
  * : minimal eigenvalue of L;
    
  * : maximal eigenvalue of L;
    
  * : the second smallest eigenvalue of L. We compute it also if L is not positive semi-definite;
    
  * : the smallest non-zero eigenvalue of L;
    
  * : multiplicity of eigenvalue 0 of L. We compute it also if L is not positive semi-definite.




  
Structural and Normalized Matrices
The features computed for the structural adjacency, degree and Laplacian matrices and for the normalized adjacency and Laplacian matrices are the same computed for their weighted counterparts. The main difference is that the structural and the normalized matrices are positive-semidefinite and there is no need to shift the eigenvalues to compute the Gini index.



  
Graph Structure in EmbIsing Domain
In the EmbIsing domain, we compute also two features related to the size of the Embedded Ising graph.
In particular, we compute:

    
  * : number of nodes of the Embedded Ising graph, which corresponds to the number of qubits used to solve an Ising problem with the Quantum Annealer;
    
  * : number of chains of nodes in the Embedded Ising graph; a chain represents a single node of the Logical Ising graph, a single variable of the original Ising problem.




  §.§.§ Matrix Structure (MatStruct)

In this domain, we compute the features related to the values of the QUBO matrix Q.
In particular, we compute:

    
  * : Gini index of the values of Q. We subtract the minimal value of Q from all the elements to guarantee that the values are positive;
    
  * : HHI of the values of Q. The share of an element of Q is equal to its number of occurrences inside Q, divided by n^2;
    
  * : Shannon entropy of the values of Q. The probability of an element of Q is equal to its number of occurrences inside Q, divided by n^2.




  §.§.§ Solution Space (SolSpace)

In this section, we compute features related to the eigenvalues of the Hamiltonian of the problem H_p, related to a QUBO problem.
A generic eigenvalue of H_p is λ_i.
Assume that λ_min is the minimal eigenvalue of H_p. Remember that, if the QUBO problem has n variable, H_p has 2^n eigenvalues.
In this domain, we compute:

    
  * : Gini index of the eigenvalues of H_p. To have all positive values, eigenvalues are shifted by subtracting λ_min to all of them;
    
    
  * : HHI of the eigenvalues of H_p. We use the value 
    
    S(λ_i) = λ_i-λ_min/∑_j=1^2^n(λ_j - λ_min)

    as the share of an eigenvalue λ_i;

    
  * : HHI of the eigenvalues of H_p, where identical eigenvalues are considered together. The share of the eigenvalue λ_i is equal to the sum of the shares of the eigenvalues equal to λ_i, computed according to the formula <ref>.
    
    
  * : Shannon entropy of the eigenvalues of H_p. We use as probability of eigenvalue λ_i the value described in Equation <ref>;
    
  * : the minimal eigenvalue of H_p;
    
  * : the eigenvalue of the first quartile of H_p;
    
  * : the median eigenvalue of H_p;
    
  * : the eigenvalue of the third quartile of H_p;
    
  * : the maximal eigenvalue of H_p.





  §.§.§ Normalized Multiplicity (NorMul)

In this section, we compute features related to the multiplicity of the eigenvalues of the Hamiltonian of the problem H_p, related to a QUBO problem. Assume that there are m different eigenvalues of H_p.
The multiplicity of the eigenvalue λ_i of H_p is equal to μ_i. We call normalized multiplicity π_i of λ_i the value:

    π_i = μ_i/∑_j=1^m μ_j

In this domain, we compute the following features:

    
  * : Gini Index of the normalized multiplicities of the eigenvalues of H_p;
    
  * : HHI of eigenvalues of H_p, where we use their normalized multiplicities as shares;
    
  * : Shannon entropy of the eigenvalues of H_p, where we use their normalized multiplicities as probabilities;
    
  * : the normalized multiplicity of the smallest eigenvalue of H_p.




  §.§.§ 25%-SolSpace and 25%-NorMul

The features computed in these domains are the same computed in the domain SolSpace and NorMul.
The only difference is that here, if consider the eigenvalues in ascending order, we consider only the eigenvalues of H_p in the first quartile.



 §.§ Component Sets


Component sets gather all the features computed on the same mathematical objects, but in different domains.
In these study, all the component sets contain features computed in both the LogIsing and EmbIsing domains.
For every component set, we use the notation  to refer to the feature , written in lowercase letters, computed in the domain  on the mathematical object related to the component set.
The component sets we consider are:

    
  * Bias: it contains the features related to the bias;
    
  * Coupling: it contains the features related to the coupling;
    
  * Laplacian: it contains the features related to the Laplacian matrix;
    
  * StructAdj: it contains the features related to the structural adjacency matrix;
    
  * StructLap: it contains the features related to the structural Laplacian matrix;




 §.§ 
Complexity in Features Computation

The major complexity in the computation of the features is given by the computation of the Hamiltonian of the problem, H_p.
To do so it is in fact necessary to compute a 2^n real-valued vector, with n number of QUBO variables, according to Equation <ref>. This of course is related only to the small instances.

For what concerns the features related to the large instances, the bottleneck lies in computing all the minor-embeddings of the instances: minor-embedding is, indeed, an NP-Hard problem.

Once H_p and the minor-embedding of the instances have been computed, all the features are easily computed without any particular complexity
.



§ HYPERPARAMETERS OF THE SOLVERS



  
Tabu Search

We optimize the number of restarts of the algorithm.
The default number of restarts is 100000.
Table <ref> contains the optimal hyperparameters found for each problem class.





  
Simulated Annealing
We optimized the number of  and the  type.
The default value of the number of  is 1000, while the defualt  is .
Notice that we solve the instances also with Simulated Annealing with no hyperparameters optimization and we obtained better results.
Table <ref> contains the optimal hyperparameters found for each problem class.





§ ADDITIONAL RESULTS ON THE EFFECTIVENESS OF SOLVERS










 Paper Title: Experimentally probing Landauer's principle in the quantum many-body regime.






