Paper Title: An iterative transversal CNOT decoder.







Universal Quantum Ltd, Gemini House, Mill Green Business Estate, Haywards Heath, RH16 1XQ, United Kingdom


Google Inc., Santa Barbara, 93117 CA, USA


Sussex Centre for Quantum Technologies, University of Sussex, Brighton, BN1 9RH, United Kingdom
Universal Quantum Ltd, Gemini House, Mill Green Business Estate, Haywards Heath, RH16 1XQ, United Kingdom

§ ABSTRACT 
Modern platforms for potential qubit candidates, such as trapped ions or neutral atoms, allow long range connectivity between distant physical qubits through shuttling. This opens up an avenue for transversal logical CNOT gates between distant logical qubits, whereby physical CNOT gates are performed between each corresponding physical qubit on the control and target logical qubits. However, the transversal CNOT can propagate errors from one logical qubit to another, leading to correlated errors between logical qubits. We have developed a multi-pass iterative decoder that decodes each logical qubit separately to deal with this correlated error. We show that under circuit-level noise and only 𝒪(1) code cycles, a threshold can still persist, and the logical error rate will not be significantly degraded, matching the sub-threshold logical error rate scaling of p^d/2 for a distance d rotated surface code. 


An iterative transversal CNOT decoder
    Winfried K. Hensinger
    
=====================================





§ INTRODUCTION

Quantum error correction (QEC) is essential for realising fault-tolerant quantum computation. The threshold theorem states that if the error rate per quantum gate is below a certain threshold, arbitrarily long quantum computations become possible <cit.>. The surface code, known for its high error threshold and ability to correct both bit-flip and phase-flip errors through local interactions, is a leading QEC candidate. It arranges qubits on a two-dimensional lattice and uses stabilizer measurements for error detection. The minimum weight perfect matching (MWPM) algorithm is an effective decoding method for the surface code <cit.>.





Quantum computing hardware varies in connectivity capabilities. Superconducting qubits typically have fixed, limited connectivity which is imposed by the chip topology, while platforms like trapped ions, neutral atoms, and photonics allow for long-range connectivity. For example, in the trapped ion paradigm, ion shuttling operations enable arbitrary connectivity <cit.>. The variability of connectivity influences QEC implementation and efficiency across different systems. 

Lattice surgery is a common technique for performing logical operations in the surface code while relying only on nearest neighbour operations. It involves merging and splitting logical qubits to facilitate operations such as the logical CNOT gate. The time complexity of lattice surgery scales with the code distance, typically requiring 𝒪(d^3) volume, where d is the code distance. One possible implementation requires a time complexity of 2d and three logical qubits, one of which is an ancilla patch for mediating the merging and splitting operations <cit.>. Note that all Clifford operations can be done in 0 time given sufficient qubits <cit.>.

The transversal CNOT gate offers an alternative approach. It requires performing a physical CNOT operation between every corresponding physical qubit of the logical qubits. For N data qubits per logical qubit, N CNOT operations are needed, often involving long-range interactions. The transversal CNOT deterministically maps Pauli errors between patches: X errors are mapped from control to target, and Z errors are mapped from target to control. This mapping, if uncorrected, leads to spurious detection events which will greatly degrade logical qubit performance if standard decoding methods like MWPM are used.

Recent research has focused on developing specialized decoders for transversal CNOT gates <cit.>. Our work significantly expands upon a concept briefly mentioned[As a long footnote.] in <cit.>, by developing a multi-pass iterative decoder that addresses the correlated errors spread by the transversal CNOT. This decoder processes each logical patch separately in multiple passes to correct these correlated errors.

Modern qubit platforms supporting long-range connectivity make transversal CNOT gates feasible, potentially improving resource estimates for quantum computing. Our multi-pass iterative decoder demonstrates that, even under circuit-level noise and only 𝒪(1) code cycles, a threshold can persist without significant degradation of the logical error rate. This matches the sub-threshold logical error rate scaling of p^⌊ d/2 ⌋ for a distance d rotated surface code.


This paper presents an analysis of our multi-pass iterative decoder for transversal CNOT gates, including its design, implementation, and performance characteristics. 

Specifically, we illustrate a simple method to separately decode two different patches of surface codes that had undergone transversal CNOT operations, we assume:

    
  * error-free and instantaneous transversal CNOT operations and
    
  * standard circuit-level depolarising noise model on all other gates - SD6 <cit.>.


We argue that the error-free transversal operation should not significantly impact the overall behavior of the logical error rates, as the realistic transversal CNOT error is expected to contribute only a marginally higher circuit-level noise error rate <cit.> for some hardware platforms <cit.>. Furthermore, the idealized transversal CNOT operation suffices for the primary purposes of our investigation here, i.e. the performance of a new approach to decoding. A detailed analysis into hardware realistic connectivity error models and its impact on the performance of the transversal CNOT operation relative to lattice surgery is in preparation <cit.>.

Our work has significant implications for resource estimation in quantum advantage applications, where there exists a trade-off space between run times and the number of magic state distillation factories. Previous work <cit.> has assumed lattice surgery operations and demonstrated the benefits of fast code cycle times when other factors are held constant. For instance, trapped ion hardware, with code cycle times potentially three orders of magnitude slower than superconducting qubits, would traditionally require substantially more physical qubits to achieve comparable quantum advantage run times <cit.>. This comparison assumes the same QEC methods and physical error rates across both platforms.

However, transversal operations offer potential improvements in both the volume scaling of magic state factories and the overhead associated with parallelized magic state consumption. By leveraging these advantages along with enhanced connectivity and higher baseline fidelities on physical gates, slower hardware architectures may be able to bridge the gap in physical qubit requirements to reach runtime parity with faster but locally-connected systems. Our findings suggest that the integration of transversal CNOT gates and specialized decoding techniques could significantly impact the resource landscape across diverse quantum computing platforms, potentially altering the balance between speed and qubit count in the pursuit of quantum advantage.





§ THE REPETITION CODE

In the following sections, we illustrate our iterative transversal CNOT decoding method with the repetition code. Due to the surface code's symmetry in suppressing Pauli-X and -Z error, it is sufficient to study the repetition code for the intuition behind our method. For simplicity, we also assume no mid-circuit errors for the visualisation of this decoding procedure. However, it should be noted that this method easily generalises to accommodate mid-circuit errors.

A distance d repetition code has code words |0̅⟩=|0⟩^⊗ d,|1̅⟩=|1⟩^⊗ d and can be represented with the following stabiliser generators, 𝒢, and logical operators X̅ and Z̅:

    𝒢=⟨⋃_j=0^d-2 Z_jZ_j+1⟩,  X̅ = ∏_i=0^d-1 X_i  ,  Z̅ = Z_k  ,

where Z_j(X_j) are Pauli-Z(X) operator acting on physical qubit j. If we were to repeatedly measure the stabiliser generators of this code, we can detect and correct for Pauli-X errors only. In the particular example in figure <ref>, we initialise the repetition code and measure the parity checks for 3 rounds before subsequently measuring all the data qubits. We can then construct the syndrome of this quantum memory experiment <cit.> (in this case a classical code). For t=1,2, the syndrome data/detector clicks can be constructed by the calculating the modulo 2 difference (⊕) of the same parity check measurement separate by one step in time. 



Figure <ref> is a syndrome graph for this quantum memory experiment with 3 rounds of syndrome extractions. The horizontal edges at each time slices of this graph represents the qubits and vertical edges between time slices correspond to the possibility of measurement errors. Detectors are the nodes of this graph, carrying the syndrome information and will be coloured if the parity of the measurements associated with the detected differs from that expected. 



§ CORRELATED ERRORS

A CNOT operation inherently spread errors between the control and target modes. X errors will flow from the control to target modes (shown in equation <ref>) and Z errors will flow from the target to control mode (see equation <ref>).


    qubit  q[2];
    
    X q[0];
    cnot q[1] | q[0];
    
    cnot q[1] | q[0];
    X -;
    
    
    qubit  q[2];
    
    X q[1];
    cnot q[1] | q[0];
    
    cnot q[1] | q[0];
    X q[1];



    qubit  q[2];
    
    Z q[0];
    cnot q[1] | q[0];
    
    cnot q[1] | q[0];
    Z q[0];
    
    
    qubit  q[2];
    
    Z q[1];
    cnot q[1] | q[0];
    
    cnot q[1] | q[0];
    Z -;


For simplicity, in this discussion we focus on the X errors only and note that by symmetry, the argument is valid for the Z errors with the control and target modes reversed. The transversal CNOT will deterministically map X errors from the control to target patch. If this propagated error is not corrected for, it will cause spurious detection events appearing on the target node right after the transversal CNOT.  Ultimately, this has the overall effect of  a degraded logical error rate. However, if we can first decode the control patch, and have a suitable guess for the locations of the X errors, we can use that to cancel out the propagated X error on the target patch. The motivation behind our decoder is to use a conventional memory experiment decoder for a single logical qubit memory experiment, which decodes each logical qubit separately, updates the syndrome data and Pauli-frame of each individual logical qubit and then propagates that information before decoding again iteratively. This process repeats until the Pauli-frames converge to a stable state. The number of iterations required is a function of the circuit structure, where simple circuits only ever require one iteration. 



§ ITERATIVE DECODER FOR THE REPETITION CODE

We shall illustrate our approach with two examples concerning the repetition code. The first example (Example 1) involves a single transversal CNOT, requiring only a single-pass decoder, whereas the multiple CNOT version (Example 2) requires multiple rounds of iterative decoding. 

We use the following notation and conventions to separate different detectors and edges by colour.

    
  * “Natural” errors (edges) and detectors (nodes) are coloured  red, 0.2!
    [scale=1,every node/.style=minimum size=1cm,on grid]        
            [xshift=0,every node/.append style=yslant=0,xslant=0,yslant=0,xslant=0]
            [white,fill opacity=0.0] (0,0) rectangle (5,1);
            [fill=none,shape=circle,draw=none] (N0T0) at (0,0) ;
            [fill=red,shape=circle,draw=black] (N1T0) at (2,0) ;
            [fill=none,shape=circle,draw=none] (N2T0) at (4,0) ;
    
            [-,line width=0.1cm,red,opacity=1] (N0T0) edge node  (N1T0);
            [-,line width=0.01cm,black,opacity=1] (N1T0) edge node  (N2T0);
    
    
    , they are the errored edges and detectors that occur genuinely, and are not due to the propagation effects of the CNOT.

    
  * Propagated errors (edges) and detectors (nodes) are coloured  brown, 0.2!
    [scale=1,every node/.style=minimum size=1cm,on grid]        
            [xshift=0,every node/.append style=yslant=0,xslant=0,yslant=0,xslant=0]
            [white,fill opacity=0.0] (0,0) rectangle (5,1);
            [fill=none,shape=circle,draw=none] (N0T0) at (0,0) ;
            [fill=brown,shape=circle,draw=black] (N1T0) at (2,0) ;
            [fill=none,shape=circle,draw=none] (N2T0) at (4,0) ;
    
            [-,line width=0.1cm,brown,opacity=1] (N0T0) edge node  (N1T0);
            [-,line width=0.01cm,black,opacity=1] (N1T0) edge node  (N2T0);
    
    
    , they are the errored edges and detectors that are caused solely by the propagation of errors due to the CNOT.

    
  * Decoded matched edges are coloured  blue, they are the edges returned by the MWPM decoding algorithm, 
    0.2!
    [scale=1,every node/.style=minimum size=1cm,on grid]        
            [xshift=0,every node/.append style=yslant=0,xslant=0,yslant=0,xslant=0]
            [white,fill opacity=0.0] (0,0) rectangle (5,1);
            [fill=none,shape=circle,draw=none] (N0T0) at (0,0) ;
            [fill=red,shape=circle,draw=black] (N1T0) at (2,0) ;
            [fill=none,shape=circle,draw=none] (N2T0) at (4,0) ;
    
            [-,line width=0.1cm,red,opacity=1] (N0T0) edge node  (N1T0);
            [-,line width=0.01cm,black,opacity=1] (N1T0) edge node  (N2T0);
            
             [-,line width=0.1cm,blue,opacity=1] ((N0T0.east) + (0,0.5)) edge node  ((N1T0) +(0,0.5));
    
    
    , in this case the blue decoded edge had correctly corrected for the red corrupted edge.


    
  * The stored propagated edges are coloured  green, they are the recorded edges perceived to be propagated by the error spreading effects of the CNOT, 
    0.2!
    [scale=1,every node/.style=minimum size=1cm,on grid]        
            [xshift=0,every node/.append style=yslant=0,xslant=0,yslant=0,xslant=0]
            [white,fill opacity=0.0] (0,0) rectangle (5,1);
            [fill=none,shape=circle,draw=none] (N0T0) at (0,0) ;
            [fill=brown,shape=circle,draw=black] (N1T0) at (2,0) ;
            [fill=none,shape=circle,draw=none] (N2T0) at (4,0) ;
    
            [-,line width=0.1cm,brown,opacity=1] (N0T0) edge node  (N1T0);
            [-,line width=0.01cm,black,opacity=1] (N1T0) edge node  (N2T0);
            
             [-,line width=0.1cm,green,opacity=1] ((N0T0.east) + (0,0.5)) edge node  ((N1T0) +(0,0.5));
    
    
    , in this case the green propagated edge had correctly corrected for the brown propagated edge.


The core part of the method relies on propagating Pauli errors and detection events between the patches. For example with X errors, the Pauli frame of the control patch directly before the CNOT operation is determined, and propagated in the round following the CNOT into a separate stored list corresponding to the target patch. Next the associated detectors of the control Pauli Patch are propagated modulo 2 into the detector pattern of the target patch. The procedure acts to recover the detector pattern corresponding to only the natural errors while the Pauli errors determined to be propagated are stored separately. The logical observable is determined by combining the detector pattern decoded Pauli frame with the propagated Pauli frame. 

Each CNOT is addressed in this manner sequentially. For more complex circuits, like those with alternating directions of CNOT operations, more than one iteration may be required, where each iteration involves addressing every CNOT in the circuit, first undoing the impact of this particular CNOT from the prior iteration, and then again performing the propagation of Pauli errors and detectors. The termination clause is when the combined Pauli frame for each logical qubit converges to a stable state, i.e. equality between this iteration and the prior. For circuits with two alternating direction CNOT operations, at most two iterations are ever required. A flowchart illustrating the iterative method is found in appendix <ref>.



 §.§ Example 1: Single pass decoder

In figure <ref>s and <ref>s, chronologically in time from figure <ref> to figure <ref>, we show a method to modify syndromes and record propagated edges to decode a single logical transversal CNOT acting between two logical qubits (left and right).

Firstly, in figure <ref>, we see a detector pattern, and have annotated the propagated detectors and edges with the brown nodes and edges respectively, and also labelled the errored qubits with red edges for illustrative purposes. However in an actual experiment, this information is not available.



In figure <ref>, we decode the left and right logical qubits and record the blue decoded edges in figure <ref>. In this case, decoding has chosen some edges incorrectly.



In order to decode correctly, we reset the blue decoded edges on the right patch, then, the predicted Pauli-frame for the left logical qubit is propagated to the right logical qubit in figure <ref>, removing spurious detection events and recording the propagated edges in green  at t=1. We then proceed to decode the right logical qubit with MWPM and record its decoded edges in figure <ref>. 

With this syndrome modification and Pauli-frame propagation procedure, we are able to correctly identify the “natural errors” and the propagated errors with all the spurious detection events removed. For an instantaneous and error-free transversal CNOT operation, the logical error rate of failure after the application of a transversal CNOT should be as close as possible to the memory experiment case as possible. This can be interpreted as an enhanced memory experiment with a transversal CNOT sandwiched in between patches.



 §.§ Example 2: Multi-pass decoder

We will now show a more complicated example which includes two alternating transversal CNOTs. Multiple iterations of frame propagation are required, and we see that this naturally follows from the mapping behaviour of the transversal CNOT. In the case of two alternating transversal CNOT's, we observe that an X error spreading from control to target in the first instance will be reflected back after the second transversal CNOT. If not correctly decoded, this would lead to an amplification of errors that will severely affect the overall logical error rate. 



  §.§.§ 1^st iteration

In the first iteration, we follow similar methods outlined in Example 1. The detector pattern from experiment can be seen in figure <ref>, the series of two alternating transversal CNOTs with a round of syndrome extraction in between leads to the brown propagated edges and nodes. We first decode the left logical qubit and record its blue decoded edges in <ref>. We then propagate the Pauli-frame at t=0 in the left patch to the right patch to remove spurious events, in this case no data (spatial) edges are matched in the left patch at t=0, hence no spurious events are propagated over to the right. These are the incorrectly decoded edges on the left, we shall see how iterating the process in the 2^nd iteration solves this.




Then, we decode the logical qubit on the right, leading to the blue decoded edges in figure <ref>. Again these matched edges are incorrect, we will still propagate the frame over from the right at t=1 to the left at t=2, removing spurious events and recording the green decoded edges (figure <ref>). At the end of the first iteration, we arrive at a series of matched edges as shown in figure <ref>. The pattern of detection events has changed, and this triggers another round of decoding.






  §.§.§ 2^nd iteration

It's clear that a single iteration of frame propagation will not be sufficient to find good decoded edges in this case. In the second iteration, we reset the magneta edges on the left qubit and decode again leading to new blue edges at t=0 (figure <ref>). Propagate the Pauli-frame over to t=1 of the logical qubit on the right, removing the spurious events and record the newly propagated green edges (figure <ref>). 



Next, the right logical qubit is decoded, leading to blue edges at t=1 in figure <ref>. Before propagating the frame from t=1 on the right to left, we need to remove the previously propagated edges and spurious events at t=2 on the left from the first iteration (see figure <ref>). We then reset the decoded blue edges on the left, before following the conventional frame propagation shown in figure <ref> to <ref>. 





After the second iteration, we have correctly identified all the propagated edges and errored edges. Please refer to figure <ref> in appendix <ref> for an outline of the process.



§ SURFACE CODE SIMULATIONS

The procedure illustrated in the previous sections can be easily generalised to patches of surface code after applications of transversal CNOT operations. We aim to study and simulate the minimally interesting cases. Firstly, we characterise a transversal CNOT operation memory experiment with one round of syndrome extraction before and after it. Then we study the next minimally interesting case with 2 alternating transversal CNOTs, with the iterative procedure and varying number of rounds of syndrome extraction between the CNOTs. All the logical error rates show close agreement with the memory experiment logical error rates.

We use Stim <cit.> to construct the detector error model and sample from the syndrome extraction circuits undergoing the SD6 circuit-level noise model. We then subsequently use PyMatching to decode each logical qubits' syndromes separately with an equivalent single patch memory experiment matching graph.

We represent arbitrary number of rounds of syndrome extraction that is performed as follows:

    [on/.style=red, very thick]
        [on/.style=style=red, control style=very thick]
        qubit |0̅⟩ q;
        
        box x SE (q[0]);
        box Û (q[0]);
        box y SE (q[0]);
        measure q;
        setstyle  q[0];
          ,
 
the quantum circuit in equation <ref> shows one logical qubit undergoing x rounds of syndrome extraction(s), before a unitary operation Û, followed by another y rounds of syndrome extraction(s) before reading it out in the computational basis.



 §.§ Single transversal CNOT






In figure <ref> we analyse the single CNOT case, where only one iteration is ever required. We show close agreement between the CNOT circuit with propagating decoder versus the memory equivalent, which corresponds to two logical qubits with no CNOT operations and the same number of total rounds, i.e. two rounds, decoded with standard independent MWPM. This represents a lower-bound on performance quality. The scaling relationship with physical error rate is preserved. 



 §.§ Two alternating transversal CNOTs






In this section we investigate the performance of the iterative CNOT decoder as a function of the number of rounds of syndrome extraction between successive alternating transversal CNOT operations. We show that in figure <ref> where only a single round of error correction is performed between, we still maintain the same scaling relationship with physical error rate, and a comparable threshold. We note that the discrepancy between the transversal CNOT case versus memory appears to grow with increasing code distance. 

For figure <ref> we instead include two rounds of syndrome extraction, and now this growing discrepancy with increasing distance is no longer evident. This difference may be attributable to the space-time edges that span across the syndrome extraction rounds where there is a degenerate choice of round occurrence in relationship to the location of the CNOT operations (discovered independently by <cit.>). The impact of this is mitigated by moving to two rounds and beyond. 



In figure <ref> we plot for the three round case, and see the same overall scaling relationships as the two round case. Of course, the ideal performance of a transversal CNOT decoder would be the minimal time complexity with the minimal final error rate. Moving from one round to two rounds improved our final logical error rate, e.g. for distance 9, physical error rate 1E-3, the final error rate improved from 4.3E-6 to 2.4E-6. In contrast, moving from two rounds to three rounds actually slightly worsened our final error rate, i.e. from 2.4E-6 to 2.6E-6. This is because there are opposing forces, i.e. the iterative propagating decoder has some additional resiliency to time component edges with increasing round count separating CNOT operations, but each additional round added itself has some additional contribution of error. For all of our investigations so far, the minimal error case is found at two rounds of QEC separating the transversal CNOT operations. The optimal choice will ultimately include the desired target error rate, and a minimization of total volume, i.e. the distance required, and the number of rounds between CNOT operations to reach the target. 





§ DISCUSSIONS AND OUTLOOK

Our iterative decoder addresses the challenge of managing correlated errors resulting from transversal CNOT operations. The relevance of this work is supported by the feasibility of long-range two-qubit gates between physical qubits, as demonstrated in hardware platforms such as trapped ions <cit.>.

A key outcome of this study is the finding that two rounds of QEC between alternating transversal CNOTs lead to the optimal final error rate, independent of the code distance. This result confirms that the time complexity of the transversal CNOT is 𝒪(1). Our results show that the iterative decoder maintains the expected scaling of logical error rates with code distance and physical error rates under typical circuit-level noise models.

The practical implications of these findings are significant for particular quantum computing platforms. Incorporating transversal CNOT gates and the specialized decoding technique can reduce the physical qubit overhead required for fault-tolerant operations. For example, the lattice surgery CNOT implementation has a volume complexity of 𝒪(d^3), whereas the transversal CNOT has a volume complexity of 𝒪(d^2). The integration of transversal CNOTs would enhance the efficiency of magic state distillation processes and parallelized magic state consumption, thus reducing the overall resources required for quantum advantage applications in the fault-tolerant regime. This has the potential to bridge the gap in performance and resource requirements between different hardware architectures, such as superconducting qubits and trapped ions, making it feasible for slower, more connected systems to achieve comparable operational efficiency to their faster, locally-connected counterparts.

Further exploration of hardware-specific error models and their impact on the performance of the transversal CNOT is essential. A detailed analysis of the performance of the transversal CNOT with hardware-realistic connectivity error models, and its contrast to the lattice surgery alternative, is in preparation <cit.>. Additionally, an extension to the quantum advantage resource estimation process to include the benefits of transversal CNOT operations is also in preparation <cit.>. In this latter upcoming work, physical qubit requirements to reach target runtime will be estimated for different connectivity and code cycle assumptions.

In summary, the multi-pass iterative decoder for transversal CNOT gates offers an approach to utilize the benefits associated with high-fidelity long-range qubit connectivity. By optimizing the number of QEC rounds required and the resulting error rate, this method can improve the performance of quantum hardware with such connectivity available.




§ ACKNOWLEDGMENTS

We want to thank Niel De Beaudrap for his insights into analysing the iterative nature of the transversal CNOT decoder. We also acknowledge discussions with Craig Gidney, Michael Newman and Lucy J. Robson. We also want to thank all our colleagues at Universal Quantum Ltd for their continued support. Part of the numerical results were presented at the ICAP 2024 Satellite Conference (Brighton, UK) on 22 July 2024. Winfried K. Hensinger acknowledges support from Innovate UK (project number 10004857) and the University of Sussex.




§ ITERATIVE DECODING FLOWCHART






 Paper Title: Properties of Krylov state complexity in qubit dynamics.



APS/123-QED






 Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India




§ ABSTRACT 
We analyze the properties of Krylov state complexity in qubit dynamics, considering a single qubit and a qubit pair. A geometrical picture of the Krylov complexity is discussed for the single-qubit case, whereas it becomes non-trivial for the two-qubit case. Considering the particular case of interacting Rydberg atoms, we show that the Krylov basis obtained using an effective Hamiltonian minimizes the complexity compared to that obtained from the original Hamiltonian. We further generalize the latter property to an arbitrary Hamiltonian in which the entire Hilbert space comprises two subspaces with a weak coupling between them.    



Properties of Krylov state complexity in qubit dynamics
    Rejish Nath
    August 5, 2024
=======================================================







§ INTRODUCTION
 
The concept of complexity has found various implications in physics. For instance, in computation, it can be the number of resources one requires, like that of bits/qubits, operations, or the depth of the circuits, etc. <cit.>. For implementing a multi-qubit unitary operation, it can be the minimum number of predefined elementary unitary operations <cit.>. It can also be interpreted as the minimum geodesic distance between the identity and a target unitary operator in an operator space with an appropriately chosen metric, as in the case of  Nielsen complexity <cit.>. Similarly, one can define the complexity of a quantum state, which can be the minimum number of unitary operations from a predefined set of operators required to synthesize the given target state from a reference state <cit.>. The latter also led to studies for finding the minimal geodesics using various metrics <cit.>. The concept of complexity also finds applications in black holes, where correspondence between the growth of an interior volume of a black hole and growth of the complexity of its dual quantum state on the boundary is conjectured, termed as the holographic complexity <cit.>. Despite all these developments, the above complexity measures suffer from ambiguities, for instance, the choice of metric in the case of the Nielsen complexity, in choosing the set of elementary unitary operations for achieving a target unitary operation from the
identity operator or a target quantum state from an initial state. 

Subsequently, a new measure of complexity was put forward for states <cit.>, motivated by a similar measure for operators <cit.>, called the Krylov complexity <cit.>. It is quantified as the depth or spread of an initial state in its Krylov basis <cit.> over its time evolution. The advantages of the Krylov complexity are twofold. First, given an initial state and a Hamiltonian, the complexity is unambiguously defined via the Krylov basis with a calculation using the Lanczos algorithm <cit.>. Second, the Krylov basis has been shown to minimize the spread of an initial state among all choices of ordered bases <cit.>. The Krylov complexity further gained attention in studying quantum chaos and integrability <cit.> and found applications in quantum phase transitions <cit.>, black holes <cit.>, neutrino oscillations <cit.>, modular Hamiltonians <cit.>, quenched systems <cit.>, non-Hermitian systems <cit.>, periodically-driven systems <cit.>, and others <cit.>. Also, there are efforts to establish a connection between Krylov complexity and other complexity measures. For instance, it is shown that the time average of the Krylov complexity can be used to set an upper bound for the Nielsen complexity of the corresponding unitary operator using a specific metric that is defined using the Krylov basis <cit.>. A unified framework for describing the operator and state complexity was discussed in <cit.>, which has generalized this measure to mixed states and reduced density matrices <cit.>. There are efforts to interpret the Krylov complexity geometrically <cit.>, and a correspondence between the Krylov complexity and the wormhole length is found <cit.>. 
In this work, we study the Krylov complexity of a single qubit and a two-qubit system in search of fundamental insights into the Krylov complexity. While it has been shown in <cit.> that the Krylov complexity or its square root cannot act as a measure of distance between states in general, we find that the square root of the complexity does, indeed, relate to the distance between time-evolved states for a single qubit. However, once we consider two non-interacting qubits, which could have been described independently, we find that such a geometrical interpretation is no longer possible precisely because the overall Krylov complexity is more than just a sum of the individual complexities. We further consider a pair of two-level Rydberg atoms <cit.> which exhibit correlated dynamics under Rydberg blockade  <cit.>. The doubly excited states are inhibited in the Rydberg blockade dynamics and an effective two-level picture emerges. In that case, one expects the dynamics of Krylov complexity to be that of a single two-level system, i.e., it periodically oscillates between zero and one. However, we observe that the Krylov basis obtained using the original Hamiltonian exhibits a complexity having an amplitude larger than one. In contrast, we find an ordered basis, obtained as the Krylov basis of an effective Hamiltonian, that minimizes the complexity. We further generalize this aspect for an arbitrary Hamiltonian for which the entire Hilbert space comprises of two subspaces with a weak coupling between them. 

The paper is organized as follows: in section <ref>, we review the Krylov state complexity and some of its properties. In section <ref>, we study the single qubit and provide a geometrical interpretation of the Krylov complexity. We extend the study to two qubits in section <ref>, considering both uncoupled and coupled qubits. The particular case of a pair of Rydberg atoms is discussed in Sec. <ref>. We generalize the minimization of the Krylov complexity in Sec. <ref>. Finally,  we summarize our results in Sec. <ref>. 





§ KRYLOV STATE COMPLEXITY

For a closed system described by a Hamiltonian Ĥ, the time evolution of an initial state |ψ_0⟩ is 

    |ψ (t) ⟩ = e^-iĤt |ψ_0 ⟩ = ∑_n=0^∞(-it)^nn! |ψ_n ⟩,

where |ψ_n ⟩ = Ĥ^n |ψ_0⟩. Employing the Gram-Schmidt process to the set of states {ψ_0, Ĥψ_0, Ĥ^2ψ_0, ...} generates an ordered, orthonormal basis, 𝒦={K_1, K_2, ..., K_n} for the part of the Hilbert space that the time-evolution explores. The new basis 𝒦 is called the Krylov basis <cit.>. The zeroth element in the Krylov basis is the initial state itself, i.e., |K_0⟩=|ψ_0⟩. The other Krylov basis states (n>1) are constructed recursively via the Lanczos algorithm <cit.> from the initial state and the Hamiltonian, i.e., via |K_n+1⟩ =b_n+1^-1|A_n+1⟩, where 

    |A_n+1⟩ = (Ĥ - a_n) |K_n⟩ - b_n |K_n-1⟩

with a_n=⟨ K_n|Ĥ|K_n⟩ and b_n= ⟨ A_n|A_n⟩^1/2 are the Lanczos coefficients. Note that b_0=0. By construction, the dimension of the Krylov space may be smaller than the full Hilbert space. 

Given an initial state, we can define a cost function on all ordered bases, ℬ = {|B_0⟩, |B_1⟩, , |B_D-1⟩}, of the Hilbert space, given by

    𝒞_ℬ (t) = ∑_n=0^D-1 c_n | ⟨ψ (t)|B_n⟩|^2 = ∑_n=0^D-1 c_n P_ℬ (n,t)

where c_n is a monotonically increasing function of n with c_n ≥ 0, P_ℬ (n,t) = | ⟨ψ (t)|B_n⟩|^2, and D is the dimension of the Hilbert space. The state or spread complexity is obtained when c_n = n and is minimized for the Krylov basis, atleast in the vicinity of t=0 <cit.>. The latter is called the Krylov complexity and is given by <cit.>, 

    𝒞_𝒦 (t) = ∑_n = 0 n |⟨ψ (t)|K_n⟩|^2.

By construction, 𝒞_𝒦 (t=0)=0 and in general, 𝒞_𝒦 (t=0)≥ 0. 

Interestingly, a given initial state only evolves through states with the same Lanczos coefficients. To show that, consider two different initial states, |ψ_0⟩=|K_0⟩ and |ψ_0^'⟩ = e^-iĤt |ψ_0 ⟩, connected by a unitary time evolution. We can see that a_0^' = ⟨ψ_0^'|Ĥ|ψ_0^'⟩ = ⟨ K_0|e^iĤtĤ e^-iĤt|K_0⟩ = a_0 and b_1^' = ⟨ A_1^' | A_1^'⟩^1/2 = b_1 since |A_1^'⟩ = e^-iĤt (Ĥ - a_0) |K_0⟩ = e^-iĤt |A_1 ⟩. Similarly, we can show that a_1^' = a_1, b_2^' = b_2, |K_1^'⟩ = e^-iĤt |K_1⟩, and thereby, |A_2^'⟩ = e^-iĤt |A_2⟩. Following the same, we get |A_n+1^'⟩ = e^-iĤt|A_n⟩ assuming that |A_j^'⟩ = e^-iĤt |A_j⟩, a_j^' = a_j and b_j^' = b_j for all j ≤ n. Thus, |ψ_0^'⟩ has the same set of Lanczos coefficients as |ψ_0⟩ and hence, the Krylov complexity <cit.>.

In the Krylov basis, the Hamiltonian satisfies,

    Ĥ |K_n⟩ = a_n |K_n⟩ + b_n |K_n-1⟩ + b_n+1 |K_n+1⟩,

and forms a tridiagonal matrix, which resembles the Hamiltonian of a tight-binding model on a chain with a site-dependent hopping amplitude b_n between n and n-1 sites. a_n acts as a local energy offset or a chemical potential. Now, the Krylov complexity 𝒞_𝒦 (t) can be interpreted as the expectation value of the position on this lattice, where
the origin is at the first site [n=0 in Eq. (<ref>)]. In other words, the Krylov complexity measures how far the state has evolved away from the initial site in the above tight-binding model.




§ SINGLE QUBIT


In this section, we discuss the properties of the Krylov complexity in the dynamics of a single qubit. Let the eigenstates of the Hamiltonian be |±⟩ with eigenenergies ±ω/2 so that the Hamiltonian in the energy eigenbasis takes the form Ĥ = (ω/2) σ̂_z (where we have set ħ = 1). The initial state |ψ_0⟩ = α |+⟩ + β |-⟩ forms the first Krylov basis state |K_0⟩. The second one is orthogonal to the initial state and is |K_1⟩ = β^* |+⟩ - α^* |-⟩, where α and β are the probability amplitudes. |K_1⟩ is defined up to a global phase factor. The Hamiltonian of a qubit in the Krylov basis can be written as (after neglecting constants), 

    Ĥ=(a_0-a_1)/2σ̂_z+b_1σ̂_x,

where a_0= (|α|^2 - |β|^2) ω/2, a_1=-a_0, b_1=|α||β|ω and σ̂_x, y are the Pauli spin-1/2 matrices. 

A general qubit state in Krylov basis can be written as |ψ(t)⟩=c_0(t)|K_0⟩+c_1(t)|K_1⟩, where c_0,1 is the probability amplitude of finding the qubit in the corresponding Krylov basis states. The Krylov complexity of a qubit for the initial state |ψ_0⟩ is obtained as <cit.>

    𝒞_𝒦 (t) = |c_1(t)|^2=4 |α|^2 |β|^2 sin^2 ( ω t2),

 satisfying 0 ≤𝒞_𝒦 (t) ≤ 1 with its maximum value of 4 |α|^2 |β|^2 and a time-averaged value of 2|α|^2 |β|^2. The maximum value of one is attained when the initial state is an equal superposition of energy eigenstates, i.e., when |α|^2=|β|^2. In terms of Lanczos coefficients, the Krylov complexity becomes,

    𝒞_𝒦 (t) = 4b_1^24b_1^2 + (a_0 - a_1)^2sin^2 ( t√(4b_1^2 + (a_0 - a_1)^2)2),

which is symmetric in interchange between a_0 and a_1. Thus, an initial state and the corresponding orthogonal state of a qubit exhibit same complexity dynamics. If the initial state is stationary or when the two levels are degenerate, 𝒞_𝒦 (t) remains zero, and those cases are disregarded from further discussion. Once the energy eigenvalues of the Hamiltonian are kept the same, the initial state determines the amplitude of the oscillation of 𝒞_𝒦 (t), i.e., how far the state is evolved away from the initial state. 





 §.§ Shannon entropy and inverse participation ratio


Now we look at the Shannon entropy S_ Sh(t) and the inverse participation ratio Π(t) computed using the Krylov basis <cit.> and provides the relation with the Krylov complexity. The Shannon entropy determines how a given quantum state is spread over the Hilbert space spanned by the Krylov basis <cit.>, whereas Π(t) is typically used to analyze the localization/delocalization properties of quantum dynamics <cit.>, expecting  a simple relation with the Krylov complexity. They are defined as follows,

    S_ Sh=-P_nlog_2 P_n 
    Π=1/∑_n P_n^2-1,

where P_n(t)=|⟨ψ(t)|K_n⟩|^2. In terms of Krylov complexity, we get,

    S_Sh   =   -𝒞_𝒦log_2 𝒞_𝒦 - (1 - 𝒞_𝒦) log_2 (1 - 𝒞_𝒦) 
    Π   =   12𝒞_𝒦^2 - 2 𝒞_𝒦 + 1-1.

 The dependence of S_Sh and Π on 𝒞_K is shown in Fig. <ref>. When 𝒞_K=0 or 1, the system is in a Krylov basis state and both S_Sh and Π vanish, indicating the localization in the Krylov space. When 𝒞_𝒦=0.5, they attain the maximum value of unity, indicating the delocalization. Above relations indicate a non-trivial functional dependence of S_ Sh(t) or Π(t) on 𝒞_𝒦.








 §.§ Geometrical interpretation


Here, we establish a geometrical interpretation of the Krylov state complexity for a single qubit. The Krylov complexity itself does not act as a measure of distance between states <cit.>. In particular, it is shown that 𝒞_K and some functions of the Krylov complexity do not satisfy the triangle inequality, i.e. 𝒞_𝒦 (t_3 - t_1) ⪇𝒞_𝒦 (t_3 - t_2) + 𝒞_𝒦 (t_2 - t_1). Now, we show that in the case of a single qubit, the square root of the Krylov complexity does act as a measure of distance between the states connected by unitary time-evolution and satisfies the triangle inequality. 




The state vector |ψ⟩ is mapped to a point, α⃗_|ψ⟩, in a three-dimensional parameter space via the density matrix written in the energy eigenbasis as

    ρ̂(t) = |ψ (t)⟩⟨ψ (t)| = ℐ2 + α⃗_|ψ⟩·σ̂,

where ℐ is the identity matrix. Henceforth, we drop the subscript |ψ⟩ in α⃗_|ψ⟩ for convenience. For an initial state |ψ_0⟩ = cos (θ_0/2) |+⟩ + sin (θ_0/2) e^iϕ_0 |-⟩ at t=0 and under the unitary evolution given by the Hamiltonian, Ĥ = (ω/2) σ̂_z, the components of the vector α⃗ (t) are obtained as,

    α_x    = sin(θ_0)2cos (ϕ_0 + ω t),
    α_y    = sin(θ_0)2sin (ϕ_0 + ω t), 
    α_z    = cos(θ_0)2.

Thus, the time-evolution of an initial state |ψ_0⟩ represents a circular motion in the α_x-α_y plane, keeping θ(t)=θ_0, as shown in Fig. <ref>. At this point, we define the distance between two states as the Euclidean distance between the corresponding points in the α⃗- space and obtain,

    |α⃗ (t_2) - α⃗ (t_1)|^2 = sin^2 θ_0 sin^2 ( ω (t_2 - t_1)2) = 𝒞_𝒦 (t_2 - t_1),

where the Krylov complexity is independent of ϕ_0. Strikingly, the square root of the Krylov complexity can be identified as the magnitude of displacement of the state vector from the initial state.