These authors contributed equally to this work: 
mailto:stefan.aimet@gmail.comstefan.aimet@gmail.com, mailto:amintajik.physics@gmail.comamintajik.physics@gmail.com
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

These authors contributed equally to this work: 
mailto:stefan.aimet@gmail.comstefan.aimet@gmail.com, mailto:amintajik.physics@gmail.comamintajik.physics@gmail.com
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
Department of Physics and Astronomy, and Stewart Blusson Quantum Matter Institute,
University of British Columbia, V6T1Z1 Vancouver, Canada

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Institute of Theoretical and Computational Physics, University of Crete, 71003 Heraklion, Greece
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Dipartimento di Fisica,  Università di Pavia, 27100 Pavia, Italy
Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

§ ABSTRACT 
Landauer’s principle bridges information theory and thermodynamics by linking the entropy change of a system during a process to the average energy dissipated to its environment. Although typically discussed in the context of erasing a single bit of information, Landauer’s principle can be generalised to characterise irreversibility in out-of-equilibrium processes, such as those involving complex quantum many-body systems. 
Specifically, the relationship between the entropy change of the system and the energy dissipated to its environment can be 
decomposed into changes in quantum mutual information and a difference in relative entropies of the environment. 
Here we experimentally probe Landauer’s principle in the quantum many-body regime using a quantum field simulator of ultracold Bose gases. Employing a dynamical tomographic reconstruction scheme, we track the temporal evolution of the quantum field following a global mass quench from a Klein-Gordon to a Tomonaga-Luttinger liquid model and analyse the information-theoretic contributions to Landauer’s principle for various system-environment partitions of the composite system. Our results agree with theoretical predictions, interpreted using a semi-classical quasiparticle picture. Our work demonstrates the potential of ultracold atom-based quantum field simulators to experimentally investigate quantum thermodynamics.



Experimentally probing Landauer's principle in the quantum many-body regime
    Jens Eisert
    Received ..... .., ....; accepted ......, ....
===========================================================================

Information theory and thermodynamics constitute foundational pillars of modern technology, underpinning our understanding of modern computers and heat engines, respectively. While these disciplines may seem distinct, they are intricately connected. This connection is encapsulated in Landauer's principle, first articulated in 1961 <cit.>. Landauer understood that erasing a bit of information in a computer is not for free but is instead accompanied by a minimal energy cost dissipated to the environment. Subsequently, various experimental studies <cit.> have confirmed this lower bound on energy dissipation for near-reversible bit erasure. Logical operations thus imply irreversibility, posing a fundamental theoretical limitation to the design of any small-scale energy-efficient information-processing technologies <cit.>. However, extending beyond the case of bit erasure, recent influential work <cit.> has generalised the link between information theory and thermodynamics. Using a quantum statistical mechanics framework, their work reinterprets Landauer's principle to relate the entropy change of a system to the energy dissipated to its environment in very general out-of-equilibrium processes. This relationship can be quantified by a measure of process irreversibility <cit.>. Such an extension is thus particularly relevant for quantum many-body systems, where contributions to irreversibility remain an area of active research, notably in phenomena such as equilibration and thermalisation <cit.>. 

In this work, we experimentally probe Landauer's principle for an out-of-equilibrium process in the quantum many-body regime by measuring the time evolution of quantum information-theoretic quantities that characterise irreversibility. We present a crisp information-inspired interpretation of correlations present in states of quantum many-body systems in terms of entropic expressions. 

Concretely, we consider a system-environment composite initially in the state ϱ_ SE(0), evolving under a global unitary time evolution U to the state ϱ_ SE(t)=Uϱ_ SE(0)U^†. The reduced state of the system S (environment E) is ϱ_ S(E)(t)=_E(S)[ϱ_ SE(t)]. 

First, let us assume that there are no initial correlations between the system and environment, and the environment is in a thermal state γ_ E^β_ E=e^-β_ EH_ E/[e^-β_ EH_ E] at inverse temperature β_ E with respect to its Hamiltonian H_ E. In this case, the entropy production Σ(t), which is a measure of irreversibility <cit.>, can be decomposed as

    Σ(t):=I_ SE(t)+D(ϱ_ E(t)||γ_ E^β_ E)  .

Here, the quantum relative entropy

    D(ϱ_ E(t)||γ_ E^β_ E)=[ϱ_ E(t)logϱ_ E(t)]-[ϱ_ E(t)logγ_ E^β_ E]   ,

quantifies the deviation of the environment from its initial thermal state. The quantum mutual information 

    I_ SE(t)=S(ϱ_S(t))+S(ϱ_E(t))-S(ϱ_SE(t))

measures the system-environment correlations, where 

    S(ϱ)=-[ϱlogϱ]

denotes the von Neumann entropy of a state ϱ.



Now we move on to a more general scenario where the system and the environment are initially correlated, and the state of the environment deviates from thermal equilibrium. We thereby introduce the generalised entropy production ΔΣ :=Σ(t)-Σ(0) where the initial correlations and deviations from thermal equilibrium are accounted for by a resource cost Σ(0)=I_ SE(0)+D(ϱ_ E(0)||γ_ E^β_ E) <cit.>, with β_ E being an effective inverse temperature <cit.>. Landauer's principle can now be formulated <cit.> as an equality as 

    ΔΣ=β_ EΔ E_ E+Δ S=Δ I+Δ D   ,

relating the system's entropy 
change Δ S:=S(ϱ_ S(t))-S(ϱ_ S(0)) to the energy dissipated to the environment Δ E_ E:=[(ϱ_ E(t)-ϱ_ E(0))H_ E].
Alternatively, ΔΣ also decomposes into the change of quantum mutual information, Δ I=I_ SE(t)-I_ SE(0), expressing how much the system-environment correlations change along the out-of-equilibrium process, as well as the term Δ D=D(ϱ_E(t)||γ_E^β_E)-D(ϱ_E(0)||γ_E^β_E) that quantifies how much both the initial and final state of the environment are different from the reference equilibrium thermal state. We provide a more thorough motivation for the generalised entropy production ΔΣ in the Methods section. 