Triangular inequality. We now prove that this does satisfy the triangle inequality. In the following, we set t_ij = t_i - t_j for brevity. The proof of the triangle inequality then follows from:


    √(𝒞_𝒦 (t_31))   = √(𝒞_𝒦 (t_32 + t_21))
       = sinθ_0 |sin(ω (t_32 +  t_21)2)| 
       = sinθ_0 |sin(ω t_322) cos(ω t_212) + sin(ω t_212) cos(ω t_322) | 
       ≤sinθ_0 ( |sin(ω t_322) | | cos(ω t_212) | 
        + |sin(ω t_212) | | cos(ω t_322) |  ) 
       ≤sinθ_0 ( |sin(ω t_322) | + |sin(ω t_212) | ) 
       = √(𝒞_𝒦 (t_32)) + √(𝒞_𝒦 (t_21)).

Note that this is still consistent with the results shown in <cit.>, where it is argued that √(𝒞_𝒦) violates the triangle inequality for three states separated by small time intervals. This is shown in <cit.> by Taylor expanding √(𝒞_𝒦 (t)) as,


    √(𝒞_𝒦 (t))≈ b_1 t - [ (a_0 - a_1)^2 + 2 (2b_1^2 - b_2^2 ) ]24 t^3 + 𝒪 (t^5).

Depending on the initial state and its Lanczos coefficients, the coefficient of t^3 may be positive, in which case √(𝒞_𝒦 (t_1+t_2))≰√(𝒞_𝒦 (t_1)) + √(𝒞_𝒦 (t_2)), i.e., not sub-additive. However, this coefficient is negative for all initial states for the single qubit, as b_2 = 0. This necessitates looking at higher-order terms in the series expansion, or the function itself, which we have explicitly shown above does satisfy the triangle inequality. 

Finally, we remark that while our mapping to the α-parameter space appears to be dependent on our initial choice of basis (the energy eigenbasis), we show below that a change in basis only results in mapping to different points in the parameter space and does not affect the displacement. We consider a unitary transformation, |ψ^'⟩ = 𝒰 |ψ⟩, under which the density matrix transforms as


    ρ^' (t) = 𝒰ρ (t) 𝒰^†   =12·ℐ + α⃗^' (t) ·σ̂

Now we show that dα⃗^' 2 (t)=dα⃗^2(t)=𝒞_𝒦 (t). Considering the unitary transformation in its general form,

    𝒰 = cos (a) ·ℐ + i sin (a) (n⃗·σ̂)

where n⃗·n⃗ = 1 and 2a is an angle of rotation about an axis along n⃗. By a straightforward, albeit slightly tedious calculation using the properties of Pauli matrices, it can be shown that 𝒰 (α⃗·σ̂) 𝒰^† = α⃗^'·σ̂ where

    α⃗^' = cos (2a) α - sin (2a) (n×α) + 2 (sin^2 (a) ) (n·α) n

For brevity, we set α (t_1=0) = α_1 and α (t_2 = t) = α_2 and α_21 = α_2 - α_1, so that (dα)^2 (t) = α_21^2. Then, we have 

    (dα⃗^')^2 (t)    = [ (cos 2a) α⃗_21 - (sin 2a) (n⃗×α⃗_21) + 2 (sin^2 a) (n⃗·α⃗_21) n⃗]^2 
       = (cos^2 2a) α_21^2 + (sin^2 2a) [α_21^2 - ( n⃗·α⃗_21)^2 ] 
        + 4 sin^4 a ( n⃗·α⃗_21)^2 + 4 sin^2 a cos 2a ( n⃗·α⃗_21)^2=(dα⃗)^2 (t),

i.e., our geometric interpretation is basis-independent.   



 §.§ A two level atom


At this point, we consider a two-level atom in which the ground state |g⟩ is coupled to the excited state |e⟩ by a light field of Rabi frequency Ω and with a detuning Δ, described by the Hamiltonian (ħ=1),

    Ĥ = - Δσ̂_ee + Ω2σ̂_x,

where the operator σ̂_ab=|a⟩⟨ b| with a, b∈{g, e} and σ̂_x=σ̂_eg+σ̂_ge. The energy eigenvalues of Ĥ are E_± = ±√(Δ^2 + Ω^2)/2. In terms of the energy eigenstates |±⟩, the bare states are,
    |e⟩ = sin (θ/2) |+⟩ + cos (θ/2) |-⟩ and |g⟩ = cos (θ/2) |+⟩ - sin (θ/2) |-⟩,
where cos (θ/2) = [(√(Δ^2 + Ω^2) + Δ)/(2√(Δ^2 + Ω^2))]^1/2. If we take either |g⟩ or |e⟩ as the initial state, the Krylov complexity is the same and is

    𝒞_𝒦(t) = Ω^2Δ^2 + Ω^2sin^2 ( √(Δ^2 + Ω^2)t2),

which characterizes the well known Rabi oscillations. Using 𝒞_𝒦(t), we can compute the Shannon entropy and the inverse participation ratio using the Eqs. (<ref>) and (<ref>) as well.




§ TWO QUBITS

Now, we extend the calculations to two qubits. In particular, we consider both uncoupled and coupled qubits. For the latter, we take the case of a pair of interacting two-level Rydberg atoms.


 §.§ Non-interacting qubits



In the following, we discuss the Krylov complexity of a pair of non-interacting qubits governed by the Hamiltonian, Ĥ=Ĥ_1 + Ĥ_2, where Ĥ_1 (Ĥ_2) is the Hamiltonian of the first (second) qubit. Let the energy eigenstates of H_i be |±⟩_i and eigenvalues E_±^(i)=ω_i/2. Considering a general initial product state for the two qubits: (α_1 |+⟩_1 + β_1 |-⟩_1) ⊗ (α_2 |+⟩_2+β_2 |-⟩_2), we obtain the Krylov complexity as,

    𝒞_𝒦 (t) = 𝒞_𝒦^(1) (t) + 𝒞_𝒦^(2) (t) + ℱ(t)

where 𝒞_𝒦^(i) (t) is the Krylov complexity of the individual qubit, given in Eq. (<ref>) and  


    
    ℱ (t)    = (ω_1 cosθ_1 - ω_2 cosθ_2)^2ω_1^2 sin^2 θ_1 + ω_2^2 sin^2 θ_2𝒞_𝒦^(1) (t) 𝒞_𝒦^(2) (t) 
        + [ sin^2 θ_1 sin^2 θ_2 [(ω_1 cosθ_1 - ω_2 cosθ_2)^2 + 2 (ω_1^2 sin^2 θ_1 + ω_2^2 sin^2 θ_2)](ω_1^2 + ω_2^2 - 2ω_1 ω_2 cosθ_1 cosθ_2)(ω_1^2 sin^2 θ_1 + ω_2^2 sin^2 θ_2)][ω_-2sin(ω_+t2) - ω_+2sin(ω_-t2) ]^2,


where ω_±=ω_1±ω_2, sinθ_i = 2 |α_i| |β_i|, and cosθ_i = |α_i|^2 - |β_i|^2. See Fig. <ref> for the Bloch sphere representation of the product state of two independent qubits as they evolve independently on the Bloch sphere with two circular trajectories. As we find, Krylov complexity is the sum of Krylov complexity of each qubit and an additional term, ℱ, which is positive-valued at any instant t. If ω_1 = ω_2, the function ℱ simplifies to

    ℱ (t)= [(cosθ_1 - cosθ_2)^2sin^2 θ_1 + sin^2 θ_2] 𝒞_𝒦^(1) (t) 𝒞_𝒦^(2) (t),

which exhibits periodic oscillations and vanishes when θ_1=θ_2 as shown in Fig. <ref>(a). The amplitude or the maximum of ℱ (t) as a function of θ_1 and θ_2 for ω_1=ω_2=ω is shown in Fig. <ref>(b), and is independent of ω. The maximum amplitude is 0.5 when θ_1=π/4 and θ_2=3π/4 or vice versa. For ω_1≠ω_2, ℱ (t) is still periodic but exhibits more complex behaviour as shown in Fig. <ref> and the maximum of the peak value of ℱ (t) is found when θ_1=θ_2=π/2.

For vanishing ℱ (t), the geometric interpretation for the single qubit can be straightforwardly extended to the case of two non-interacting qubits. As the Krylov complexity, in this case, is simply the sum of the complexities of the individual qubits, the total complexity can be viewed simply as a restatement of the Pythagoras theorem. The individual complexities can represent squared distances in orthogonal directions, and the total complexity can represent the hypotenuse squared. However, when ℱ (t)≠ 0, the square root of the Krylov complexity generally does not satisfy the triangle inequality. 












Two-level atoms. For a pair of two non-interacting two level atoms coupled by light fields, the Hamiltonian is

    Ĥ = ∑_i=1^2 [ - Δ_i σ̂^i_ee + Ω_i2σ̂_x^i ].

The two atom bare states are |gg⟩, |ge⟩, |eg⟩ and |ee⟩. For a global driving, i.e., when Δ_1 = Δ_2 = Δ and Ω_1 = Ω_2 = Ω, the states |ge⟩ and |eg⟩ are degenerate, and only the symmetric state |+⟩=(|ge⟩ + |eg⟩)/√(2) is relevant to the dynamics and the antisymmetric state |-⟩=(|ge⟩ - |eg⟩)/√(2) can be disregarded. In the case of global driving, we obtain the Krylov complexity 𝒞_K^α (t) for an initial product state |α⟩ as, 

    𝒞_𝒦^gg (t) = 2Ω^2Δ^2 + Ω^2   sin^2 (√(Δ^2 + Ω^2)t2), 
    𝒞_𝒦^ge (t) = 2Ω^2Δ^2 + Ω^2   sin^2 ( √(Δ^2 + Ω^2) t2) 
       + 2Δ^2 Ω^2(Δ^2 + Ω^2)^2sin^4 ( √(Δ^2+Ω^2)t2),
    
𝒞_𝒦^ee (t)=𝒞_𝒦^gg(t) and 𝒞_𝒦^eg (t)=𝒞_𝒦^ge(t). For the symmetric state, we get 

    𝒞_𝒦^+ (t) = Ω^2Δ^2 + Ω^2sin^2 (√(Δ^2 + Ω^2)t ) + 
    8Ω^2 Δ^2(Δ^2 + Ω^2)^2sin^4 (√(Δ^2 + Ω^2) t2),

which cannot be expressed in terms of 𝒞_𝒦^eg (t) and 𝒞_𝒦^ge (t). Although in a single qubit, the Krylov complexity is identical for initial states |g⟩ and |e⟩, in a pair of qubits, despite being entirely uncoupled, 𝒞_𝒦 is found to depend on the initial state, whether being in |gg⟩ or |ge⟩. The latter can be attributed to the function ℱ(t) in Eq. (<ref>). When Δ=0, all states are degenerate, ℱ(t) vanishes and the complexity simplifies to 𝒞_𝒦^gg = 𝒞_𝒦^ge = 2 sin^2 (Ω t/2) and  𝒞_𝒦^+ = sin^2 (Ω t).




 §.§ Interacting qubits: a Rydberg atom pair


At this point, we consider the coupling between the qubits, and in particular, we consider the case of a pair of interacting two-level Rydberg atoms. The Rydberg atoms are two-level atoms in which the ground state |g⟩ is coupled to the Rydberg state |e⟩. When both qubits are in the Rydberg state, they interact repulsively, and the governing Hamiltonian in the interaction picture is,

    Ĥ = ∑_i=1^2 [ - Δ_i σ̂^i_ee + Ω_i2σ̂_x^i ] + V_0 σ̂^1_eeσ̂^2_ee

where V_0 = C_6/R^6 is the interaction strength, and R is the distance between the two Rydberg atoms.    





  §.§.§ Rydberg blockade


For the global driving and V_0≫Ω,
the system exhibits coherent Rabi oscillations between the states |gg⟩ and |+⟩ with an enhanced Rabi frequency of √(2)Ω, resulting in an effective two-level atom or called the superatom <cit.>. The doubly excited state |ee⟩ is completely inhibited in the dynamics and is called the Rybderg blockade. We would, therefore, expect the Krylov state complexity of the two initial states |gg⟩ and |+⟩ to be identical and resemble that of a two-level system, with the complexity peaking at one and exhibiting oscillations at the enhanced Rabi frequency. However, as we show in Fig. <ref>(a), the dynamics of 𝒞_𝒦 (t) is different for these two initial states, and in particular, the amplitude of oscillation of 𝒞_𝒦 (t) for initial state |+⟩ exceeds one, although the population dynamics is identical for both initial states as shown in Figs. <ref>(b) and <ref>(c). As shown below, it is different since the Krylov basis is different for the two initial states. 



For the initial state |gg⟩, apart from |gg⟩, the Krylov basis consists of,

    |K_1⟩^gg   = 1√(2)(|ge⟩ + |eg⟩) = |+⟩
    
        |K_2⟩^gg   = |ee⟩
    
        |K_3⟩^gg   = 0,

with Lanczos coefficients a_0 = a_1 = 0, a_2 = V_0, and b_1 = b_2 = Ω/√(2). Note that only the first two Krylov basis states (|K_0⟩^gg and |K_1⟩^gg) are relevant in the blockade regime. In that case, we can use Eq. (<ref>) and obtain the Krylov complexity as 𝒞_𝒦 (t) = sin^2 ( Ω^' t2) with Ω^' = √(2)Ω, the enhanced Rabi frequency. 

In contrast, for the initial state |+⟩, the Krylov basis comprises of |K_0⟩^+=|+⟩ and,

    |K_1⟩^+   = 1√(2)(|gg⟩ + |ee⟩) 
    
        |K_2⟩^+   = 1√(2)(-|gg⟩ + |ee⟩) 
    
        |K_3⟩^+   = 0.

 with Lanczos coefficients a_0 = 0, a_1 = a_2 = V_0/2, b_1 = Ω, and b_2 = V_0/2. Here, the first three Krylov basis states are needed to describe the blockade dynamics, indicating that 𝒞_𝒦(t) can go beyond the value of one, as shown by a dashed line in Fig. <ref>(a). It is surprising given that we observe coherent Rabi oscillations between |gg⟩ and |+⟩, as can be seen in Fig. <ref>(b) and <ref>(c), we expect to see identical Lanczos coefficients and Krylov complexity for both the states, as discussed in section <ref>. Using an ordered basis consisting of  {|+⟩, |gg⟩,|ee⟩, |-⟩}, the spread complexity dynamics starting from |+⟩ is found to be identical to the Krylov complexity obtained for the initial state |gg⟩ [solid line in Fig. <ref>(a)]. It implies that for the initial state |+⟩, another ordered basis exists for which the spread complexity becomes minimal, but not the Krylov basis constructed using the original Hamiltonian. Below, we show that the new ordered basis for which the spread complexity is minimum can be obtained as the Krylov basis obtained using a zeroth order effective Hamiltonian.   
 
The Krylov complexity attains even a larger amplitude (∼ 3) under Rydberg blockade [see the dashed line in Fig. <ref>(a)] if the initial state is |eg⟩ or |ge⟩. As we see below, all four Krylov states become relevant to characterize the dynamics and give an amplitude of three. The population dynamics shown in Fig. <ref>(b) is characterized by the periodic oscillations between |ge⟩ and |eg⟩ via |gg⟩. It implies that we can choose an ordered basis {|ge⟩, |gg⟩, |eg⟩, |ee⟩} such that the spread complexity amplitude is two [see the solid line in Fig. <ref>(a)], making it less than that of the Krylov complexity. For the initial state, |K_0⟩ = |ge⟩, we get

    |K_1⟩^ge   = 1√(2)(|gg⟩ + |ee⟩) 
    
        |K_2⟩^ge   = 1√(2)Ω_V(-V_0 |gg⟩ + 2Ω |eg⟩ + V_0 |ee⟩) 
    
        |K_3 ⟩^ge   = 1Ω_V(-Ω |gg⟩ - V_0 |eg⟩ + Ω |ee⟩,

 where Ω_V = √(2Ω^2 + V_0^2). The corresponding Lanczos coefficients are a_0 = 0, a_1 = V_0/2, a_2 = V_0^3/(2Ω_V^2), b_1 = Ω/√(2), b_2 = Ω_V/2 and b_3 = V_0^2Ω/(√(2)Ω_V^2). All four Krylov basis vectors have significant projections to |gg⟩, |eg⟩, or |ge⟩, making them all relevant for the blockade dynamics and resulting in a larger Krylov complexity. 

 The above results indicate that if the exact dynamics occur only in a subset of the full Hilbert space, the spread complexity is generally not minimal for the Krylov basis. This ambiguity can be removed by truncating the original Hilbert space and using an effective Hamiltonian. For instance, in the perfect blockade scenario, where the population in |ee⟩ is zero, we can arrive at an effective Hamiltonian (zeroth order) for V_0≫Ω <cit.>, 
 
    Ĥ_ eff = Ω2(σ̂_gg^1σ̂_x^2+σ̂_x^1σ̂_gg^2).

 Using the truncated Hilbert space and the effective Hamiltonian, we now compute the new Krylov basis for a given initial state and find that the modified Krylov complexity is minimal. For instance, for the initial state |ge⟩, using the effective Hamiltonian, we obtain the Krylov basis as |ge⟩, |gg⟩, |eg⟩, |ee⟩} and the Krylov complexity as 𝒞_𝒦 (t) = 2 sin^2 (Ω t/2√(2)), which has an amplitude of two not three. This aspect of reduced Hilbert space and an effective Hamiltonian to minimize the Krylov complexity is generalized in Sec. <ref>.





  §.§.§ Rydberg-biased freezing


Now, we examine the dynamics of the Krylov state complexity 
under the phenomenon of Rydberg-biased freezing, which arises from a combined effect of Rybderg blockade and an offset in the Rabi couplings of the two Rydberg atoms <cit.>. Again, we keep Δ_1 = Δ_2 = 0. When Ω_2>>Ω_1 and V_0 >> Ω_2, starting from |gg⟩, the first atom is essentially frozen in the ground state while the second atom undergoes Rabi oscillations between |g⟩ and |e⟩. In other words, Rabi-oscillations occur between |gg⟩ and |ge⟩ states as shown in Fig. <ref>(a), and |eg⟩ and |ee⟩ states do not contribute to the dynamics. 



For the initial state |K_0⟩^gg = |gg⟩, the Krylov basis states are,

    |K_1⟩^gg   = 1Ω̅(Ω_2 |ge⟩ + Ω_1 |eg⟩) 
    
        |K_2⟩^gg   = |ee⟩
    
        |K_3⟩^gg   = 1Ω̅( -Ω_1 |ge⟩ + Ω_2 |eg⟩) .

where Ω̅= √(Ω_1^2 + Ω_2^2). Even though, three Krylov basis states (|K_0⟩^gg, |K_1⟩^gg, and |K_3⟩^gg) look relevant to the dynamics, in the limit  Ω_2≫Ω_1, |K_2⟩^gg and |K_3⟩^gg≈ |eg⟩ can be disregarded. The corresponding Lanczos coefficients are a_0 = a_1 = a_3 = 0, a_2 = V_0, b_1 = Ω̅/2, b_2 = (Ω_1 Ω_2)/Ω̅≈Ω_1, and b_3 = (Ω_2^2 - Ω_1^2)/Ω̅≈Ω_2 and the Krylov complexity is obtained as 𝒞_𝒦 (t) ≈sin^2 (Ω_2 t/2), which oscillates between zero and one, as expected, in agreement with the numerical results shown in Fig. <ref>(b).

If the initial state is |ge⟩, then the Krylov basis states are given by 

    |K_1⟩^ge   = 1Ω̅(Ω_2 |gg⟩ + Ω_1 |ee⟩)  
    
        |K_2 ⟩^ge   = 1Ω_V Ω̅(-V_0 Ω_1 |gg⟩ + Ω̃^2 |eg⟩ + V_0 Ω_2 |ee⟩) 
    
        |K_3⟩^ge   = 1Ω_V ( -Ω_1 |gg⟩ - V_0 |eg⟩ +  Ω_2 |ee⟩ )

where Ω_V = √(Ω_1^2 + Ω_2^2 + V_0^2) here. The Lanczos coefficients are obtained as a_0 = 0, a_1 = V_0 Ω_1^2/Ω̅^2, a_2 = V_0(Ω_2^2 Ω_V^2 - Ω_1^2 Ω̅^2)/(Ω_V^2 Ω̅^2), a_3 = V_0Ω_1^2/Ω_V^2, b_1 = Ω̅/2, b_2 = Ω_1 Ω_2 Ω_V/Ω̅^2, and b_3 = √(Ω̅^22 + Ω_1^2 Ω_2^2 Ω_V^2Ω̅^2 - Ω_1^2 Ω̅^2(Ω_2^2 + V^2)Ω_V^4 - Ω_1 Ω_2 Ω_VΩ̅).
In the limit V_0 ≫Ω_2 ≫Ω_1, the Krylov basis states become |K_1⟩^ge→ |gg⟩, |K_2⟩^ge→ |ee⟩ and |K_3⟩^ge→ |eg⟩. Thus, the Krylov complexity oscillates between zero and one and is identical to what we obtained for initial state |gg⟩. 



§ MINIMIZATION OF THE STATE COMPLEXITY


It is proved that among the available ordered bases, the cost function in Eq. (<ref>) is minimized in the Krylov basis obtained from a given initial state and the original Hamiltonian, such that for 0 ≤ t ≤τ for some τ > 0, 𝒞_𝒦 (t) ≤𝒞_ℬ (t) <cit.>, where ℬ is any ordered basis other than the Krylov basis. However, in Sec. <ref>, we have shown that the Krylov basis computed using the zeroth order effective Hamiltonian minimized the state complexity rather than that obtained using the original Hamiltonian under the Rydberg blockade. Here, we discuss this in a more general context. We consider the Hamiltonian, Ĥ = Ĥ_A + Ĥ_B + Ĥ_AB with 

    Ĥ_A    = ∑_i,j = 1^N_A c_ij^A |a_i⟩⟨ a_j| 
    Ĥ_B    = ∑_i,j = 1^N_B c_ij^B |b_i⟩⟨ b_j| 
    Ĥ_AB   = ∑_i=1^N_A∑_j=1^N_B( d_ij |a_i⟩⟨ b_j| +  h.c. )

where A and B are orthogonal subspaces of the Hilbert space, spanned by {|a_i⟩| 0 ≤ i ≤ N_A } and {|b_j⟩| 0 ≤ j ≤ N_B } respectively. The coefficients c_ij^X (X = A, B)  represent the matrix elements of the Hamiltonian in the subspace X, while d_ij  provide the coupling between the two subspaces. We assume that the subspaces A and B are energetically well-separated and are only weakly coupled between them, i.e.,

    |d_ij| ≪min({ℰ_B}) - max({ℰ_A}).

Our results then suggest that whenever the dynamics of an initial state that lies completely in one of the subspaces, say A without any loss of generality, can be described by an effective Hamiltonian on A, the state complexity is minimized by the Krylov basis obtained using the effective Hamiltonian rather than the complete Hamiltonian. This effective Hamiltonian is Ĥ_A at zeroth order (as was the case in Sec. <ref>), or could be computed by including effects of Ĥ_AB at higher orders (see <cit.> and references therein, for instance). We discuss below the case when Ĥ_eff = Ĥ_A.  

Let 𝒦_A and 𝒦 represent the Krylov basis obtained using the effective Hamiltonian, Ĥ_A, and the complete Hamiltonian, Ĥ, respectively. The first basis state in both 𝒦 and 𝒦_A is the initial state |ψ_0⟩. We denote the basis states in 𝒦_A as |K_A,j⟩ and the Lanczos coefficients obtained using Ĥ_A as a_A,j and b_A,j. We further assume that the first M states in 𝒦 and 𝒦_A are identical, i.e. |K_A,j⟩ = |K_j⟩ for 0 ≤ j ≤ M-1. It also implies that the first M Krylov basis states in 𝒦 reside solely in the subspace A. We further assume that M is smaller than the size of 𝒦_A; if not, both 𝒦_A and 𝒦 result in identical Krylov complexity. 

Following the Lanczos algorithm, we obtain

    b_M |K_M⟩ = b_A,M |K_A,M⟩ + Ĥ_AB |K_A,M-1⟩,

where we have used Ĥ_B |K_A,M-1⟩ = 0, Ĥ_A |K_A,M-1⟩ = a_A,M-1|K_A,M-1⟩ + b_A,M |K_A,M⟩ + b_A,M-1 |K_A,M-2⟩, which is obtained using Eq. <ref>, and |K_A,j⟩ = |K_j⟩ for 0 ≤ j ≤ M-1. The latter implies we have identical Lanczos coefficients for j ≤ M-1. Eq. (<ref>) can be rewritten as |K_M⟩ = α |K_A,M⟩ + (1 - α)e^iλ |b⟩, up to a global phase with 0 < α≤ 1, where the state |b⟩ contains states from B-subspace, which is populated by Ĥ_AB. For α≠ 1, |K_A,M⟩ is not exactly identical to any state in 𝒦 and must therefore be a superposition of two or more states in 𝒦. 

Given that the dynamics is taking place only in A in this weakly-coupled limit, when the M+1'th basis state, i.e., |K_A, M⟩ becomes relevant to the dynamics (after the first M basis states, which contribute equally to the spread in both 𝒦 and 𝒦_A), 𝒦_A is more efficient than 𝒦 in minimizing the spread of the state, as the overlap of the state with |K_A, M⟩ will necessarily involve at least one extra state in 𝒦. Thus, we find that at least up to this point, 𝒞_𝒦_A (t) ≤𝒞_𝒦 (t).

In a larger Hilbert space, the initial state in A may couple to other states in A through virtual processes involving the subspace B, that may otherwise not feature in the Krylov basis obtained from Ĥ_A. By following the same line of reasoning above, we expect that the Krylov basis obtained using the effective Hamiltonian instead, which takes into account such higher-order couplings to B, will further minimize the spread complexity compared to 𝒦_A. 

This discrepancy arises because the contribution from different subspaces on a Krylov basis may not always reflect their importance in the dynamics. This is because the construction of the Krylov basis only involves the couplings between different states and does not directly feature the energy separation between them, which plays a vital role in the dynamics. Returning to our example of the Rydberg blockade, the two subspaces can be identified as A = span({|gg⟩, |ge⟩, |eg⟩}) and B = {|ee⟩}. Then we have


    Ĥ_A   = Ω2(σ̂_gg^1σ̂_x^2+σ̂_x^1σ̂_gg^2) 
    Ĥ_B   = V_0 σ̂_ee^1 σ̂_ee^2 
    Ĥ_AB   = Ω2(σ̂_ee^1σ̂_x^2+σ̂_x^1σ̂_ee^2)

The c_ij^X's (X = A, B) can be read off from above, while the non-vanishing d_ij's are equal to Ω/2. Diagonalizing Ĥ_A and Ĥ_B, we obtain ℰ_A = {0, ±Ω/√(2)} and ℰ_B = {V_0}. Thus, |d_ij|/( min( ℰ_B) - min( ℰ_A) ) ∼𝒪(Ω/V_0), which was equal to 0.01 for our choice of values. Note, however, that d_ij is identical to the non-vanishing off-diagonal matrix elements in Ĥ_A, as a result of which the Krylov basis states from both subspaces appear on an equal footing in the Krylov basis of |+⟩ and |ge⟩, even though |ee⟩ does not take part in the dynamics in the blockade regime. 

As we conclude this section, we note that while we only considered two isolated subspaces A and B in the limit of very weak coupling between them, we expect the discussion above to be equally applicable for more than two subspaces as well, where the coupling between two distinct subspaces is much smaller than the minimum energy gap between them to restrict the dynamics to only one subspace effectively.



§ SUMMARY


We analyzed the Krylov state complexity in the quantum dynamics of a single qubit and a pair of qubits. In the single qubit case, we demonstrated that the square root of the Krylov complexity measures the distance between time-evolved states by explicitly constructing an associated parameter space on which the states in the Hilbert space are mapped. In the case of two non-interacting qubits, we found that the total Krylov complexity was not simply the sum of the individual complexities of the two isolated qubits as one may have expected but consisted of an additional term that only vanishes uniformly for certain initial states. We further noted that this extra term in the complexity breaks the subadditivity of the square root of the complexity, rendering it impossible to treat the Krylov complexity as a measure of distance between states in general for this system.

We further considered the case of two interacting Rydberg qubits in the blockade regime, where two atoms simultaneously occupying the excited Rydberg state is prohibited. Hence, there is a redundancy in the Hilbert space. It has an important consequence in the dynamics of the Krylov complexity. We found that the Krylov basis obtained using the original Hamiltonian does not necessarily minimize the complexity, but the same obtained using an effective Hamiltonian describing the reduced Hilbert space minimizes it. 





We wish to acknowledge J. Bharathi Kannan for useful discussions. S.S. acknowledges funding support from the Junior Research Fellowship (JRF) awarded by the University Grants Commission (UGC), India. We further acknowledge DST-SERB for the Swarnajayanti fellowship (File No. SB/SJF/2020-21/19), MATRICS Grant No. MTR/2022/000454 from SERB, Government of India, National Supercomputing Mission for providing computing resources of “PARAM Brahma” at IISER Pune, which is implemented by C-DAC and supported by the Ministry of Electronics and Information Technology and Department of Science and Technology (DST), Government of India, and acknowledge National Mission on Interdisciplinary Cyber-Physical Systems of the Department of Science and Technology, Government of India, through the I-HUB Quantum Technology Foundation, Pune, India.












 Paper Title: Using low-cost Blu-Ray Optical Pickup Units for Measurement of Single Photon Emission from NV-Centers.



Global convergence of a modified BFGS-type method based on function information for nonconvex multiobjective optimization problems
    
		Ying Xue yang 
	
    August 5, 2024
==================================================================================================================================





§ ABSTRACT  
This work presents a cost-effective method for collecting single photons emitted from single nitrogen-vacancy centers in nanodiamonds. Conventional components of a confocal laser-scanning microscope, such as microscope objectives and the piezo translation stages, are replaced by two affordable Blu-ray optical pickup units. A Hanbury Brown and Twiss setup is used to identify single photon emission. The proposed approach is inexpensive and simple and lowers the entry-level to single photon research for quantum technologies. This enables student lab experiments or demonstration experiments at schools and shows that efficient sources of quantum light can be made from standard components compatible with established industry processes.






§ INTRODUCTION

In the developing field of quantum technologies, single photon sources (SPSs) have attracted considerable interest due to their role in various quantum-based applications, including quantum communications, cryptography, computing, and metrology <cit.>. These applications rely significantly on the controlled generation and coupling of single photons. Among the various methods investigated for single photon generation, the use of single nitrogen-vacancy centers (NV-centers) in diamonds has proven to be one of the promising and versatile solutions, since they exhibit efficient photon emission at room temperature, photostability, and offer easy ways to manipulate its spin quantum state <cit.>.

There are different approaches for interfacing single photons, such as using high-NA microscope objectives with piezo positioning systems for accurate sub-micrometer positioning <cit.>, direct coupling to fibers <cit.> or employing special optical elements <cit.>, although these methods are usually expensive to produce or build.
The goal of this work was to develop a cost-effective solution based on a Confocal Laser Scanning Microscope (CLSM) setup for the efficient optical coupling of single quantum emitters based on NV centers. A major challenge is to identify an alternative option for precise and selective control of individual diamonds without relying on expensive piezoelectric technology and replacing expensive optical components like high-NA microscope objectives. 
 In this work, a commercially available optical pickup unit (OPU) (Blu-Ray, PHR-803T, Toshiba Co., Tokyo, Japan), which has already been successfully used in other low-cost applications <cit.>, is used to replace the high-NA microscope objective and piezoelectric positioner. 



§ EXPERIMENTAL SETUPS






 §.§ Reference CLSM Setup


CLSMs have proven to be a powerful technique for detecting individual NV centers making it an ideal choice for investigating individual nanodiamonds on a coverslip. 

To get more detailed information about the diamond samples and to get a comparison for a potential 2D scan with the low-cost setup, a reference CLSM is built (see Fig. <ref>). Therefore, a single frequency diode-pumped laser (Cobolt 08-DPL) with a good beam quality (M^2 < 1.1) is expanded by a Galilean telescope (Thorlabs ACN127-030-A + Thorlabs LA1708-AB) with a magnification of  M=6.6̅  from approx. ≈ 0.6  mm to approx. ≈ 4  mm in diameter to fully illuminate the lens system inside the OPU. The beam is reflected by a dichroic beamsplitter (Thorlabs DMLP550R) and focused on the sample through a microscope objective (Nikon CFI Plan Apo Lambda 60XC) with a high NA of 0.95. To eliminate the laser- and background light, the collected fluorescence is spectrally filtered by the dichroic beam splitter, two long-pass (Thorlabs FELH0550) and one short-pass (Thorlabs FESH0750) filter. Following this, a spatial frequency filter (30   m pinhole,  2  × Edmund Optics  48774  f=100 mm) is used to eliminate out-of-focus light. To perform a 2d confocal scan of a sample, a piezo translation stage (PiezoSystem Jena Tritor 100, open loop) is used to allow movements with nanometer precision and ensure accurate positioning. For the Hanbury Brown and Twiss (HBT) experiment, two actively quenched and cooled SPAD (Laser Components COUNT T-series) and the correlation electronics (TimeTagger 20, Swabian Instruments) are utilized. Furthermore, a scientific CMOS camera (Hamamatsu Orca Fusion BT) for widefield microscopy for pre-aligning the nanodiamond sample slide is integrated into the beam path.



 §.§ Low Cost Single Photon Coupling


NV-centers can exist in the NV^- negative or NV^0 neutral charge state and both can act as a source of single photons. The NV-centers can efficiently be excited using a laser at a wavelength of 532  nm. The fluorescence spectrum for an ensemble of NC-centers inside a 100  m diamond is shown in figure  <ref> (a). Both charge states can be identified in the sample by the zero phonon line at 575  nm <cit.> for the NV^0-center and 638 nm  <cit.> for the NV^--center. At room temperature, the emission is broader ranging from 630   nm to 800  nm. The lifetimes of the two charge states in nanodiamonds differ slightly. The NV^- centers has a lifetime of 12 ns compared to 21  ns for the NV^0 center <cit.>. This makes the NV^- center brighter and more easily to be detected. A spectrum of a bulk diamond with 1 m in diameter containing NV centers in both charge states was captured using a spectrometer (Ocean HDX spectrometer) in the described optical setup and is exemplarily depicted in Figure <ref> (a) (red).

As a low-cost approach, and as a replacement for the microscope objective to collect the photons, the integrated lens from a PHR-803T Blu-Ray pickup is to be used (see Fig. <ref> (a)). As shown in Figure <ref>, it is integrated into the existing setup using a flip mirror. The aspheric lens has a large NA of about 0.6 at 650 nm <cit.>. Transmission measurements for this optic were taken using a spectrophotometer (PerkinElmer Lambda 1050), which shows that the lens has an anti-reflective coating and high transmission, particularly in the spectral range of the NV center emission (see Fig. <ref> (a) (black)). The OPU has a smaller transmission at the pump laser wavelength and can be compensated by increased laser power.





 §.§ Scanning OPU Setup


To be able to perform spatial scans without employing piezo actuators, the internal positioning mechanisms of the PHR-803T are utilized for the XY-scan. The optics of a single OPU can be moved along the horizontal and vertical direction and can also be tilted in the horizontal direction. The operation principle is similar to voice-coil actuators.
Five individual coils are directly fixed to the lens holder.  Two permanent magnets near the coils create an external static magnetic field and allow the optics to be positioned precisely. For the laboratory demonstrator, the OPUs were clamped in optical mounts and placed opposite each other. In order to realize the 2D scan, it is necessary to install two PHR-803T at an angle of 90^ ∘ to each other, as seen in Figure <ref> (b) <cit.>. A solid 3D-printed housing is intended for future setups.
  

Since the lens is mounted in the center position and is held by four springs, it can deflect in two directions.  Therefore, the direction of the currents within coils needs to be controlled. To ensure this, DC-motor control operational amplifiers (TCA0372) were used <cit.>, which provide an output current of up to one ampere. The input voltages for the scan are provided by the 16-bit data digital-to-analog converter (DAC, RedLab 3103 USB), which was also used to provide the input voltages for the piezo driver in the reference setup. In the DC motor circuit used here, the virtual ground is at ≈ 2  V output voltage of the DAC. Here, the lens is in the center position and can be deflected bidirectionally with a control voltage between 1  to  3  V, resulting in coil voltages between -1  to 1  V and coil currents between -200  to  200  mA.  The deflection in the horizontal direction in  Figure <ref> (b) was measured by continuously increasing the current through the coils and tracking it with a manual translation stage. The deflection has a high linearity in both directions around the center position of the lens.  




One of the factors limiting the resolution of a CSLM is the size of the exciting laser spot on the sample. For the nanodiamonds to be excited individually, the distribution on the slide and thus the minimum distance between the individual diamonds needs to agree with the diameter of the laser spot. Since these OPUs are not optimized for the green wavelength of the excitation laser and no information about the focal length is available, an optical approach was chosen to determine the focus size. We positioned a CMOS image sensor (Raspberry Pi High Quality Camera Module) at various distances behind the lens. Gaussian fits through the center of the mode images (an example is shown in Figure <ref> (c)) were used to measure diameters at different positions. Subsequently, the transverse Gaussian beam profile w(z) = w_0 √(1 + ( z λ/π w_0^2)^2) was fitted for all measured diameters, as shown in Figure <ref> (c). From this fit, a beam radius in the focus of w_0 = 1.66 ± 0.74  m can be estimated. The parameters for spin coating should therefore be selected so that the diamonds are at least 2 · w_0 apart. The mode profile pictured in Figure <ref> (c) appears suboptimal, suggesting that the smallest achievable focus diameter may not be reached, possibly due to aberration in the OPUs imaging system, since it is not optimized for our external pump wavelength of 532 nm. Nevertheless, the estimated spot diameter is sufficient for our intended measurements.



 §.§ Samples Preparation


 Fused silica serves as an ideal coverslip substrate for nanodiamond adherence due to its minimal autofluorescence. This reduces unwanted background signals and ensures a better signal-to-noise ratio, which is very important considering the small fluorescence signal of single NV centers.

To achieve a uniform deposition of nanodiamonds (Adamas Nanotechnologies NDNV40nmLw10ml) on the coverslip, a multi-step process is employed. A concentration of 1:100 nanodiamonds in deionized water (DI-water) is explored, with a solution comprising 1 mg/ml diamonds in DI-water and 0.3  wt% PVA to enhance adhesion and distribution. The coverslips undergo pretreatment with a 2% Hellmanex solution in DI-water, as suggested in <cit.>, utilizing ultrasonic bath exposure at 50^∘ C for 10 minutes to enhance cleaning. This step minimizes contamination from residues, oils, and particles while improving nanodiamond adhesion. The ultrasonic bath also aids in dispersing nanodiamonds in DI-water and PVA, ensuring a homogeneous particle distribution. Subsequently, a spin-coating technique is employed, where 30  l of nanodiamond suspension is placed at the coverslip center and rotated at 3500  rpm. This process utilizes centrifugal force to create a thin, uniform nanodiamond film, offering precise control over coating density through spin speed and duration adjustments.



§ CONFOCAL SCAN AND MEASUREMENTS OF THE NV CENTER



The 2D scanning, focusing, and data acquisition are controlled for both setups using a Python script.
The fluorescence intensity plots generated from the 2D scans indicate the quality of the control of the pickup coils. The bright spots in the image of the low-cost system (see Fig. <ref>(b)), are slightly oval with a lower signal-to-noise ratio compared to those in the reference setup (see Fig. <ref> (a)). This may imply that the wire resistance of the pickup coils varies slightly from coil to coil, resulting in differences in the strength of linear displacement with the control voltages.  The reduced contrast observed in the scan obtained from the low-cost setup compared to that obtained from the reference setup can be attributed to the increased autofluorescence emanating at 532 nm excitation wavelength from the polymeric aspheric lens of the OPU (see Fig. <ref> (e)). Unfortunately, the autofluorescence occurs in the same spectral region as the emission spectrum of the NV center, thus reducing the signal-to-noise ratio.