Expressed in this form, Landauer's principle serves as a means of tracking changes in information-theoretic quantities, which compose generalised entropy production ΔΣ in two different ways via a single equation. Herein lies the power at the heart of thermodynamics: just as knowing the precise microscopic degrees of freedom, such as the position and momentum of each particle in a classical gas, is not useful for drawing meaningful conclusions about its dynamics, knowing the density matrix of a quantum many-body system is similarly not descriptive. In traditional thermodynamics, macroscopic variables like changes in pressure or volume effectively characterise the dynamics. Similarly, analysing the information-theoretic quantities in Landauer's principle allows for a simple characterisation of the irreversibility in the out-of-equilibrium dynamics of a quantum many-body system.

So far, there has been no systematic experimental pursuit of probing Landauer's principle in the quantum many-body regime. Here, we address this gap by measuring the time evolution of the different terms in Eq. (<ref>) in a (1+1)D quantum field simulator of tunnelling-coupled ultracold Bose gases. This experimental platform has proven to be a suitable testbed for simulating one-dimensional quantum field theories. Previous studies using this setup have examined the area law of quantum mutual information in thermal equilibrium <cit.> 
and the propagation of second- <cit.> or higher-order correlations <cit.>. 

In this experiment, two parallel ultracold clouds of  atoms (illustrated in Fig. <ref>) are confined in highly anisotropic magnetic traps produced by an atom chip <cit.>. In the axial direction (z), the clouds are confined in a parabolic trap superposed with an optical dipole potential to produce hard walls, while in the radial directions (x,  y), by a double-well trap with an adjustable barrier in between, created by RF dreessing <cit.>. The single-particle tunnelling rate can be modified by the amplitude of a radio-frequency field created by two parallel wires on the atom chip.

The bosonic quantum field operator for each condensate can be written using the phase-density representation as ψ_n(z) = √(ρ_n(z))e^iθ_n(z), where θ_n and ρ_n denote the phase and density of the respective condensate, indexed by n=1,2. We prepare the Bose-Einstein condensates in a global thermal state of a massive Klein-Gordon Hamiltonian 

    H_ KG   = ∫_0^L dz [ g_ 1Dδρ^2(z) + ħ^2 n_ 1D/4m (∂_zφ(z))^2 
        + ħ J n_ 1Dφ^2(z) ]

which models low-energy phononic excitations. Here, ħ is the reduced Planck constant, m is the atomic mass, g_ 1D is the effective one-dimensional atomic interaction strength, L=49 is the axial length of the condensates, n_ 1D≈ 70^-1 is the average linear density and J≈ 2π× 0.8 Hz is the tunnelling rate introduced before. The operators φ(z) = θ_1(z) - θ_2(z) and δρ(z) = [ρ_1(z) - ρ_2(z)]/2 represent the relative phase and relative density, respectively, and satisfy the commutation relation [φ(z), δρ(z')] = iδ(z-z'). 

To initiate out-of-equilibrium dynamics in the experimental system and measure the evolution of information-theoretic quantities, we rapidly quench J to zero by ramping up the barrier between the condensates within approximately 2. This change corresponds to a global mass quench of the Klein-Gordon Hamiltonian. The condensates then evolve independently under the post-quench Tomonaga-Luttinger liquid Hamiltonian <cit.> (H_KG with J=0) for times t, up to 65.



 

For every time step t, we turn off all the traps and let the atoms fall freely for 15.6. The clouds will expand and interfere, resulting in absorption pictures similar to the one shown in Fig. [fig:setupandtomography]<ref>fig:setupandtomographyb, allowing us to measure the spatially resolved relative phase φ(z) between them. Since the detection process is destructive, the measurements are repeated to gather statistics. In our current experimental setup, relative density fluctuations δρ are not directly measurable, which prompted the development of a dynamical tomographic reconstruction technique <cit.> to access all of the elements of the covariance matrix 

    Γ(t)=[ Γ_φφ(t) Γ_φρ(t); Γ_ρφ(t) Γ_ρρ(t);         ]  .

Here, the elements are defined as [Γ_φφ(t)]_m,n=⟨φ(z_m, t)φ(z_n, t)⟩, [Γ_ρρ(t)]_m,n=⟨δρ(z_m,t)δρ(z_n,t)⟩, and [Γ_φρ(t)]_m,n=[Γ_ρφ(t)]^T_m,n=⟨1/2{φ(z_m, t),δρ(z_n, t)}⟩, all on a discrete grid with N pixels (m,n∈{1,…,N}), which are determined by resolution constraints of our imaging system, thus introducing an ultraviolet (UV) cutoff.

The phase–phase correlations for different evolution times after the quench can be measured directly from the extracted relative phase profiles. Assuming that the short-time dynamics after the quench are governed by the Tomonaga–Luttinger liquid (TLL) Hamiltonian, as time progresses, the initial eigenmodes of the relative density transform into the phase quadrature, and the phase quadrature transforms into the relative density. This transformation allows us to extract information about these eigenmodes by fitting the initial second-order correlation functions of phase–density, and density–density with the observed evolution of the phase–phase correlations in momentum space. 

The schematics in Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc illustrate our dynamical tomographic reconstruction scheme. We follow the technique in Ref. <cit.> for various input intervals with varying starting points. By scanning the starting points of these input intervals throughout the trapping times, we can reconstruct the full covariance matrix at every time t, as depicted in Fig. <ref>. The interval length (32.5 close to L/c ≈ 27) is selected to be sufficiently long for the slowest eigenmode to acquire enough dynamical phase for a stable reconstruction, yet short enough to prevent mode interactions from affecting the reconstruction. A detailed overview of this reconstruction process is provided in the Methods section.