The detection of single photons with low-cost optics can be demonstrated by measuring the anti-bunching behavior and comparing it with the reference setup.
For the red-marked diamonds of the reference and low-cost scans (see Fig. <ref> (a), (b)), the g^2(τ)-functions were measured. Surprisingly, during the measurement, the PHR-803T coils showed significantly less drift than the open-loop piezo stages of the reference setup.  An antibunching effect can be seen with both setups (see Fig. <ref> (c), (d)). 
The fact that the g^(2)(0) function reaches a value below 0.5 is a strong indicator of the presence of a single NV center.




§ CONCLUSION

To summarise, the development of mechanically stable 3D confocal scanning for measuring single photon emission is a significant advance in terms of cost efficiency and applicability in this field. By using affordable components such as two OPUs,  that cost about 30   in sum, this approach stands in strong contrast to conventional setups, like our reference setup presented here, with expensive high-NA microscope objectives and piezo stages costing routinely more than 10k .  
The development of a low-cost single photon collection system based on OPUs holds the potential to advance quantum technologies and promote their widespread adoption in industrial applications. This would also benefit schools and universities, for which quantum experiments are often too expensive and complex. The integrated lenses and voice coil actuators for positioning ensure a compact and alignment-friendly design for further experiments in single photon source designs and laser-scanning confocal microscopes.






§ DISCLOSURES
 The authors declare no conflicts of interest.



§ DATA AVAILABILITY
 The data that support the findings of this study are available from the corresponding author upon reasonable request.


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 Paper Title: Shadow Hamiltonian Simulation.









Shadow Hamiltonian Simulation
    
Rolando D. SommagoogleGoogle Quantum AI, Venice, CA 90291, USA
  
Robbie Kinggoogle
  
Robin Kotharigoogle
  
Thomas O'Briengoogle
  
Ryan Babbushgoogle

    
==============================================================================================================================================================



§ ABSTRACT 
We present shadow Hamiltonian simulation, a framework for simulating quantum dynamics using a compressed quantum state that we call the “shadow state”. The amplitudes of this shadow state are proportional to the expectations of a set of operators of interest. The shadow state evolves according to its own Schrödinger equation, and under broad conditions can be simulated on a quantum computer. 
    
We analyze a number of applications of this framework to quantum simulation problems. This includes simulating the dynamics of exponentially large systems of free fermions, or exponentially large systems of free bosons, the latter example recovering a recent algorithm for simulating exponentially many classical harmonic oscillators. Shadow Hamiltonian simulation can be extended to simulate expectations of more complex operators such as two-time correlators or Green's functions, and to study the evolution of operators themselves in the Heisenberg picture.














§ INTRODUCTION


One of the most important applications of quantum computers is  simulating the dynamics of quantum systems, often referred to as the Hamiltonian simulation problem. The standard presentation of this problem involves an initial quantum state |ψ(0)⟩ that is easy to prepare on a quantum computer and a Hamiltonian H that describes the interactions in the system. The goal is to produce the state at a later time t>0, |ψ(t)⟩, which satisfies the Schrödinger equation: d/ d t|ψ(t)⟩=-iH|ψ(t)⟩.
A plethora of Hamiltonian simulation algorithms exist, with various complexities in terms of the relevant problem parameters, including the time t, the error ϵ to which the final state is prepared, and intrinsic properties of the Hamiltonian such as the magnitude of interaction strengths (cf. <cit.>). Under reasonable assumptions, these complexities are logarithmic in the dimension of the Hilbert space, thus providing an exponential advantage over classical algorithms for this problem, which might require time polynomial in the dimension.

In applications of standard Hamiltonian simulation, once the final state |ψ(t)⟩ has been prepared, one is often interested in obtaining some physical property, such as the expectation of a particular observable. One example is the ground state energy of the system, which can be obtained by measuring the expectation of H or by performing quantum eigenvalue estimation (cf. <cit.>). Another example could be the computation of an order parameter in a condensed matter system that can be obtained by measuring a two-body correlation function <cit.>. Many of these properties can be extracted from a limited set of expectations, such as those of products of a few Pauli operators in qubit systems, or those of products of a few fermionic operators in fermionic systems, among others. 
While access to |ψ(t)⟩ allows us to obtain the  expectation of any observable on this state, for certain problems, it might suffice to prepare a different quantum state that does not encode all the information encoded by |ψ(t)⟩, but instead only encodes information about certain properties.
Doing so could result in significant computational savings; or, it could potentially let us use resources differently, 
for example, enabling the simulation of quantum systems of
sizes that are significantly larger than those that can be studied with standard Hamiltonian simulation approaches. 

Based on this observation, we introduce a framework to perform quantum simulation 
that we call “shadow Hamiltonian simulation”. Rather than storing the entire time-dependent state in quantum memory during the quantum computation, we encode certain expectations in the amplitudes of a quantum state of smaller dimension, which we refer to as the   
“shadow state”. We will see that, under a broad set of conditions, the shadow state evolves unitarily under its own Schrödinger equation, which captures the Heisenberg-picture dynamics of the observables.
For many examples, this evolution can be simulated efficiently on a quantum computer.
Furthermore, we will see that a shadow state is a projection of |ψ(t)⟩⊗|ψ(t)⟩[We use |ψ⟩ to denote the state whose amplitudes are complex conjugates of the amplitudes of |ψ⟩ in the computational basis.] on a certain subspace specified by the operators whose expectations are encoded, hence creating a “shadow” of the state.


To provide an idea of what type of problems can be 
efficiently solved using the shadow Hamiltonian simulation approach, we apply it to free-fermion and free-boson systems, qubit systems, and other systems with a Lie algebraic structure that arises naturally within this context. In the first two cases, by encoding only the expectations of products of fermionic or bosonic operators of constant degree, 
we can leverage the exponential dimension of 
the shadow state to simulate systems of exponential size. 
These instances can be relevant for first-quantization simulations, 
where increasing the number of modes, corresponding to the number of points in a discretized grid, allow for better approximations of the states <cit.>. They can also be relevant for simulating disordered quantum systems, where classical methods perform poorly.
We can then use shadow states to compute, for example, energies associated with fermionic or bosonic modes at any time, including expectations of Hamiltonians that are interacting (i.e., they are not-free fermion or free-boson Hamiltonians),  and other interesting quantities that would require exponential resources using standard Hamiltonian simulation methods.
Moreover, using a classical–quantum correspondence, we are able to reproduce the results of Ref. <cit.> on simulating classical oscillators, which can alternatively be interpreted as an example of shadow Hamiltonian simulation using a classical–quantum correspondence.

The principles behind shadow Hamiltonian simulation can be generalized in a way that allows encoding the expectations of more complex operators, such as two-time correlators, in the amplitudes of a different quantum state. 
This is convenient for exploring other dynamical properties of the system, yet without requiring the preparation of the full system's state. 
Additionally, while shadow Hamiltonian simulation is mainly designed to track the time-dependent expectations of certain operators, it can be easily extended to track the evolution of these operators determined by Heisenberg's equations of motion, without making any reference to a system's state. This enables other efficient quantum computations, such as finding signatures of operator spreading (e.g., light cones) or scrambling <cit.>, for which classical algorithms can take exponential time.

Shadow Hamiltonian simulation relaxes the requirements of standard Hamiltonian simulation in the same way that shadow state tomography relaxes the requirements of full quantum state tomography <cit.>. 
That is, in the traditional state tomography problem, we are given copies of an unknown state and need to learn a full classical description of the state, from which we can obtain the expectation of any observable. 
This task requires exponentially many (in the system size) copies of the unknown state in general. In contrast, in shadow tomography, the goal is to learn expectations of a limited set of observables  on the state, a task that can be shown to be much easier than traditional state tomography. 
Nonetheless, both problems are different: while shadow tomography concerns the problem of learning some classical description of a quantum state, 
shadow Hamiltonian simulation is related to the problem of evolving expectations in time. 



§ FORMALISM




 §.§ Shadow states







Let S={O_1,…,O_M} be a set of M < ∞ operators and suppose we are only interested in a set of expectations ⟨ O_m ⟩, for all m ∈ [M]:={1,…,M}, with respect to a (potentially mixed) quantum state ρ. We would like to store a compressed version of ρ that does not necessarily allow us to reconstruct it, but does allow us to recover certain quantities that depend on the ⟨ O_m ⟩'s only.
We call this the “shadow state” of ρ with respect to the set S and define it as follows.



    Let ρ be a quantum state and S={O_1,…,O_M} be a set of operators. The shadow state of ρ with respect to S is  the M-dimensional (pure) unit state
    
    |ρ; S⟩ = 1/√(A)[ ⟨ O_1 ⟩⋮⟨ O_M ⟩ ]
            = 1/√(A)∑_m=1^M ⟨ O_m ⟩|m⟩,

    where A = ∑_m ⟨ O_m⟩^2 and, for m ∈ [M], ⟨ O_m ⟩ is the expectation of the operator O_m with respect to ρ. For finite-dimensional systems, ⟨ O_m ⟩ = (ρ  O_m). 



 
In the definition, we implicitly assume that at least one expectation ⟨ O_m ⟩ is nonzero and that the expectations are bounded, or otherwise the definition of shadow state would be irrelevant. For a given set S, the same shadow state can correspond to infinitely many physical states ρ, which occurs when S is not a complete set of operators. 
In this case, |ρ; S⟩ contains less information than ρ, in the sense that a classical description of ρ cannot be reconstructed from a classical description of |ρ; S⟩ (whereas |ρ; S⟩ is fully specified by ρ). 
Similarly, not every M-dimensional state corresponds to a valid system's state ρ, since shadow states contain certain structure determined by S.
While |ρ; S⟩ is a state of finite dimension M, we do not make any assumptions on the dimension of ρ. 
This allows us to consider infinite-dimensional systems as well, such as bosonic systems.  



For example, when S={1,X,Y,Z}^⊗ n is the set of all 4^n products of Pauli operators acting on n qubits, and if ρ=|ψ⟩⟨$| is ann-qubit pure quantum state, then|ρ;S⟩will be a2n-qubit state sinceM=4^n. While the state|ρ;S⟩looks unfamiliar, we show in <Ref> that, up to a simple “Bell rotation”, this is the state|ψ⟩⊗|ψ⟩where|ψ⟩is the complex conjugate of|ψ⟩. For this choice ofS, which spans the complete set of operators acting onℂ^2^n, the shadow state contains all the information contained in|ψ⟩. In fact, the shadow state is even more powerful than|ψ⟩in this case, in the sense that there are problems that can be solved with only one copy of|ψ⟩⊗|ψ⟩while still needing exponentially many copies of|ψ⟩alone to solve (<Ref>). 

Many applications studied in this article, however,  consider shadow states that contain less information than|ψ⟩. 
As a simple illustrative example where this occurs, consider a one-qubit state and the set of Pauli operatorsS={_2,X,Y}, where_2is the2×2identity. Then, 
any state that is polarized along theZdirection, i.e.,ρ= 1 /2 (_2 + ⟨Z ⟩Z), satisfies⟨X ⟩= ⟨Y ⟩=0, and results in the same shadow state, prohibiting the recovery of the full stateρfrom|ρ;S⟩in this example. 



A different example is whenSis the set of quadratic fermionic operators of the forma^†_j a^_k, wherea^†_jcreates a fermion in thej^thmode anda^_kannihilates a fermion in thek^thmode. In this case,|ρ;S⟩contains less information thanρ, which again follows from the fact that distinct fermionic states can  give rise to exactly the same values for the expectations⟨a^†_j a^_k⟩. In this example, however, the shadow state lets us encode exponentially many such expectations if we allow forMto be exponentially large. This enables the computation of certain properties of systems involving exponentially many fermionic modes, as we discuss in <Ref>. Similar observations apply to bosonic systems. 




One common task that can be solved with shadow states
is the estimation of an arbitrary linear combination of the expectations⟨O_m ⟩, which reduces to an overlap estimation problem. Another common task is that of sampling from a distribution where probabilities are(m) ∝|⟨O_m ⟩|^2, which can be done by
measuring the shadow state in the computational basis.
These tasks are related to other problems, like estimating energies of a subset of  modes in fermionic and bosonic systems of exponential size, which are discussed in <Ref>.




Having established the notion of shadow state, we proceed to discuss the problem of simulating quantum dynamics within this formalism. 





 §.§ Shadow Hamiltonian simulation




In standard Hamiltonian simulation, we are given a HamiltonianHand the goal is to map the (possibly mixed) initial state of the system,ρ(0), to  the final stateρ(t). The operation that implements this mapping is the evolution operatorU(t)determined byH. In shadow Hamiltonian simulation, we are additionally given a set of operatorsS, and the goal is instead to map the initial shadow state|ρ(0);S⟩to the final shadow state|ρ(t);S⟩. The challenge is to construct an efficient method that implements this mapping, which is different fromU(t)in general.

For illustration, we return to the example ofSbeing the set of all Pauli products acting onnqubits; the shadow state of a pure state|ψ(t)⟩is a simple Bell rotation of|ψ(t)⟩⊗|ψ(t)⟩. 
Despite the shadow state being more powerful than the original state in this example, the shadow Hamiltonian simulation problem is not much harder than standard Hamiltonian simulation. If we have an efficient quantum circuit to implement the unitaryU(t)that maps a pure state|ψ(0)⟩to|ψ(t)⟩, then we can also map|ψ(0)⟩⊗|ψ(0)⟩to|ψ(t)⟩ ⊗|ψ(t)⟩using the unitaryU(t)⊗U(t), whereU(t)is the complex conjugate ofU(t). 
Note that we can create an efficient circuit forU(t)by complex conjugating all the elementary gates appearing in the circuit that implementsU(t). But, more generally, it is not always the case that|ρ(0);S⟩can be easily mapped to|ρ(t);S⟩. Even worse, the vector|ρ(0);S⟩might not contain enough information to completely determine the vector|ρ(t);S⟩at a later time. 


For it to be possible to obtain|ρ(t);S⟩from|ρ(0);S⟩, even in principle, the expectations of the operators inSat timetshould be a function of the initial expectations at time 0. This occurs when the HamiltonianHleaves the space of operators spanned bySinvariant, in the sense that the commutation relations betweenHand the operators inSyield a linear combination of operators inSonly.



    Let H be a Hamiltonian and S={O_1,…,O_M} be a set of operators. We say that H and S satisfy the invariance property if for all m ∈ [M] we have
   
    [H,O_m]= -∑_m'=1^M h_mm' O_m' ,

   where h_mm' are coefficients.
Let H_S be the M × M matrix whose (m,m') entry is h_mm'. Then, we can write <ref> as

    [H,O⃗]= -H_S O⃗ ,

where O⃗ := (O_1,…,O_M)^T is a vector of operators. 




The invariance property ensures that the shadow Hamiltonian simulation problem is solvable, but the map that takes|ρ(0);S⟩to|ρ(t);S⟩need not be simple in general. Our first result is that if the matrixH_Sis Hermitian, then the shadow state of the  physical stateρ(t)that evolves withH,  evolves according to a Schrödinger equation but with the HamiltonianH_S. 


    Let H be a Hamiltonian and S={O_1,…,O_M} be a set of operators, such that H and S satisfy the invariance property. Let ρ(t) be the state of the quantum system obtained by evolving ρ(0) under H for time t.
    The shadow state |ρ(t);S⟩ satisfies the differential equation:
    
    d/ d t|ρ(t);S⟩=-iH_S|ρ(t);S⟩  .

    Furthermore, if H_S is Hermitian,  <ref> is a Schrödinger equation.

Let ρ(t) = U(t) ρ(0) U^†(t)=∑_ℓ p_ℓ|ψ_ℓ(t)⟩⟨$|, where thep_ℓ's are probabilities.
The amplitudes of|ρ(t);S⟩are proportional to⟨ O_m(t) ⟩=∑_ℓ p_ℓ⟨ψ_ℓ(t)| O_m |ψ_ℓ(t)⟩, 
and Schrödinger's equation and the invariance 
property imply 

    d/ d t⟨ O_m(t) ⟩    = 
      ∑_ℓ p_ℓ( d/ d t⟨ψ_ℓ(t)|)  O_m|ψ_ℓ(t)⟩ +  ⟨ψ_ℓ(t)|  O_m ( d/ d t|ψ_ℓ(t)⟩)
       = i∑_ℓ p_ℓ⟨ψ_ℓ(t)| [H,O_m]|ψ_ℓ(t)⟩
       = -i∑_m'=1^M h_mm'⟨ O_m'(t) ⟩ ,

which is also <ref>.



Hence, we can perform shadow Hamiltonian simulation on a quantum computer
ifH_Sis Hermitian and can be efficiently simulated. For finite-dimensional systems, this occurs
when the operators inSare orthogonal.


    Let the system be finite-dimensional and the operators in S be orthogonal, satisfying (O_m^† O_m')=λδ_mm', for some λ >0. Then H_S is Hermitian. 

   Under the orthogonality condition, the entries of
   H_S satisfy
    
    h_mm' = -1/λ(O^†_m' [H, O_m]) = -1/λ([O^†_m, H] O_m')_(take Hermitian conjugate) = - 1/λ(O^†_m [H, O_m'])_(cyclic property of trace) = h_m'm.




While orthogonality of the operators inSsuffices forH_Sto be Hermitian, this is not a necessary requirement in general. Indeed,
<Ref>
applies even if the operators inSare not linearly independent or even ifρ(t)is infinite-dimensional. 
 In <Ref>, we discuss an example of a system of free bosons, or coupled quantum harmonic oscillators, where the underlying Hilbert space is infinite-dimensional. The HamiltonianHis quadratic inb^†_jandb^_k, the creation and annihilation bosonic operators, respectively.  
For this specific example,
<Ref> generalizes the result in
Ref. <cit.>, which considers a quantum algorithm for simulating classical harmonic oscillators, but for simulating quantum oscillators in this case.


In general, the invariance property alone does not imply thatH_Sis Hermitian. 
However, it is sometimes possible to take a particular linear combination of theO_m's to fix this; that is, we can use the property[H, AO⃗]=- A H_SO⃗for anM × Mmatrix Aand take specific linear combinations of theO_m's to obtain a Hermitian matrix.
This provides some flexibility for shadow Hamiltonian simulation, and we will use this observation for simulating bosonic systems in <Ref>.
Also, note that <ref> applies even if H_Sis not Hermitian.  It might be possible
to apply quantum algorithms for simulating differential equations to prepare the shadow states in this case, where the evolution is not unitary (cf. <cit.>). These analyses are outside the scope of this article.

Additionally, since quantum circuits can be interpreted as time evolution with a time-dependent Hamiltonian, our results can be extended to address this case. 
More precisely, letG=G^L… G^1be a quantum circuit described as a sequence of gatesG^j,j ∈ [L]. The analogous property to the invariance property in this case is 

    (G^j)^† O_m G^j=∑_m' g^j_mm' O_m',

for allm ∈ [M]andj ∈ [L], whereg^j_mm'∈ℂ. These coefficients define anM × Mmatrix G^j_Sfor eachj. If these matrices are also unitary—the analogous condition to requiring H_Sbe Hermitian—then we can simulate them and perform maps between the corresponding shadow states.








 §.§ Beyond shadow Hamiltonian simulation




The basic idea underlying shadow Hamiltonian simulation
can be generalized and applied to other related problems. 
We discuss two such problems in <Ref>. In one case we show how to encode the expectations of more complex operators, these being products of time-dependent operators, in the amplitudes of another quantum state. Each of these operators evolves according to the Heisenberg picture with their own Hamiltonians. This provides a more general result, in which quantities like two-time correlators can be encoded. In another case, we show how to encode time-evolved operators in quantum states that can be simulated, without making reference to the state of the system. The approach here is similar to shadow Hamiltonian simulation, and by measuring the corresponding states we could obtain properties of the evolved operators, such as the size of their light cones as a function of time.






§ APPLICATIONS


In this section, we first apply shadow Hamiltonian simulation to quantum systems obeying different statistics, and depict
problems that can be solved efficiently using this formalism.
Subsequently, we demonstrate how similar ideas can be applied to related problems: encoding two-time correlators or expectations of products of operators in a quantum state, 
and encoding a time-dependent operator, subject to Heisenberg's equations of motion, in a quantum state.






 §.§ Free fermions




Free-fermion systems, which are modeled by quadratic fermionic Hamiltonians, appear ubiquitously in quantum chemistry and physics. Examples include spin models like the transverse-field Ising model <cit.>, BCS superconductivity <cit.>, integer quantum Hall effect <cit.>,
and other cases described by mean-field theories. 
Fornfermionic modes, the number of degree-kfermionic operators is(n^k), and in this case we could use shadow states to encode the expectations of operators acting on systems that are exponentially large, e.g.,n=2^r. These instances can arise in, for example, approaches based on first-quantization <cit.>.



To map between shadow states, 
the first goal is to provide a method for shadow Hamiltonian simulation of these systems that is efficient, i.e., of complexity polynomial inror(n), and polynomial in the evolution timet. We will show this is possible for  free-fermion systems under broad conditions.
The Hamiltonian of a free-fermion system is 

    H = ∑_j,k=1^n α_jk a^†_j a^_k + β_jk a^_j a^_k -β_jk a^†_j a^†_k  ,

wherea^†_j(a^_j) are fermionic creation (annihilation) operators satisfying anticommutation relations{a_j,a_k}={a^†_j,a^†_k}=0and{a_j,a^†_k}=δ_jk. The coefficientsα_jk,β_jk∈ℂare “interaction strengths” and satisfyα_kj=α_jk. To ease the exposition, we equivalently assume a presentation ofHin terms of2nMajorana operators, which are Hermitian and defined by

    c_2j-1 := a^†_j + a^_j  ,    c_2j :=i (a^†_j - a^_j)  ,    j = 1,…,n  .

These satisfy the anticommutation relations

    {c_j,c_k} = 2δ_jk ,    j,k = 1,…,2n  .

Hence,  <ref> becomes

    H = ∑_j,k=1^2nγ_jk c_j c_k  ,

where the interaction strengthsγ_jk∈ℂdefine a2n × 2nHermitian matrixΓ.[Without loss of generality, we can assume Γ to be purely imaginary.] 
Majorana operators are orthogonal and are mapped to products of Pauli operators acting onnqubits using the Jordan-Wigner transform <cit.>. Furthermore, it is known that quadratic fermionic operators 
are closed under taking commutators.[Quadratic fermionic operators span the Lie algebra 𝔰𝔬(2n); see <Ref>.]
More precisely, the relevant commutation relations are given by

    [c_jc_k,c_l]=2δ_lk c_j - 2δ_lj c_k,

which follow from <ref>.
This implies thatHand the setSof degree-kfermionic operators satisfy the invariance property in <Ref> for anyk ≥ 1.
Thus,  we may choose our operator setSto be given by monomials  of Majorana or creation and annihilation operators; examples areS={c_j}_1 ≤ j ≤ 2n,S={c_jc_k}_1 ≤ j < k ≤ 2n,S={c_jc_kc_lc_m}_1≤ j< k<l<m≤ 2n,S={a_j^†a^_k, a^_ja^_k, a_j^†a_k^†}_1 ≤ j ≤ n, 1≤ k ≤ n, or, whenHis fermion number-conserving,S={a_j^†a^_k}_1 ≤ j ≤ n, 1 ≤ k ≤ n.
In the other limit, we could consider all possible products of fermionic operators (up to degree-2n), but this case is not too interesting
as it is equivalent to standard Hamiltonian simulation that provides access to the full state|ψ(t)⟩; see <ref>.
All these sets satisfy the invariance property; however, for H_Sto be Hermitian, some specific linear combinations of these operators might be necessary to make them orthogonal.


As an illustrative example, we chooseS={c_jc_k}_1 ≤ j < k ≤ 2n, and note that thec_jc_k's are readily orthogonal.
The amplitudes of the shadow state|ρ(t);S⟩are proportional to the expectations⟨ c_j c_k(t) ⟩=(ρ(t)  c_j c_k), which in this case coincide with the entries
of the so-called 1-RDM (reduced density matrix) of the evolved system's stateρ(t)<cit.>.
To prepare the shadow state at timetfrom that at time 0, we would like to simulate the HamiltonianH_Sin <ref> efficiently.
We can do this with standard Hamiltonian simulation methods whenH_Sis sparse and when we have efficient access to its nonzero entries. That is, in the sparse-access model, given a row index ofH_S, we can efficiently locate and compute the nonzero entries for that row (cf. <cit.>).
SinceH_Sis obtained from the commutation relations betweenHandS, <ref> implies that a sufficient property for having efficient sparse access toH_Sis having efficient sparse access to the matrixΓ. This occurs when the graph underlying the interactions in <ref> is of bounded degreed, and holds even if the number of fermionic modes, and hence the dimension ofH_S, is exponentially large ((n)).
For this example, the dimension ofH_SisM=(n^2), its sparsity isd' ≤ 2d(at most2dquadratic operators inHdo not commute with a givenc_jc_k, and the commutator[c_jc_k,c_lc_m]between two quadratic operators is another quadratic operator or zero), and its largest entry in magnitude satisfiesH_S_max≤ 2 max_jk |γ_jk|=2Γ_max.
A query toH_Scan be constructed with(d)queries toΓ: any row ofH_Sis labeled by some(j,k)corresponding to an operator inS, and all of the neighbors ofjandkin the graph can be learned with at most2dqueries toΓ. Hence, H_Scan be simulated efficiently under these assumptions, with complexity(n)and polynomial int Γ_max, using known Hamiltonian simulation approaches. 


For shadow Hamiltonian simulation to be efficient,
we also need a procedure that prepares the initial
shadow state|ρ(0);S⟩efficiently.
This is not always possible, but for many interesting states that can be expressed succinctly, efficient quantum circuits might be constructed. One example is the preparation of the shadow state corresponding to the bare vacuumρ(0)=ρ= | vac⟩⟨$|.
The bare vacuum is the pure state containing no fermions, so it is destroyed by all annihilation operators: a_j  | vac⟩ =0 for all j ∈ [n].
In this state, the expectations of quadratic Majorana operators are

    ⟨ c_j c_k ⟩ = 
            i     (j,k) = (2l-1,2l)  for l ∈ [n] ,
    
            0    otherwise.

The corresponding shadow state with respect to S is (up to a global phase)

    |ρ;S⟩ = 1/√(n)∑_l=1^n |2l-1,2l⟩ .

This state has a simple form, i.e., it is an equal superposition over basis states and can be prepared on a quantum computer in
time (log(n)) using elementary gates.
It is equivalent to the shadow state corresponding to the completely filled state, which is the state with one fermion in every mode.
This is implied by the fact that operators in S change sign under a particle-hole transformation that maps a^†_j ↦ a^_j and a^_j ↦ a^†_j <cit.>. 
More generally, for this choice of S={c_jc_k}_1≤ j<k≤ n, the shadow state of any ρ is invariant under the particle-hole transformation (up to a global sign). This property can be useful to identify other shadow states that can also be prepared efficiently.
Indeed, other examples
of shadow states that can be prepared efficiently are those corresponding to fermionic product states ∏_j ∈ I a^†_j | vac⟩ that contain one or no fermions in each mode j ∈ [n], where I ⊆ [n] is a subset of modes. 
These shadow states can be obtained from <ref>
by simply changing some signs of the amplitudes, which can be done efficiently if we have access to an oracle that specifies I efficiently. The shadow states of
more general fermionic Gaussian states (Slater determinants) |ϕ⟩ could be prepared, for example, by means of Thouless's theorem <cit.>, which asserts
that |ϕ⟩ can be obtained as e^-iH̃∏_j ∈ I a^†_j | vac⟩. 
Here, H̃ is a free-fermion and number-conserving Hamiltonian.
If H̃
can be accessed efficiently as explained above, then the Hamiltonian H̃_S required for mapping corresponding shadow states
can also be implemented efficiently. 
In general, there will be other shadow states that can be prepared efficiently, and which do not necessarily correspond to fermionic Gaussian states.




Given the shadow state of an exponentially large fermionic system, there are various quantities of interest which can be extracted from the shadow state.
As an example, consider the problem of estimating
the energy ⟨ H_J ⟩ corresponding to a subset J ⊆ [2n]
of Majorana modes, where H_J:=∑_j ∈ J,k ∈ Jγ_jk c_j c_k. This problem can be reduced to an overlap estimation problem, since

    ⟨ H_J ⟩ = G √(A)⟨ψ_J |ρ ; S⟩ ,

where |ψ_J⟩:=1/G∑_j ∈ J,k ∈ Jγ_jk|j,k⟩  is a unit quantum state that encodes the interaction strengths γ_jk corresponding to J, and G =[∑_j∈ J,k∈ Jγ_jk^2]^1/2 is a normalization constant.  The constant A is also for normalization, i.e., |ρ;S⟩=∑_jk⟨ c_jc_k ⟩|j,k⟩ / √(A).
The overlap ⟨ψ_J |ρ;S⟩ can be obtained within additive precision ϵ using Heisenberg-limited methods, which require the unitary that prepares |ρ;S⟩ and its inverse (1/ϵ) times <cit.>. 
For instances where the set J is extensive so that |J| is proportional to n, and when the matrix Γ of interaction strengths is sparse and bounded, then G=(√(n)Γ_ max). Also, A ≤ n in general, since this is maximized for fermionic Gaussian states, including the bare vacuum specified by <ref>. Computing the (intensive) property ⟨ H_J ⟩/(n Γ_ max) is then a problem that can be solved efficiently, given access to shadow states with respect to the set S of quadratic fermionic operators. 

More generally, we can consider shadow states that encode 
the expectations of degree-k fermionic operators, k>2, and with them compute linear combinations of these expectations. This 
would allow us to estimate energies of interacting fermionic systems. For example, a standard problem in fermionic simulations involves preparing a fermionic Gaussian state and then computing properties of an interacting fermionic Hamiltonian that involves quartic fermionic operators. 
The fermionic Gaussian state can be prepared through an evolution of a simple state with a free-fermion Hamiltonian. Using shadow Hamiltonian simulation, we can prepare the shadow state of this Gaussian state efficiently using the techniques above, even if we allow for encoding expectations of quartic operators (see also <Ref>). 
Estimating the energy of the interacting system also reduces to an overlap estimation problem that uses this shadow state.








  §.§.§ Comparison to first-quantization


First-quantization methods enable the simulation of exponentially many fermionic modes as well <cit.>. 
Consider a free-fermion Hamiltonian whose terms are number-conserving, i.e., it is a linear combination of 
a^†_j a^_k only. In the subspace of a single fermion, the Hilbert space dimension is n, and H can be represented as an n × n matrix.
Simulating this Hamiltonian in that subspace can be done efficiently if we have efficient sparse access to this matrix.
This simulation approach will produce a quantum state that corresponds to the single-fermion evolved state.
This state can be used to compute properties of the 1-RDM at any time; note that the entries of the 1-RDM for  any fermionic state, including those without a well-defined number of fermions and non-Gaussian states, 
can be faithfully reproduced and evolved by free-fermion Hamiltonians simply by working in the single-fermion subspace.
First-quantization methods can be used to simulate two or more fermions, which would require increasing the number of registers and the dimension accordingly, to compute properties of the 2-RDM and so on.

First-quantization and shadow Hamiltonian simulation coincide in the special case where the Hamiltonian is free-fermion and number preserving, and the system contains only a single fermion. In this case, the shadow state is equivalent to |ψ⟩|⟩, where |ψ⟩ is the first-quantized wavefunction.

However, there are important differences between the two techniques. First of all, the two approaches depart from one another when there are many fermions in the system. Secondly, shadow Hamiltonian simulation can deal with free-fermion Hamiltonians that are not number preserving, and it can also encode the expectations of non-number-preserving operators such as a_i a_j and a_i^† a_j^† in the corresponding shadow states, using a proper choice for S. On the other hand, first quantization can simulate interacting fermionic Hamiltonians, as long as they are number-preserving; shadow Hamiltonian simulation breaks down in this case, since the invariance property is not satisfied.









 §.§ Free bosons


Free-boson systems are also prevalent in physics, being able to describe physical
phenomena such as  superfluidity <cit.>, quantum optics <cit.>, and more. 
Free-boson systems can also be understood as a collection 
of coupled quantum harmonic oscillators, which are quantum particles evolving under the influence of quadratic potentials. Like in the prior example, we are interested in simulating exponentially large systems efficiently, in time (n) and (t).


A free-boson system is described by a quadratic Hamiltonian of the form

    H = ∑_j,k=1^n α_jk b^†_j b^_k + β_jk b^_j b^_k +β_jk b^†_j b^†_k  ,

where b^†_j (b^_j) are the bosonic creation (annihilation) operators satisfying the canonical commutation relations [b^_j,b^†_k]=δ_jk, [b^†_j,b^†_k]=[b^_j,b^_k]=0. The coefficients α_jk∈ℂ and β_jk∈ℂ are “interaction strengths” that satisfy α_jk=α_kj. 
In contrast to the free-fermion case, the Hilbert space dimension is now infinite. We equivalently assume a presentation of H in terms of 2n operators associated with  position and momentum operators, which are Hermitian and defined via

    Q_j ∝1/√(2) (b^†_j + b^_j)   ,     P_j ∝i/√(2) (b^†_j - b^_j)  ,    j=1,…,n  .

(The constants of proportionality can depend on j.)
Hence, <ref> becomes
quadratic in the Q_j's and P_j's. If we let Y⃗:=(P_1,P_2,…,Q_1,Q_2,…)^T be the vector of 2n operators, we can write H=1/2Y⃗^T ΓY⃗, where Γ is a 2n × 2n Hermitian matrix of interaction strengths γ_jk, and we can assume these to be real, i.e., Γ=Γ.


It is well known that quadratic bosonic operators are closed under commutation.[Quadratic bosonic operators span the symplectic Lie algebra 𝔰𝔭(2n); see <Ref>.] 
More precisely, the commutation relations between generalized coordinates can be  compactly  expressed as  [Y_j,Y_k]=iΩ_jk, for j,k ∈ [2n]. Here,
  Ω_jk is the entry of the symplectic matrix Ω=[      0 -_n _n      0 ], where _n is the n × n identity. 
 This implies

    [Y_j Y_k,Y_l]=iΩ_kl Y_j  + iΩ_jl Y_k  .
 
Then, the set of degree-k  bosonic operators also satisfy the invariance property of <Ref> for any k≥ 1. Like in the fermionic case, this means that we can take our set of operators to be given by low-degree products of the bosonic operators Q_j and P_j.


As a special case we could choose S={P_1,P_2,…,Q_1,Q_2,…}. However, with this choice, one immediately arrives at the issue that H_S is not Hermitian, since
[H,Y⃗]=-iΩΓY⃗ from <ref>,
and -iΩΓ is not Hermitian in general. 
This operator is only Hermitian in the special case of number-conserving Hamiltonians, where all β_jk in <ref> vanish, and this case was recently addressed in Ref. <cit.>.
To circumvent this issue in our example, we consider a decomposition Γ=Γ_1Γ_2 for some Γ_1 of dimension 2n × M and Γ_2 of dimension M × 2n, where M ≥rank(Γ). 
Then, transforming
Y⃗↦O⃗:=Γ_2 Y⃗
changes the coefficients in the commutators such that the invariance property reads
[H,O⃗]=-iΓ_2 ΩΓ_1 O⃗.
The set of operators is now S:={O_1,…,O_M},
and we will be interested in instances where iΓ_2 ΩΓ_1 ≡ H_S is Hermitian.

One example is obtained when the matrix of interaction strengths is additionally positive semidefinite, i.e., Γ≽ 0.
It is possible to find a factorization where Γ= B B^† in this case, so that O⃗:= B^†Y⃗, H=1/2O⃗^T O⃗,
and -i B^†Ω B is Hermitian, i.e., H_S=i B^†Ω B= H_S^†. Note that 
to prepare the shadow state with respect to this S, we would like to simulate
this H_S efficiently, and this might require accessing
B or B^† efficiently. 
Reference <cit.> describes a way to do this for a collection of coupled classical oscillators; the same result applies to this case if the Hamiltonian is that of exponentially many coupled quantum harmonic oscillators, expressed as

    H=  ∑_j (P_j)^2/2m_j+ κ_jj/2  (Q_j)^2 +∑_k>jκ_jk/2 (Q_j-Q_k)^2  .

Here, m_j>0 and κ_jk≥ 0 for all j ∈ [n], k ∈ [n], are the masses and spring constants.
Given sparse access to matrices M and K
that specify the m_j's and κ_jk's, the matrices B,  B^†, and H_S can be accessed efficiently.
This occurs when the graph underlying the interactions in the system is of bounded degree d.
The factorization in Ref. <cit.> is such that M is at least the number of terms in <ref>. In this case, the vector of operators O⃗ =  B^†Y⃗ is the one with
the first n entries being the P_j/√(m_j)'s, corresponding to the square roots of the first n terms  in <ref>, and the remaining (n^2)
entries being either the √(κ_jj)Q_j's or the √(κ_jk)(Q_j-Q_k)'s, corresponding to the square roots of the remaining terms in <ref>.
The complexity of simulating the H_S obtained through this factorization of Γ is almost linear in 
t √(ℵ d), where ℵ:=max_j,kκ_jk/min_j m_j, and (n). This simulation is efficient and discussed in Ref. <cit.> in great detail.



Hence, choosing S={O_1,…,O_M} for this example, where the O_m's are defined above, the shadow state at any time with respect to S can be expressed as

    |ρ(t);S⟩   =1/√(A)∑_m=1^M ⟨ O_m(t) ⟩|m⟩≡1/√(A)(∑_j=1^n 1/√(m_j)⟨ 
        P_j (t) ⟩|j⟩|j⟩+  .
       . √(κ_jj)⟨ 
        Q_j (t) ⟩|j+n⟩|j+n⟩ +∑_n+1 ≤ j<k ≤ 2n√(κ_jk)⟨ 
        (Q_j(t) -Q_k(t)) ⟩|j⟩|k⟩)  .

The expectations are with respect to the system's state ρ(t) at time t.
To prepare |ρ(t);S⟩ efficiently, we also 
require the initial shadow state |ρ(0);S⟩ to 
be efficiently preparable. This is not always possible, however, there are examples where this can be done. 
For illustration, consider the case of a bosonic product state ρ where
⟨ Q_j(0) ⟩=0 and 1/√( m_j)⟨ P_j (0)⟩ is the same constant for all j ∈ [n]. 
Then, |ρ(0);S⟩=1/√(n)∑_j=1^n |j⟩|j⟩
is  a simple superposition state that can be prepared with (log(n)) elementary gates. Like in the case of free-fermion systems, other shadow states
can be considered by building upon this example.



As an application that uses the shadow state in <ref>,
we consider the problem of estimating a semiclassical approximation to the (rescaled) kinetic or potential energies
of a subset of masses or springs. This is essentially the same problem discussed in Ref. <cit.>, and the reason why this becomes a semiclassical approximation here is because the expectation of a quadratic bosonic operator is not equal to the product of expectations of single operators in general. 
This estimation problem can be solved efficiently, also requiring 
(1/ϵ) preparations |ρ(t);S⟩ and its inverse, where ϵ>0 is the allowed error. 