The quadratic form of the pre- and post-quench Hamiltonians allows us to work within the framework of Gaussian quantum information theory <cit.>. In this framework, the covariance matrix Γ captures all the accessible information about the state of the composite system during the dynamics, from which we can extract the information-theoretic quantities of interest in Eq. (<ref>). As has been demonstrated in various experiments <cit.>, the quadratic approximation of the Hamiltonian accurately captures the dynamics for the considered time scales. Even if the true dynamics deviate from the Gaussian regime and violate Tomonaga-Luttinger liquid theory, a Gaussian extremality argument presented in the Supplementary Material justifies the Gaussian tomography scheme and provides bounds on the information-theoretic quantities. Thus, based on our technique of dynamical tomographic reconstruction of covariance matrices, Landauer's principle can also be meaningfully experimentally probed for interacting models.

Having experimentally reconstructed the post-quench time evolution of the covariance matrices, we partition the one-dimensional field of length L into two distinct subregions and split the covariance matrix accordingly (Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc). Since the quantum field is isolated from its surroundings, one subregion serves as the system S with length L_S, while the other subregion functions as the environment E with length L_E = L - L_S. We probe Landauer's principle (Eq. (<ref>)) for various system-environment bipartitions by calculating β_ EΔ E_ E, Δ S, Δ I and Δ D. The results are presented in Fig. <ref> as a function of time for different subregion size ratios and in Fig. <ref> as a function of subregion size for different times. The time scales are shown in units of ct/L, where c=√(g_ 1Dn_ 1D/m)≈ 1.8  / is the speed of sound. Overall, a very good fit of the data (circles) is obtained compared to the theoretical calculations (shaded areas), which use the lowest N=7 modes and consider the imaging resolution of the experiment. The low-lying modes already capture the dynamics of the continuum theory very well. The error bars representing the 68% confidence intervals are obtained via bootstrapping <cit.> with 999 samples, considering the uncertainty in the tunnelling rate and the estimated initial global temperature. The effective inverse temperature of the environment β_ E with respect to the post-quench Tomonaga-Luttinger liquid Hamiltonian is computed using field-theoretic simulations by constructing a global thermal state of the Klein-Gordon Hamiltonian with the estimated initial global temperature from the experimental data. 

First, we examine the decomposition of the generalized entropy production ΔΣ into β_ EΔ E_ E and Δ S. The term associated with the energy dissipated to the environment, β_ EΔ E_ E, reveals that a small amount of energy flows into (or out of) the environment for small (or large) systems around ct/L_ S=1. However, this term contributes only minimally to the irreversible post-quench dynamics of the quantum field. In contrast, the entropy change of the system, Δ S, dominates the dynamics and shows a clear linear increase up to ct/L_ S<1, followed by more pronounced growth.

In comparison, the second decomposition involving Δ I and Δ D sheds a slightly different focus on the out-of-equilibrium dynamics. The largest contribution to ΔΣ, the change in quantum mutual information, Δ I, exhibits similar behaviour to Δ S but with a higher magnitude. This increased magnitude is accounted for by the additional entropy change in the environment. However, Δ I competes with the term proportional to the change of the environment, Δ D, which accounts for both the entropic and energetic changes to the environment. This makes Δ D a particularly interesting quantity to characterise the out-of-equilibrium dynamics of the environment experimentally. In our case, Δ D increases negatively, mirroring otherwise the behaviour of Δ S, due to the small energetic contribution. 

To interpret the results, we need to consider the effects of Neumann and Dirichlet boundary conditions on the quantum field, as shown with field-theoretic simulations in Fig. <ref>. These boundary conditions may be contrasted with the curved
backgrounds giving rise to effective boundary conditions discussed in Refs. <cit.>. The experimental system exhibits Neumann boundary conditions (∂_zφ(z)|_z=0, L=0) due to the vanishing particle current at the edges. Plots contrasting the scaling with subregion size are shown in the Supplementary Material. 

However, we first consider simulations for Dirichlet boundary conditions (φ(z)|_z=0, L=0), which are simpler to understand. Here, the well-established semi-classical quasiparticle picture <cit.> offers a clear and intuitive model for the post-quench dynamics of global mass quenches <cit.>, depicting the propagation of short-range initial correlations via ballistically moving quasiparticles. Following the homogeneous global quench, the initial Klein-Gordon thermal state becomes a non-equilibrium state with excess energy relative to the post-quench Tomonaga-Luttinger liquid Hamiltonian, which governs the time evolution. This initial state acts as a source of quasiparticle pairs emitted globally from every point across the length of the composite system. Each pair in the bulk consists of two quasiparticles correlated with each other and moving in opposite directions at the same speed. For our set of parameters, due to the finite initial correlation length, there exists a small localised region where quasiparticles are correlated.

In this quasiparticle picture, the change of quantum mutual information Δ I between the system and environment is proportional to the number of pairs of quasiparticles that are shared between the two different subregions. The spread of correlations of short-range interacting models is described by a linear effective light cone originating at the system-environment boundary. The increase in correlations is thus proportional to the section of the light cone, or in other words, it is proportional to the distance between the quasiparticles of the pair emitted at the system-environment boundary (Fig. [fig:DBC_main]<ref>fig:DBC_mainc). If the system is smaller than the environment, once the first quasiparticle of this boundary pair reaches the edge of the system at ct/L_ S=1 and undergoes reflection, the section of the system-environment boundary pair light cone remains constant. The behaviour of Δ I transitions to a plateau value. From a system perspective, the composite system appears to have locally equilibrated to a generalised Gibbs ensemble steady state <cit.>. For finite-size composite systems, the plateau eventually ends again at ct/L_ E=1 when the other quasiparticle of the boundary pair reaches the edge. The distance between the two quasiparticles decreases until they meet again at ct/L=1, explaining the phenomenon of recurrences <cit.> (with respect to the correlation functions). 