To avoid this semiclassical approximation, we can consider instead a set of quadratic operators where now S={O_m O_m'}_1 ≤ m ≤ M,1≤ m' ≤ M, and each O_m is the same as in the prior example, being linear in the P_j's and Q_j's. This operator set also satisfies the invariance property. With this choice we have
    |ρ;S⟩ =1/√(A)∑_m,m'⟨ O_m O_m'⟩|m,m'⟩, where A>0 is for normalization.
It can be shown that the Hamiltonian used to prepare the new shadow state at time t from that at time 0 is H_S ⊗_M + _M ⊗ H_S, where H_S is the Hamiltonian described above, and which can be simulated efficiently under the assumptions. (Indeed, in <ref> we provide a general result that proves this is the correct Hamiltonian in this example.) 
The energy of a subset of oscillators or springs
is ⟨ H_I⟩=1/2∑_m ∈ I⟨ (O_m)^2 ⟩, where I ⊆ [M] denotes the subset.  (Recall that each 1/2(O_m)^2 corresponds to a term in <ref>.)
This problem can also be reduced to an overlap estimation problem, since

    ⟨ H_I⟩ =1/2√(A |I|)⟨ψ_I |ρ;S⟩ ,

where |ψ_I⟩=1/√(|I|)∑_m ∈ I|m,m⟩ is a unit quantum state of nonzero amplitudes only if m=m'. The overlap ⟨ψ_I |ρ;S⟩ can be obtained within additive error ϵ with (1/ϵ) uses of the unitary that prepares |ρ;S⟩ and its inverse <cit.>. If I is extensive, then 
|I| is proportional to n. However, the normalization constant A can be Ω(n^2) in general since the number of operators in S is now Θ(n^2)  and all their expectations can be nonzero. (Note that M ∝ n if the interactions are of bounded degree.)
This is inconvenient since, to be efficient, the quantum algorithm  would output an estimate of ⟨ H_I⟩/n^3/2, which approaches zero in the large n limit. Nevertheless, it is still possible to consider instances of the problem where the shadow state satisfies A=( n), giving an efficient quantum algorithm to estimate ⟨ H_I⟩/n. This quantity is (1) if ⟨ H_I⟩ is extensive. 
Examples of such shadow states are those corresponding to low-energy states ρ, where only (n) expectations ⟨ O_m O_m'⟩ contribute significantly to the normalization constant.
Moreover, if S contains bosonic operators of degree k>2,
we can perform other calculations like estimating the energies
of interacting bosonic systems in the corresponding states.


We note that our shadow Hamiltonian simulation approach
for this example is clearly different from other approaches aimed at simulating bosonic field theories formulated in the Schrödinger picture approach <cit.>. 
Here, we did not perform any cutoff on the space dimension or other reductions to encode an infinite dimensional system in one that is finite dimensional, and the simulation is  exact, in the sense that we provided a recipe to prepare |ρ(t);S⟩ for exponentially many bosonic modes.
This, however, comes at the expense of only being able to 
simulate the expectations of a restricted set of operators, which are linear combinations of (low-degree) products of the Q_j's and P_j's. At the same time, it is possible
to consider larger degree products of the operators  to encode more information of the states. 
We leave such investigations open. 






  §.§.§ From classical to quantum harmonic oscillators


There is an inherent relation between the quantum algorithm of
Ref <cit.> for simulating exponentially many coupled classical oscillators and shadow Hamiltonian simulation of quantum oscillators. 
That quantum algorithm shares similar features in that we encoded the coordinates (position and momentum) of the oscillators in the amplitudes of a quantum state, which can be shown to evolve unitarily. While those are classical variables, it is simple to show that the dynamics of such a classical system is similar to the dynamics of the expectations of the corresponding operators of exponentially many coupled quantum oscillators.  This can be seen as a consequence of Ehrenfest's theorem, which relates classical and quantum evolution, applied to quadratic bosonic systems. 

More precisely, in classical Hamiltonian mechanics, the generalized coordinates q_j and p_j evolve according to

    q_j/ t={q_j,H_ c}_ PB  ,   p_j/ t={p_j,H_ c}_ PB ,    j =1,…,n  ,

where {a,b}_ PB is the Poisson bracket and H_c is the classical Hamiltonian; for coupled classical oscillators, H_c is similar to <ref>, after replacing the operators by the classical coordinates, i.e., Q_j ↦ q_j and P_j ↦ p_j.
For these quadratic classical Hamiltonians, the Poisson brackets are linear in q_j and p_j, which follows from {q_j,p_k}_ PB=δ_jk  and {q_j,q_k}_ PB={p_j,p_k}_ PB=0.
The equations of motion for the classical coordinates
are then similar to those for the corresponding quantum operators in the Heisenberg picture, being a consequence of canonical quantization where {q_j,H_ c}_ PB→i [Q_j,H] and {p_j,H_ c}_ PB→i [P_j,H]. 
Considering the expectations ⟨ Q_j(t)⟩ and ⟨ P_j(t)⟩, these evolve in the same way than the classical coordinates q_j(t) and p_j(t) evolve. As a consequence, the same quantum algorithm in Ref. <cit.>
can be used to simulate an exponentially large system of quantum harmonic oscillators, where the amplitudes of the shadow states are the expectations of the corresponding linear combinations of Q_j and P_j, as described. In the other direction, shadow Hamiltonian simulation can give techniques for simulating other classical systems that follow a similar classical–quantum correspondence.







 §.§ Qubits


We apply shadow Hamiltonian simulation to systems composed of n qubits.
The simplest example, which could be viewed as “free qubits”, in analogy with the previous sections on free fermions and free bosons, is when the Hamiltonian H is a 1-local operator; that is, the Hamiltonian is a sum of terms, each acting non-trivially on only one qubit. In particular, such a Hamiltonian can always be written as a sum of operators in S={𝕀, X_1,Y_1,Z_1,…,X_n,Y_n,Z_n}, where the subscript of an operator denotes the qubit that it acts on. 

It is easy to see that the invariance property is satisfied
and that H_S is Hermitian due to the orthogonality of Pauli operators. But note that the dimension of the shadow state is only 3n+1, so we can encode a system of exponentially many qubits into a shadow state of exponential dimension, which uses only (log(n)) many qubits. Due to the commutation relations of the Pauli operators, any operator in S can only have a nonzero commutator with at most two terms in the Hamiltonian (i.e., those operators that act on the same qubit as the chosen operator and are different from it) and hence the nonzero entries of the resulting Hamiltonian H_S are easy to compute if the terms in H are easy to compute. For example, [H,X_1] will be a linear combination of Y_1 and Z_1 only, and hence H_S will be a 2-sparse Hamiltonian. Standard Hamiltonian simulation approaches can then be used to simulate H_S efficiently.
The shadow states corresponding to some initial n-qubit states is also easy to prepare. For example, if ρ
is the all-zero state ρ=|0⟩⟨^|⊗ n, the only nonzero expectations are ⟨𝕀⟩ = 1 and ⟨ Z_j ⟩ = 1 for all j ∈ [n].
The corresponding shadow state with respect to S is
a superposition state that can be prepared with (log(n)) gates. Other shadow states can also be prepared efficiently, even for cases where the states ρ are entangled.
A downside with this example is that the qubits in the system do not interact, and hence the system might not be very interesting. 



A different example is when we consider the set S of all n-qubit Pauli operators. Specifically, for i,j∈{0,1}^n, let P_ij := X_1^i_1Z_1^j_1⊗⋯⊗ X_n^i_nZ_n^j_n and let S:={P_ij:i,j∈{0,1}^n}. Pauli operators are orthogonal and |S|=4^n. For a state ρ, the shadow state with respect to S is

    |ρ;S⟩ = 1/√(A)∑_i,j∈{0,1}^n⟨ P_ij⟩|i,j ⟩ ,

where ⟨ P_ij⟩ = (ρ P_ij) is the expectation. 
The normalization constant is A=∑_ij|⟨ P_ij⟩|^2 = 2^n  (ρ^2), being proportional to the purity of ρ. In particular, when ρ=|ψ⟩⟨$|
is a pure state,A=2^n.


LetV_Sbe the “Bell rotation”, which maps between the Bell basis and the standard basis: 
    
    V^†_S |i,j⟩↦ (𝕀_N ⊗ P_ij) 1/√(N)∑_k =0^N-1|k,k⟩  =(𝕀_N ⊗ P_ij) (|00⟩+|11⟩/√(2))^⊗ n ,
    
    whereN=2^n. Hence,V_SmapsnBell pairs to states in the computational basis. 
 As we show in <Ref>, the shadow state of a pure state is|ρ;S⟩ = V_S |ψ⟩⊗|⟩, which is a special case of a more general result. The unitaryV_Sis efficient to implement and an example is given in <Ref>.



SinceV_Scan be efficiently implemented, we can prepare the shadow state whenever we have an efficient circuit to prepare|ψ⟩. We can also construct an efficient circuit for|⟩by complex conjugating all the elementary gates in the circuit for preparing|ψ⟩. Similarly, it is possible to perform Hamiltonian simulation or unitary evolution on the shadow state, since the shadow state corresponding toU|ψ⟩is simply(U⊗U) |ψ⟩⊗|⟩. The unitaryU⊗Uhas an efficient circuit whenUdoes.




 §.§ Lie algebras



The case of Lie algebras is a natural one to consider 
in shadow Hamiltonian simulation because the invariance property is automatically satisfied. More precisely,
subspaces of operators on a Hilbert space sometimes form Lie algebras under the commutator, often corresponding to some physical symmetry in the system. We have seen three examples of this already: the set of quadratic fermionic operators span the Lie algebra𝔰𝔬(2n), quadratic bosonic operators span the symplectic Lie algebra𝔰𝔭(2n), and Pauli operators acting onnqubits span the Lie algebra𝔲(2^n). 
Other examples are spin systems with associated Lie algebras𝔰𝔲(2)or, more generally,𝔰𝔲(n).
This motivates us to consider the general case of Lie algebras.

(Lie algebra)
    A Lie algebra𝔤 is a vector space with a Lie bracket binary operation [·,·] : 𝔤×𝔤→𝔤 satisfying
    
  *  (Bilinearity) [x,y+z] = [x,y] + [x,z] and [x+y,z] = [x,z] + [y,z].
        
  *  (Alternating) [x,x] = 0.
        
  *  (Jacobi) [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.
    

The following analyses focus on the specially interesting case where the Lie bracket is the commutator:[x,y]=xy-yx. 
If the HamiltonianHof the system belongs to a Lie algebra𝔤, this automatically implies that any discrete setSof operators in the Lie algebra𝔤satisfy the invariance property in <Ref>. This provides a unifying framework for a broad class of situations where shadow Hamiltonian simulation is applicable.
In the specific case where the dimension of the Lie algebra is polynomial in the system size, shadow Hamiltonian simulation is able to tackle quantum systems that are of exponential size. 

The structure constants of a Lie algebra are important in shadow Hamiltonian simulation.
(Structure constants)
    Given a basis of anti-Hermitian operators {O_1,…,O_M} of a Lie algebra 𝔤, the structure constants of 𝔤 are given by f_jkl in the equations
    
    [O_j,O_k] = ∑_l=1^M f_jkl O_l  .


Let𝔤be a real Lie algebra of dimensionM<∞, with an orthogonal basis of anti-Hermitian matrices{O_1,…,O_M}, satisfying (O^†_m O^_m') ∝δ_mm'. Given a HamiltonianHsuch thatiH ∈𝔤, we can write down the conditions under which efficient sparse access toH_Scan be obtained:

   i) The structure constants f_jkl of 𝔤 are d'=(M)-sparse, i.e., given j and k at most d' structure constants are nonzero, and we have efficient sparse access to these constants.
    
   ii) The maximum number of terms p in H which do not commute with a given O_j is at most p=(M), and we can efficiently compute these terms given any j ∈ [M].

These conditions are sufficient but not necessary. 
Note that the entries of H_Sare determined by the structure constants and the interaction strengths in the Hamiltonian, which isH=i∑_m=1^M α_m O_m,α_m ∈ℝ. In particular, H_S_max≤ pmax_m |α_m| ×max_j,k,l|f_jkl|, and H_Sis at mostd ≤ p d'sparse. An example where these assumptions
are satisfied is the case of free fermions in <Ref>.





Certain shadow states
can also be prepared efficiently in this case.
Assume further thatis compact and semisimple. 
There is a special lowest-weight state| lw⟩that is a generalization of the bare vacuum state in <Ref> and has the following properties: it is an eigenstate of all diagonal operators, and it is annihilated by all lowering operators, when consider a particular decomposition ofreferred to as the Cartan-Weyl decomposition.
Ifh_i,i ∈ [r], are the diagonal operators, thenh_i | lw⟩= e_i | lw⟩, and the shadow state of| lw⟩is 

    |ρ;S⟩= 1/√(A)∑_i=1^r e_i |i⟩ .

The eigenvaluese_iare purely imaginary andA=∑_i |e_i|^2is a normalization constant, which can often be computed efficiently. The shadow state can be prepared efficiently if thee_i's can be described succinctly;
this turns out to be the case in many examples like <ref> for≡𝔰𝔬(2n). 

Other examples of efficiently preparable shadow states
can be those corresponding to “generalized coherent” states of, obtained ase^| lw⟩, wheree^is a unitary in the group induced by. These are the Gaussian fermionic states in <ref> when≡𝔰𝔬(2n). 
See Refs. <cit.>
for more details on Lie algebras and coherent states.














 §.§ Green's functions and other correlators


IfHandSsatisfy the invariance property, so doesHand the operator set constructed from arbitrary degree products of the operators inS. This observation can be extended to encode expectations of products of different time-dependent operators in the amplitudes of a quantum state. 
This extension is most relevant in the Heisenberg picture, where the time-dependent expectation of an operatorOon stateρ(t)is alternatively described as the expectation of a time-dependent operatorO(t)on the initial stateρ(0). The evolution of the operator is determined by Heisenberg's equations:/ t O(t) = i [H,O(t)], whenHis time-independent. 
More generally, we can consider products of operators, each at a different time in the Heisenberg picture, e.g.O_1(t) O_2(t'). 
The following result
extends shadow Hamiltonian simulation 
to this scenario. 



Let H be a Hamiltonian and S={O_1,…,O_M} a set of operators.
Assume that H and S satisfy the invariance property. Let ρ be the initial state of the system and O_m(t), m ∈ [M], be the corresponding time-evolved operators. Define the (pure) state

    |ρ;S (t,t')⟩ = 1/√(A)[ ⟨ O_1(t) O_1(t') ⟩⟨ O_1(t) O_2(t') ⟩⋮⟨ O_M(t)  O_M-1(t')⟩⟨ O_M(t)  O_M(t')⟩ ] ,

where ⟨ O_m(t) O_m'(t')⟩ are the expectations with respect to the initial state ρ. Then, |ρ;S(t,t')⟩ satisfies the differential equations:

    ∂/∂ t|ρ;S(t,t')⟩   = -i ( H_S ⊗_M) |ρ;S(t,t')⟩ , 
    ∂/∂ t'|ρ;S(t,t')⟩   = -i (_M ⊗ H_S) |ρ;S(t,t')⟩ .

    Let O⃗(t) =(O_1(t),…,O_M(t))^T and consider the vector of M^2 operators O⃗(t) ⊗O⃗(t'). Heisenberg's equations imply
    
    ∂/∂ tO⃗(t) ⊗O⃗(t') =i [H, O⃗(t)] ⊗O⃗(t') = -i( H_S O⃗(t)) ⊗O⃗(t') = -i( H_S ⊗_M) (O⃗(t) ⊗O⃗(t'))  .

    A corresponding equation can be obtained applying ∂/∂ t'.
    We can compute the expectation with respect to the initial state ρ on either side 
    of this equation and obtain <ref>, since we can alternatively express |ρ;S(t,t')⟩ as the vector 1/√(A)⟨O⃗(t) ⊗O⃗(t') ⟩.
<ref> allows us to encode two-time correlators and Green's functions in the amplitudes of a quantum state. To prepare|ρ;S(t,t')⟩from|ρ;S(0,0)⟩efficiently, it suffices to have efficient sparse access to H_S. The procedure involves evolving with H_Sacting on the first register for timetand evolving with the same H_Sacting on the second register for timet'.
In addition, it is often possible to prepare the initial state|ρ;S(0,0)⟩efficiently, as in the examples discussed above where the set of initial expectations contains structure that can be expressed succinctly.

Importantly, <ref> can be generalized in two directions. First, we can encode higher order correlation functions⟨ O_m_1(t_1) … O_m_q(t_q) ⟩, where all theO_m_j's are in the same operator setSand evolve underH. For this case there areqregisters and, to evolve the expectations, we apply the Hamiltonian H_Son each register for the corresponding time. This gives a straightforward generalization of
<ref> to the case ofq>2. 
Second, the operators in the product can belong to different operator setsS_1,…,S_q, and these can evolve with their own distinct HamiltoniansH_1,…,H_q. Assume that eachH_jandS_jsatisfy the invariance property.
Then, to evolve the expectations of products
of  time-dependent operators, we apply a Hamiltonian H_S_1on the first register for timet_1, 
a Hamiltonian H_S_2on the second register for timet_2, and so on. Each H_S_jis obtained from the commutation relations betweenH_jand the operatorsS_j, as in <ref>.










 §.§ Operators in the Heisenberg picture


In shadow Hamiltonian simulation, we encode the expectations of certain operators with respect to the evolved stateρ(t), in the amplitudes of a different state.
However, it is possible to extend the method to encode a representation of the time-evolved operators themselves in the Heisenberg picture, rather than their expectations, without making reference to the state of the system. 
We present a quantum algorithm for this problem.


Consider the operator setS={O_1,…,O_M}and letZ:=∑_m=1^M z_m O_mbe another operator, wherez_m ∈ℂfor allm ∈ [M].
The time-evolved version in the Heisenberg picture isZ(t)and, if the HamiltonianHis time-independent, Heisenberg's equation of motion is/ tZ(t) =  i[H,Z(t)].
IfHandSfurther satisfy the invariance property of <ref>, we can expressZ(t)=∑_m=1^M z_m(t) O_m, wherez_m(t) ∈ℂ. These coefficients evolve according to

    / tz_m(t)  = - i∑_m'=1^M h_m'm z_m'(t)  ,

which is a direct consequence of Heisenberg's equation of motion. The coefficientsh_m'mare the entries of the matrix H_Sin <ref>.

Note that <ref> is also a Schrödinger equation when H_Sis Hermitian, and it is natural to consider the operator-vector mappingZ(t) ↦|Z(t)⟩∝∑_m=1^M z_m(t) |m⟩. In this way, we can encode the time-evolved operator in a quantum state, and  <ref> gives rise to Hamiltonian evolution:

    / t|Z(t)⟩  = - i H_S|Z(t)⟩ .

A quantum algorithm that prepares|Z(t)⟩involves then preparing the initial state|Z(0)⟩and performing time evolution with the Hamiltonian H_Sfor timet. The algorithm is efficient when these two steps can be implemented efficiently.
This could let us probe many interesting properties of the system, such as operator growth and operator scrambling. For example,
in the case whereSconsists of all products of Pauli operators onnqubits, and whenZ(0)is a local operator, we can use the algorithm to measure the support of the evolved operator (related to the Hamming weight of|m⟩). Similar to <ref>, we can also consider the evolution of products of operators and two-time correlators by encoding them in quantum states.