On the other hand, Δ I behaves differently for Neumann boundary conditions. Neumann boundary conditions introduce a zero mode, whose variance does not evolve harmonically like the other momentum modes, but quadratically (phase diffusion <cit.>). The global fluctuations of the zero mode contribute to entropic quantities <cit.> and as they increase with time after a quench they eventually dominate over the contribution of other modes. Since the quasiparticle picture cannot capture this zero-mode feature, its predictions for ct/L_ S>1, when a plateau value should be reached, need to be modified under the present Neumann boundary conditions. The above features of the dynamics also apply to the change of system entropy Δ S. In addition, the zero mode restricts the validity of the tomography scheme to ct/L<1. 

The energetic contribution, β_ EΔ E_ E, can also be explained using the quasiparticle picture. Initially, β_ EΔ E_ E remains constant because quasiparticles traveling between the system and the environment carry equal energy, resulting in zero net energy flux. However, a finite-size energy flow becomes apparent when quasiparticles from the edge of the system cross the system-environment boundary. The translational invariance of the homogeneous quench is broken due to the higher energy density at the edges of the composite system for Neumann boundary conditions (the reverse is true for Dirichlet boundary conditions, where the energy density is lower at the edges). This edge region has a size of the order of the initial correlation length. The resulting behaviour of the change of the environment Δ D reflects both the entropic and energetic fluxes. 

In this study, we have experimentally probed Landauer's principle in the quantum many-body regime following a global mass quench in an ultracold atom-based quantum field simulator. By reconstructing the dynamics of the state of the composite system, we examined the information-theoretic quantities related by Landauer's principle, which we interpret using a semi-classical quasiparticle picture. Our approach underscores the general utility of Landauer's principle for characterising the irreversibility of out-of-equilibrium dynamics in quantum many-body systems. The way we express correlations in terms of entropic quantities with an 
information-theoretic meaning can be viewed as a vehicle to capture many-body correlations in 
non-equilibrium. The Gaussian extremality argument may offer a pathway to extend our methodology to capture features of out-of-equilibrium processes in interacting models, despite the challenges posed by non-Gaussian effects. Looking forward, progress has already been made towards investigating a local quench involving two Bose-Einstein condensates of different temperatures being joined together <cit.>. This protocol represents a crucial primitive towards developing a quantum field thermal machine <cit.>, potentially allowing for Landauer erasure as a mechanism to reduce the entropy of a subregion in the quantum many-body regime, hence functioning as an effective cooling mechanism. Our current work demonstrates the potential of ultracold one-dimensional gases as testbeds for quantum thermodynamics in the many-body regime, where complexity, quantum effects, and finite-size effects play a crucial role.


We thank Bernhard Rauer for conducting the experimental measurements. In addition, we are grateful to Marcus Huber, Patrick Emonts, Ivan Kukuljan, Mauro Paternostro, Philippe Faist, Florent Goulette, Marek Gluza, and Igor Mazets for helpful discussions and comments. This work has been supported by the DFG Research Unit FOR 2724 on `Thermal machines in the quantum world', the FQXi, the Quantum Flagship (`Millenion' and `PasQuans2'), the Einstein Research Unit, the BMBF, Berlin Quantum, and the ERC-AdGs `Emergence in Quantum Physics' and 
`Delineating the boundary
between the computational
power of quantum and
classical devices'.

S.S. acknowledges support by
the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No.  101030988. 



§ METHODS




 §.§ Landauer's principle

Consider a system S and an environment E with a bipartite Hilbert space ℋ=ℋ_S⊗ℋ_E. For an initial product state ϱ_SE=ϱ_S⊗γ_E^β_E, where the environment is in a thermal state γ_E^β_ E=e^-β_ E H_ E/[e^-β_E H_E], the entropy production under global unitary dynamics is given by <cit.>

    Σ(t)=I_ SE(t)+D(ϱ_E(t)||γ_E)=D(ϱ_ SE(t)||ϱ_S(t)⊗γ_E^β_ E)  .

Since it is expressed as a quantum relative entropy, Σ is non-negative due to Klein's inequality <cit.>, which is a manifestation of the second law of thermodynamics. More generally, as in our work, we start with a resourceful initial state that includes both initial correlations I_ SE(0) and athermality of the environment D(ϱ_E(0)||γ_E^β_ E). While defining temperature for a non-equilibrium state is ambiguous, we determine the effective inverse temperature of the environment β_ E by demanding that the environment's energy is equal to that 
of a Gibbs state, such that [ϱ_ EH_ E]≡[γ_E^β_ EH_ E] <cit.>. This choice also minimises the athermality term. Thus, we define the resource cost of the initial state <cit.>

    Σ(0)=I_ SE(0)+D(ϱ_E(0)||γ_E^β_ E)=D(ϱ_ SE(0)||ϱ_S(0)⊗γ_ E^β_ E).


Although there is no unique consensus on entropy production for initial non-equilibrium environment states <cit.> and initial correlations <cit.>, we define Landauer's principle Eq. (<ref>) in terms of generalised entropy production ΔΣ:=Σ(t)-Σ(0) as

    ΔΣ :=β_EΔ E_ E+Δ S=Δ I+Δ D  .

This can be derived similarly to Ref. <cit.>. One caveat is that ΔΣ may not necessarily be non-negative, which is a necessary criterion for a strictly valid definition of entropy production. However, this makes sense as ΔΣ may be considered a finite-difference version of the entropy production rate ΔΣ=Σ(t)-Σ(0)=∫_0^t dt' dΣ(t')/dt', which can also be negative <cit.>. Thus, ΔΣ intuitively measures the "extra" irreversibility associated with the protocol, excluding the initial resources.



 §.§ Experimental implementation

The experimental setup consists of a pair of tunnelling-coupled one-dimensional  quasicondensates, following a protocol implemented previously <cit.>. The magnetic fields required to trap the gas in a double-well potential are generated by an atom chip <cit.>, achieving transverse trapping frequencies of ω_⊥/2π = 1.4 kHz. After preparing a thermal equilibrium state at a temperature of 49 nK, all traps are turned off, and absorption imaging of the atoms is performed during a free-fall time of 15.6 ms to measure the two-dimensional atomic density distributions. This procedure enables us to detect the relative phase between the condensates across the length of the quasicondensate using matter interferometry (Fig.  [fig:setupandtomography]<ref>fig:setupandtomographyb). Due to the destructive nature of the detection process, many repetitions of the experiment are necessary to accumulate numerous phase correlation images throughout the time evolution, thereby also ensuring statistically accurate expectation values.



 §.§ Quantum field simulation

We have the Klein-Gordon Hamiltonian as a model for our quantum field simulator

    H_ KG=∫_0^Ldz [g_ 1Dδρ(z)^2 +ħ Jn_ 1Dφ(z)^2+ħ^2n_ 1D/4m(∂_zφ(z))^2] ,

where the length of the composite system is set to L =49 with a mean density of n_ 1D = 70^-1. The coupling strength between the particles is chosen as g_ 1D = 8.594× 10^-39kg·m^3·s^-2, and we have an atomic mass of m = 1.433× 10^-25kg.
The initial single-particle 
tunnelling rate is J= 2π× 0.76Hz. The post-quench evolution is described by the Tomonaga-Luttinger liquid Hamiltonian

    H_ TLL=∫_0^L dz[g_ 1D(δρ(z))^2+ħ^2 n_ 1D/4m(∂_zφ(z))^2
     ]  .




 §.§ Dynamical tomographic reconstruction of covariance matrices

The tomography scheme for reconstructing the initial thermal covariance matrix is detailed in Refs. <cit.>. Here, we extend this method to dynamically reconstruct the state over time. We provide a brief overview below.
Given Neumann boundary conditions ∂_zφ(z)|_z=0,L=0, the relative phase φ(z) and relative density fluctuations δρ(z) can be expanded in Fourier space in terms of cosine eigenfunctions

    f_k^φ(z) = 
    2(1/kħπ√(mg_ 1D/n_ 1D))^1/2cos(kπ/Lz)   if  k > 0 ,
    
    1    if  k = 0,
    
    
    f_k^ρ(z) = 1/L(kħπ√(n_ 1D/mg_ 1D))^1/2cos(kπ/Lz)   if  k > 0 ,
    1/L   if  k = 0.


The Hamiltonian of the post-quench Tomonaga-Luttinger liquid Eq. (<ref>) is then diagonalised as 
(indices omitted for brevity from now on)

    H=ħ u/2δρ_0^2+∑_k=1^∞ħω_k/2[δρ_k^2+φ_k^2]

with u:=2g_ 1D/ħ L and 

    ω_k:=√(g_ 1Dn_ 1D/m)kπ/L  .

The interaction strength g_ 1D is given by 

    g_ 1D:=ħω_⊥a_s2+3a_s n_ 1D/(1+2a_s n_ 1D)  ,

where a_s=5.2 nm is the three-dimensional scattering length.

Consequently, the time evolution of modes (k>0) in Fourier space is

    φ_k(t)=φ_k(0)cos(ω_k t)+δρ_k(0)sin(ω_k t),
    δρ_k(t)=δρ_k(0)cos(ω_k t)-φ_k(0)sin(ω_k t),

with non-harmonic time evolution of the zero mode

    φ_0(t)= utδρ_0(t)+φ_0(0),
    δρ_0(t)=δρ_0(0)= const   .

As mentioned in the main text and shown in Fig. <ref>, the covariance matrices can be dynamically tomographically reconstructed by measuring the relative phase profiles using matter interferometry at various times. Subsequently, the relative phase correlations are computed as
Φ^2_m,n(t)=⟨(φ(z_m,t)-φ(z_0, t))(φ(z_n,t)-φ(z_0, t))⟩ with z_0 as a reference position. 
We then expand in Fourier space, see Eq. (<ref>), and expand the time evolved operators to express the measured phase correlation function in terms of the covariance matrix elements prior to the time evolution

    Φ^2_m,n(t)=⟨(φ(z_m,t)-φ(z_0, t))(φ(z_n,t)-φ(z_0, t))⟩
    
    =∑_k,l=1^N-1f_k,l^m,n⟨φ_k(t)φ_l(t)⟩

where

    f_k,l^m,n=(f_k^φ(z_m)-f_k^φ(z_0))(f_l^φ(z_n)-f_l^φ(z_0))

and so we see that the covariance matrix Γ(t) can be obtained by fitting the coefficients f_k,l^m,n using the phase-space correlations

    Φ^2_m,n(t+Δ t)
    		=∑_k,l=1^N-1f_k,l^m,ncos(ω_k Δ t)cos(ω_lΔ t)Γ_φφ; k,l(t)
    +∑_k,l=1^N-1f_k,l^m,nsin(ω_k Δ t)sin(ω_l Δ t)Γ_ρρ; k,l(t)
    +∑_k,l=1^N-1(f_k,l^m,n+f_l,k^m,n)cos(ω_k Δ t)sin(ω_l Δ t)Γ_φρ; k,l(t)  .

By gathering statistics, we employ the same method as in Ref. <cit.>, but simply shift the window of observation.

We obtain the covariance matrix in momentum space for the first N modes that give physical (positive) occupation numbers. The imaging system is additionally accounted for by a Gaussian point spread function with standard deviation of 3μ m to estimate the initial state. In this way, we can also confirm the global initial thermal state and extract a pre-quench temperature. We use discrete cosine eigenfunctions to convert the momentum space correlators to real-space correlators, from which we compute the quantities of interest.






 §.§ Probing Landauer's principle from covariance matrices

In this section, we delineate how the quantities in Landauer's principle, as expressed in Eq. (<ref>), are computed using the full covariance matrix Γ in real space, defined on a grid of N pixels.
First, the von Neumann entropy S(Γ) is computed using the symplectic eigenvalues λ_n of the covariance matrix Γ <cit.>. These eigenvalues correspond to the eigenvalues of iΩΓ, where 

    Ω:=[    0  1_N; -1_N    0 ]

is the 
standard kernel of the symplectic form capturing the canonical commutation relations. The expression for 
the von Neumann entropy 
S(Γ) of a Gaussian state is given by

    S(Γ)=∑_n=1^N[(λ_n+1/2)ln(λ_n+1/2)-(λ_n-1/2)ln(λ_n-1/2)].


Given two full covariance matrices corresponding to different times and partitioning it into system and environment subspace covariance matrices (Fig. [fig:setupandtomography]<ref>fig:setupandtomographyc), this allows us to compute the entropy change of the system Δ S and the mutual information Δ I.

Next, we illustrate how to calculate the energy of the environment with respect to the post-quench Tomonaga-Luttinger liquid Hamiltonian. To obtain the correct environment Hamiltonian, we need to proceed with care. The field corresponding to the environment is discretely defined with N_ E=N (L_ E/L) pixels. A naive approach would discretise the continuous Hamiltonian defined over the length of the environment, but this nearest-neighbour phase coupling Hamiltonian would yield a non-linear dispersion relation, inadequately describing the physics of the environment for the continuum model used in the experiment. Instead, we start from the environment Hamiltonian of the continuous theory, diagonalise it to obtain a momentum decomposition of infinitely many modes, and then truncate the Hamiltonian <cit.> of the environment to a form with a linear dispersion relation, retaining only N_ E modes

    H_ E=ħ u_ E/2(δρ^ E_0)^2+∑_k=1^N_ E-1[(δρ_k^ E)^2+(φ_k^ E)^2]

with u_ E:=2g_ 1D/ħ L_ E and ω_k^ E:=√(g_ 1Dn_ 1D/m)kπ/L_ E. Converting back to real-space operators, we derive an effective resolution scale-dependent Hamiltonian

    H_ E=g_ 1DΔ z∑_n=N_ S^N-1δρ(z_n)^2
    +ħ^2 n_ 1D/4mΔ z∑_n,m=N_ S^N-1φ(z_n)φ(z_m)2π^2/N_ E^3κ_n,m^ E  ,

where 

    κ_n,m^ E   :=   ∑_k=1^N_ E-1{ k^2cos(kπ/N_ E(n-N_ S+1/2))
       ×   cos(kπ/N_ E(m-N_ S+1/2))}

encodes the long-range coupling of the effective Hamiltonian. Here, Δ z:=L/N is the pixel size, N_ S(E) is the number of pixels of the system (environment).

The energy of the environment can then be calculated <cit.> as [ϱ_ E H_ E]=[Γ^ EH̃_ E], where H̃_ E is the matrix expression related to H_ E=X^TH̃_ EX, with X=(φ(z_N_S), …,  φ(z_N-1), δρ(z_N_S), …, δρ(z_N-1))^T. Thus, to calculate the energetic contribution β_ EΔ E_ E, we need to obtain the inverse temperature of the environment. As indicated in the previous section and detailed in Ref. <cit.>, we have obtained the inverse temperature of the composite system with respect to the pre-quench Klein-Gordon Hamiltonian experimentally. Subsequently, we construct the thermal state corresponding to this temperature in the continuum limit simulation. We obtain the inverse temperature of the environment β_ E for different subregion sizes, by demanding the environment's energy to match that of a Gibbs state. We then interpolate the profile for each pixel. We note that the post-quench temperature is slightly higher than the temperature measured with respect to the pre-quench Hamiltonian and β_ E>0 
is nearly homogeneous across different subregion sizes.

To calculate the energy of a Gibbs state for the environment in the Tomonaga-Luttinger liquid model, we begin with the partition function Z_ E(β_ E)=Z^ zm_ E(β_ E)Z^ ho_ E(β_ E) <cit.>.
Due to the compactified nature of the field, the zero mode partition function can be expressed in terms of the Jacobi-theta function. Given our experimental parameters, it approximates to

    Z^ zm_ E(β_ E)=∑_n=-∞^∞e^-β_ Eħ u_ E/2n^2≈√(2π/β_ Eħ u_ E)  .

The harmonic oscillator partition function is

    Z_ E^ ho(β_ E)=∏_k=1^N_ E-11/2sinh(βħω^ E_k/2)  .

The energy is then E_ E(β_ E)=-∂_β_ E[logZ_ E(β_ E)]=E_ E^ zm(β_ E)+E_ E^ ho(β_ E). The zero-mode contribution to the energy is

    E^ zm_ E = 1/Z_ 0^ E∑_n=-∞^∞e^-β_ Eħ u_ E/2 n^2(ħ u_ E/2n^2 ) ,

which approximates to 1/(2β_ E) for our experimental parameters. The harmonic oscillator contribution to the energy is

    E_ E^ ho=∑_k=1^N_ E-1ħω_k^ E/2(β_ Eħω_k^ E/2)  .

Consequently, we can express then quantum relative entropy of the environment state ϱ_ E with respect to a Gibbs state at inverse temperature β_ E as

    D(ϱ_ E||γ_ E^β_ E)
    		=-S(ϱ_ E)+β_ E[ϱ_ E H_ E]+logZ_ E(β_ E)

and we obtain Δ D accordingly.






Supplementary Material













 §.§ Continuum limit and role of boundary conditions

In this section we compare the effect of Neumann boundary conditions (∂_zφ(z)|_z=0, L=0) and Dirichlet boundary conditions (φ(z)|_z=0, L=0) on the information-theoretic quantities within Landauer's principle Eq. (<ref>) in the continuum limit. The quantum field is discretised on a  sufficiently fine grid to ensure convergence, achieved when higher momentum modes no longer affect the results significantly. The simulation parameters closely match experimental values, assuming a homogeneous density profile n_ 1D.



The plots showing the scaling with subregion size at various times are presented in Fig. <ref>. The scaling of β_ EΔ E_ E can be explained by the initial symmetric energy density profile, which is higher at the edges than in the bulk for Neumann boundary conditions and lower for Dirichlet boundary conditions (Fig. [fig:DBC_main]<ref>fig:DBC_maina-b). The dominant contribution for the considered quench, Δ I, shows scaling symmetrically about L_ S/L=0.5 by construction. For Dirichlet boundary conditions, small systems (or environments) do not increase in correlations with its environment for ct/L_ S>1 as correlations reach a plateau phase in time (Fig. [fig:DBC_main]<ref>fig:DBC_maine). Neumann boundary conditions, however, actually yield Δ I to be higher for the case of a small system than for the case of system and environment being of equal size. As discussed in the main text, the zero mode causes the results for Dirichlet boundary conditions, which are well described by the quasiparticle picture, to diverge from those for Neumann boundary conditions after ct/L_ S > 1.  



 §.§ Exploiting the extremality of Gaussian states

The validity of the Gaussian tomography scheme has been confirmed by measuring the normalised averaged connected fourth-order correlation functions, as demonstrated in previous works <cit.>. However, leveraging properties of Gaussian quantum information, we argue that the information-theoretic quantities in Landauer's principle can also be estimated using our second-moment tomography scheme, even for time evolution under a non-Gaussian model with significant higher-order correlation functions. This justifies the potential use of our experimental protocol under the lens of Landauer's principle for more complex protocols and interacting models.

To begin, we define G as the Gaussian state that shares the same second moments as the true state ϱ that may be non-Gaussian, reflecting a preparation involving interactions. Central to the argument are several extremality properties of Gaussian states. Firstly, Gaussian states G have the largest von Neumann entropy S(G)≥ S(ϱ) among all states ϱ with a given covariance matrix.
Secondly, we can make use of the property

    [ϱ(t)logG(0)]=[G(t)logG(0)]  ,

as any Gaussian state can be written as a matrix exponent of a quadratic polynomial, thereby causing the matrix logarithm of the relative entropy to cancel out. Thirdly, Ref. <cit.> has demonstrated that the quantum conditional entropy is also maximised for Gaussian states, and the  inequality 

    S(ϱ_ S(t))-S(ϱ_ SE(t))≥ S(G_ S(t))-S(G_ SE(t))

is true. This is less obvious, as the expression under consideration is 
a difference of von Neumann entropies. 
The generalised entropy production of the true dynamics ΔΣ:=β_ EΔ E_ E+Δ S= Δ I+Δ D can be written as

    ΔΣ:=Δ I+Δ D
    
    =S(ϱ_ S(t))+S(ϱ_ E(t))-S(ϱ_ SE(t))-S(ϱ_ S(0))-S(ϱ_ E(0))+S(ϱ_ SE(0))
    -S(ϱ_ E(t))-[ϱ_ E(t)logγ_ E^β_ E]+S(ϱ_ E(0))+[ϱ_ E(0)logγ_ E^β_ E]
    
    =S(ϱ_ S(t))-S(ϱ_ S(0))-[ϱ_ E(t)logγ_ E^β_ E]+[ϱ_ E(0)logγ_ E^β_ E]
    
    =Δ S-[ϱ_ E(t)logγ_ E^β_ E]+[ϱ_ E(0)logγ_ E^β_ E]  ,

where we have used the unitary invariance of the von Neumann entropy. Clearly, the generalised entropy production ΔΣ^G:=β_ EΔ E_ E^G+Δ S^G=Δ I^G+Δ D^G assuming an initially Gaussian state and assuming Gaussian dynamics is given by

    ΔΣ^G:=Δ I^G+Δ D^G
    =Δ S^G-[G_ E(t)logγ_ E^β_ E]+[G_ E(0)logγ_ E^β_ E]

where Δ E_ E^G=[(G_ E(t)-G_ E(0))H_ E], Δ S^G=S(G_ S(t))-S(G_ S(0)), Δ I^G=I^G_ SE(t)-I^G_ SE(0) and Δ D^G=D(G_ E(t)||γ_ E^β_ E)-D(G_ E(0)||γ_ E^β_ E). By utilising the property in Eq. (<ref>) and considering that γ_ E^β_ E is Gaussian we can write

    ΔΣ = Δ S-[G_ E(t)logγ_ E^β_ E]+[G_ E(0)logγ_ E^β_ E]=Δ S-Δ S^G+Σ^G  .

This also implies that

    Δ E_ E = Δ E_ E^G

demonstrating that covariance matrices are entirely sufficient for capturing the energetic dynamics, even when the true dynamics are not Gaussian. This is important, as in this way, one can meausure second moments experimentally and still get bounds for the true state that may show signatures of interactions.

To proceed further, we must assume that the initial state post-quench is also Gaussian, i.e., ϱ_SE,S,E=G_SE,S,E. At this step, we have not made the additional assumption of Gaussian dynamics to bound the quantities in Landauer's principle. By the maximality of entropy for Gaussian states, we have Δ S≤Δ S^G, we quickly deduce that the generalised entropy production is upper-bounded by the Gaussian value, i.e., ΔΣ≤ΔΣ^G. Similarly, we infer that the change in mutual information is upper-bounded by the Gaussian value <cit.>, i.e., Δ I≤Δ I^G. Utilising Eq. (<ref>), we find that the change in relative entropy is lower-bounded by the Gaussian value, i.e., 

    Δ D≥Δ D^G  .

However, in both cases, these inequalities alone do not provide quantitative information on the extent to which these quantities dominate each other in the true state. To gain a more precise understanding of the relative contributions of mutual information and relative entropy in this context, a more comprehensive theoretical model and advanced experimental methods are required for an experiment modelled by an arbitrary interacting theory.









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