The analysis can be extended to time-dependent Hamiltonians. In that case, it can be shown that the 
transformation that takes|Z(0)⟩to|Z(t)⟩is the evolution operator with the Hamiltonian H_S(t-s), where the final timetis fixed andsis increased from0tot. This resembles the evolution of the operator in the Heisenberg picture that evolves backwards in time. 
Since that case is equivalent to a quantum circuit, where quantum gates can be interpreted as evolutions with time-dependent Hamiltonians, we provide the analysis for the latter.
LetG=G^L … G^1be the quantum circuit composed ofLunitary gatesG^j. The requirement for these gates andSto satisfy the invariance property is given in <ref>. Accordingly, to study the evolution of an operatorZ=∑_m=1^M z_m O_m, we define the operatorsZ_l:=(G^L+1-l)^†… (G^L)^† Z G^L … G^L+1-lfor alll ∈ [L]. AlsoZ_0:=Z, i.e.,z_0,m=z_mfor allm ∈[M]. 
TheseZ_l's are basically a (backwards) transformation ofZwhen acting with the lastlgates of the circuit, and we are interested inZ_L, which is the operator transformed by all gates.


The invariance property 
impliesZ_l = ∑_m=1^M z_l,m O_m, with coefficientsz_l,m∈ℂ. Then <ref> gives

    Z_l+1=(G^L-l)^† Z_l G^L-l=∑_m=1^M z_l,m∑_m'=1^M g^L-l_mm'O_m'  z_l+1,m=∑_m'=1^M z_l,m' g_m'm^L-l .

Let G^j_Sbe the matrix of entriesg_mm'^j.
When this is unitary—the analogue of requiring H_Sbe Hermitian—<ref> denotes a unitary transformation on the vector of coefficientsz_l,m, for eachl. We then consider the operator-vector mappingZ_l ↦|Z_l⟩∝∑_m=1^M z_l,m|m⟩. This provides an encoding of the time-evolved operatorsZ_lin a quantum state, and <ref> implies

    |Z_l+1⟩ =( G^L-l_S)^T |Z_l⟩ ,    l=0,…,L-1  .

A quantum algorithm that prepares|Z_L⟩is then a sequence of the unitaries( G^1_S)^T … ( G^L_S)^Tacting on|Z_0⟩. 
Note that applying the transpose of G^L-l_Sis the analogue
to evolving with H_S(t-s).









  §.§.§ Comparison with the transfer matrix method


The quantum algorithm shares some similarities 
with the “transfer matrix” method used for time-evolving Pauli operators acting onnqubits under the action of a quantum circuit. This is  a classical algorithm that describes the evolution of the operator, and is also applied to studying the growth of light cones and operator scrambling, even under the presence of noise <cit.>. In the noiseless case, at each step of a quantum circuit (also evolving backwards in time), the operator is represented by a vector of dimension4^n, and this is acted on by a4^n × 4^nunitary that provides the vector at the next step and corresponds to a gate in the circuit (i.e., the unitary( G_S^L-l)^T). Depending on the choice of operator basis, this unitary may also be called the super-operator <cit.> or Pauli transfer matrix <cit.>.
The quantum algorithm described in <Ref> performs essentially the same evolution, but where the4^n-dimensional vector is encoded in a quantum state of2nqubits in this case, and where each4^n × 4^nunitary matrix( G_S^L-l)^Tis implemented with a quantum circuit.






§ PROPERTIES OF THE SHADOW STATE


We now discuss some properties and alternate representations of the shadow state|ρ;S⟩for finite dimensional systems, without reference to any time-evolution problem one might wish to solve with this state.



  
Standard basis. 
Letρ = |ψ⟩⟨∈|^N × Nbe a pure quantum state of a system of dimensionN <∞, and letSbe an orthogonal basis of the space^N × N. In this case,|ρ;S⟩is related to other standard representations of the state|ψ⟩. One example is the standard basis,S_SB={O_ij:i,j=0,…,N-1}, whereO_ij = |j⟩⟨i|. Then,

    |ρ;S_SB⟩ 
        = ∑_i,j=0^N-1⟨ O_ij⟩|i,j⟩
        = ∑_i,j=0^N-1( ⟨ψ|j⟩⟨i|ψ⟩) |i,j⟩
        = ∑_i=0^N-1|i⟩⟨i|ψ⟩⊗∑_j=0^N-1|j⟩⟨j|ψ⟩
        = |ψ⟩⊗|⟩.


This is already an interesting state that is different from the “standard” representation|ψ⟩. Clearly we can create|ψ⟩given one copy of|ψ⟩⊗|⟩, but the reverse is not possible. In fact, there exist problems that can be solved using only one copy of|ψ⟩⊗|⟩that would require exponentially many copies of|ψ⟩alone to solve.


    Consider the problem of deciding if an n-qubit state |ψ⟩ satisfies |⟨ψ|⟩|≤ 1/3 or |⟨ψ|⟩|≥ 2/3, promised that one of these holds. Any quantum algorithm that solves this problem using only copies of |ψ⟩ must use Ω(2^n/2) copies, but the problem can be solved with one copy of |ψ⟩⊗|⟩ with constant success probability.

    The problem can be solved with constant success probability using one copy of |ψ⟩⊗|⟩ by using the well-known swap test. The success probability can be made arbitrarily close to 1 using a constant number of copies of |ψ⟩⊗|⟩ by standard error reduction via majority vote.

    For the lower bound, we observe that any state with all real amplitudes satisfies |⟨ψ|⟩|=1 and that a Haar-random state has |⟨ψ|⟩|=o(1) with high probability. The lower bound follows from Ref. <cit.> that shows that a binary phase state (i.e., a uniform superposition state where all phases are uniformly random in {± 1}) is indistinguishable from a Haar-random state unless we have Ω(2^n/2) copies of the state. Similarly, <cit.> shows that distinguishing a Haar-random state with real entries from a standard Haar-random state requires Ω(2^n/2) copies of the state.


The state|ψ⟩⊗|⟩is also related to another well-known alternate representation of quantum states that uses only real amplitudes. In this “real representation” of quantum states, the state|ψ⟩is represented by the quantum state|0⟩|(ψ)⟩ + |1⟩|(ψ)⟩, where|(ψ)⟩and|(ψ)⟩are the quantum states where each amplitude is replaced by the real and imaginary parts of the corresponding amplitude of|ψ⟩, respectively. Up to a Hadamard on the first qubit, this state is also|0⟩|ψ⟩+|1⟩|⟩.



  
Orthonormal bases.
For the standard basisS_SB, the shadow state is|ψ⟩⊗|⟩.
More generally, the shadow state for any orthonormal basisSis a unitary transformationV_Sapplied to|ψ⟩⊗|⟩.


    Let S = {O_ij: i,j=0,…,N-1 } be an orthonormal basis of ^N × N. Let V_S be the unitary on ^N ⊗^N defined by 
    
    V^†_S |i,j⟩ = (𝕀_N ⊗ O_ij) ∑_k =0^N-1|k,k⟩    .
    
    Then, if ρ = |ψ⟩⟨$| is pure, we have
    
    |ρ;S⟩ = V_S |ψ⟩⊗|⟩.

    First, we show that V_S is indeed unitary. Observe that
    
    ⟨i',j'| V^_S V_S^†|i,j⟩ 
            = ∑_k,k'=0^N-1⟨k',k'| (𝕀_N ⊗ O^†_i'j'O^_ij) |k,k⟩ 
            = ∑_k=0^N-1⟨k| O^†_i'j' O^_ij|k⟩ 
            = (O^†_i'j' O^_ij)
            = δ_ii'δ_jj',

    since the operators O_ij are orthogonal. We obtain
    
    ⟨ O_ij⟩ 
            =  ( |ψ⟩⟨ψ| O_ij) 
            =  ∑_k=0^N-1⟨k|ψ⟩⟨ψ| O_ij|k⟩  
            = ∑_k=0^N-1⟨k|ψ⟩⟨k| O^†_ij|⟩
            = ∑_k=0^N-1⟨k,k| (𝕀_N ⊗ O^†_ij) |ψ⟩⊗|⟩ .

    Using the definition of V_S, we get that ⟨ O_ij⟩ = ⟨i,j|V_S|ψ⟩⊗|⟩, which gives
    
    |ρ;S⟩ = ∑_i,j=0^N-1⟨ O_ij⟩|i,j⟩ 
            = ∑_i,j=0^N-1⟨i,j|V_S|ψ⟩⊗|⟩|i,j⟩ 
            = (∑_i,j=0^N-1|i,j⟩⟨i,j|) V_S|ψ⟩⊗|⟩ 
            = V_S|ψ⟩⊗|⟩.



So far we have only considered pure states. But observe that the lemma can equivalently be restated as asserting that|ρ;S⟩ = V_S |ρ;S_SB⟩. Since both sides of the equation are linear inρ, we have the straightforward corollary that|ρ;S⟩ = V_S |ρ;S_SB⟩holds for all mixed statesρ.

To understand the form of|ρ;S_SB⟩, note that <ref> equivalently implies that|ρ;S_SB⟩equalsvec(ρ)whenρ = |ψ⟩⟨$|, where vec is the operation that maps ∑_j,k a_jk|j⟩⟨k|↦∑_j,k a_jk|j⟩|k⟩ <cit.>. When ρ is mixed, |ρ;S_SB⟩ is the state whose entries are proportional to vec(ρ) with the normalization constant equal to the square root of the purity of ρ, √((ρ^2)). Alternatively, if ρ = ∑_ℓ p_ℓ|ψ_ℓ⟩⟨$|, then|ρ;S_SB⟩∝∑_ℓ p_ℓ|ψ_ℓ⟩⊗|ψ_ℓ⟩.



  
Examples of V_S. 

WhenS=S_SB, observe thatV_S≡𝕀_N^2, whereN=2^n. More interestingly, letSbe the set of Pauli operators onnqubits,{P_ij}_i,j ∈{0,1}^n, 
whereP_ijare tensor products ofnPauli operators as discussed in <Ref>. In this case, the state(𝕀_N ⊗ P_ij)1/√(N)∑_k=0^N-1|k,k⟩are tensor products ofnBell pairs. Thus the operatorV_Sis a Bell rotation and can be achieved by the sequence ofnentangling gates shown in <Ref>.



As another example, letSbe the set of Heisenberg-Weyl operators of dimensionN. This is a group ofN^2unitary matrices spanned by the generatorsXandZ, whereX|j⟩ = |j+1  N⟩andZ|j⟩=e^-i 2π j/N|j⟩. In this case, the unitaryV_Scan be constructed from the discrete Fourier transform matrix of sizeN × N, and the conditional shift operator∑_j=0^N-1|j⟩⟨⊗|X^j, both of which have efficient quantum circuits.
In these examples, the unitaryV_Swas efficient to implement as a quantum circuit. In general, this may not be the case.




  
Incomplete bases.
In many of the applications considered in this article,Sis not a complete basis of the space, but an incomplete set of orthonormal operators. In this case,|ρ;S⟩may not contain all the information contained in|ψ⟩and hence it will not simply be a unitary transformation applied to|ψ⟩⊗|⟩. However, the state|ρ;S⟩can still be viewed as a unitary transformation and projection of this state, justifying the name “shadow” state. 

More precisely, even whenSis not a complete basis, the same proof of <Ref> can be followed, except for the last equation where∑_i,j=0^N-1|i,j⟩⟨i,j|is not identity in this case: the sum is not over alli,j. This sum becomes an orthogonal projectorP_S. Thus, whenρ=|ψ⟩⟨$| is pure, we have |ρ;S⟩ = P_S V_S |ψ⟩⊗|⟩. 





§ SUMMARY AND  FUTURE DIRECTIONS





We devised a novel approach to quantum simulation
that uses other representations of quantum states, being
complementary to the traditional approach based on simulating the full quantum state.
Our shadow Hamiltonian simulation approach 
allows us to encode limited information about a system's quantum state, like the expectations of a relevant set of operators at any time, in the amplitudes of a different “shadow state”. 
In cases where the number of these operators is exponentially smaller than the space dimension of the system, we can leverage the exponential growth of the quantum computer's dimension and encode, for example, 
the expectations of operators acting on systems of exponential size (i.e., the system's dimension is doubly exponential).
By doing so we unveiled natural applications to  quantum systems like free-fermions and free-bosons  with an exponential number of modes. We described computations that can be carried efficiently via shadow Hamiltonian simulation, while these computations can be hard for traditional approaches, as preparing the full states would require exponential resources. 

Additionally, we demonstrated how the ideas underlying shadow Hamiltonian simulation can be applied to a broader range of problems. In one example, we extended these concepts and described how to encode the expectations of two-time correlators or other products of time-dependent operators in a different quantum state. Furthermore, we described an approach for encoding operators in the Heisenberg picture in quantum states, without referencing to the state of the system. These developments enable the solution to other problems; an example is studying the growth of the light cone associated with an evolving operator. 


We anticipate future directions based on similar ideas. 
For instance, we mentioned that quantum algorithms for differential equations might be used when the invariance property is satisfied but H_S is not Hermitian, and understanding when such algorithms are efficient would be important. 
Another interesting direction is the simulation of open quantum system dynamics, and whether shadow states can be used to perform these simulations more efficiently or address questions that traditional approaches are unable to resolve. Furthermore, many
classical systems have an underlying Lie algebra structure (e.g., under the Poisson bracket like in classical harmonic oscillators), and it would be interesting to discover other classical systems whose dynamics can be efficiently simulated on a quantum computer using shadow states.[One simple example is the simulation of exponentially many and non-interacting magnetic dipoles under the influence of an external field.]Also, we discussed
some applications to operator scrambling, but a more thorough characterization of what related problems can be solved efficiently would be interesting. 


Shadow Hamiltonian simulation could be simulated classically,
although such a simulation would be inefficient in the worst case. 
Nevertheless, 
properties like locality of interactions could be exploited, potentially enabling efficient classical simulations of the dynamics of certain expectations of correlators, or operators.


We also note that is possible to prove a robust version of <Ref>, which applies to cases where
the invariance property is “almost” satisfied. In this case, the commutations [H,O_m] are not necessarily
linear combinations of the operators in S, but if this “leakage” outside S is bounded and small, we can still satisfy <ref> approximately, 
potentially allowing the simulation of larger classes of systems. One interesting example could be simulating the dynamics of quantum field theories, extending the results in <Ref>. It might be possible to perform a cutoff in the degree of the operator set given by polynomials of Q_j and P_j, and attempt a simulation using shadow states, which encode less information than the full states.


In addition, there is a plethora of encodings from states and operators to other “shadow states” that one might consider.
For example, when the system's state ρ=∑_ℓ p_ℓ|ψ_ℓ⟩⟨$| is mixed, the amplitudes of|ρ;S⟩are linear inρand might not reveal anything useful (e.g., if the state is highly mixed, many expectations can become too small).
However, one can consider an alternative encoding whereρmaps to a  mixed shadow state, e.g., to each pure stateρ_ℓ:=|ψ_ℓ⟩⟨$| we assign a shadow state |ρ_ℓ;S⟩, and then take the convex combination of such shadow states, e.g.,  ∑_ℓ p_ℓ|ρ_ℓ;S⟩⟨$|. For problems where the goal is to compute expectations of quantities that are quadratic in the pure-state expectations(ρ_ℓ  O_m )'s, this type of encoding can be advantageous.



 




§ ACKNOWLEDGEMENTS



We thank Cristian Batista, Kipton Barros, David Gosset, Robert Huang, William Huggins,  Stephen Jordan, Marika Kieferova, Nicholas Rubin, and Nathan Wiebe for discussions. 


alphaurltocsectionReferences

 Paper Title: Local analysis of a single impurity on a graphene Josephson Junction.







Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.


francesco.pellegrino@dfa.unict.it
Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.
CNR-IMM, Catania (University) Unit, via Santa Sofia 64, I-95123 Catania, Italy


Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.
CNR-IMM, Catania (University) Unit, via Santa Sofia 64, I-95123 Catania, Italy


Dipartimento di Fisica e Astronomia "Ettore Majorana",  Universitá di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
INFN, Sez. Catania, I-95123 Catania, Italy.
CNR-IMM, Catania (University) Unit, via Santa Sofia 64, I-95123 Catania, Italy



§ ABSTRACT 
In this work, we investigate the local effects of a single short-range impurity on the electron system of a short ballistic graphene Josephson Junction. Within the Dirac-Bogoliubov-De Gennes approach, we systematically analyze the local density states, whose subgap energy dependence enables us to distinguish between elastic and inelastic scattering processes and identify the magnetic nature of the impurity. Furthermore, 
we observe that the spatial dependence of the local density of states is a sensitive probe of the microscopic processes resulting in subgap impurity-induced bound states. The Fourier analysis evidences the wavevectors related to the momenta of the high-transmissive channels in ballistic graphene.





Local analysis of a single impurity on a graphene Josephson Junction
    Elisabetta Paladino
    August 5, 2024
====================================================================





§ INTRODUCTION



The potential of quantum technologies is being explored by developing hybrid systems that combine different physical components with complementary functionalities <cit.>.
Within this context, gate-tunable hybrid superconducting qubits, known as gatemons, have been implemented using semiconducting nanowires <cit.>, InAs Josephson Junctions (JJs) <cit.>, two-dimensional materials <cit.>, van der Waals heterostructures <cit.> and graphene <cit.>. These devices offer reduced dissipative losses, crosstalk, and compatibility with high magnetic fields <cit.>, paving the way for the realization of fault-tolerant topological qubits based on Majorana zero modes <cit.>. High-quality graphene superconductor heterostructures with clean interfaces, obtained by encapsulating graphene in hexagonal boron nitride (hBN) <cit.>, have enabled ballistic transport of Cooper pairs over micrometer-scale lengths, gate-tunable supercurrents that persist at large parallel magnetic fields  <cit.>, and different features of two-dimensional Andreev physics <cit.>. Moreover, thanks to the low specific heat of graphene embedded in hBN, highly sensitive graphene Josephson Junction (GJJ) based microwave bolometers have been realized for circuit quantum electrodynamics applications <cit.>. Single near-infrared photon detection has also been achieved by coupling photons to localized surface plasmons of a GJJ, which can be integrated into JJ-based computing architectures as high-speed, low-power optical interconnects <cit.>. GJJs are also an excellent platform for generating exotic quantum states, such as long-lived Floquet-Andreev states generated by applying continuous microwave light without significant heating <cit.>.


The Josephson effect in heterostructures is described microscopically by the proximity effect and constructive interference between Andreev processes at the two normal-superconducting (N-S) interfaces, which lead to coherent electron-hole superpositions known as Andreev bound states (ABSs). In the short junction regime, the current-phase relation (CPR) resulting from the phase dependence of the ABSs spectrum differs from the sinusoidal CPR of the tunnel JJs <cit.>. 

Recently, there has been interest in the spatial control properties of ballistic GJJ. In fact, an experimental methodology has been developed to measure and control the real-space current density in two parallel ballistic GJJs, based on Fourier and Hilbert transformations of the magnetic field-induced modulation of the critical current <cit.>.
Tunneling spectroscopy measurements in GJJs revealed the possible presence of weakly coupled microscopic quantum dots to proximitized graphene, which act as energy filters in the tunneling process <cit.>. 
A theoretical analysis has suggested that a homogeneous dilute distribution of impurities in a GJJ can alter the critical current and the skewness of the CPR <cit.>. 

Moreover, defects in the nearby substrate can act as carrier traps and induce fluctuations in the carrier density of the graphene channel <cit.> and lead to critical current fluctuations with a 1/f spectrum <cit.>. 


Motivated by findings related to point systems weakly interacting with proximitized graphene and the latest interest in controlling the spatial properties of ballistic GJJs, we investigate the local effect of a short-range impurity on the low-energy properties of a short planar GJJ, employing an analytical approach based on the Dirac-Bogoliubov-de Gennes model <cit.>. Two paradigmatic descriptions of a single impurity are considered: the Anderson model <cit.> and the Lifshitz model <cit.>. According to the latter, an impurity interacts with the density of conduction electrons through a scalar potential <cit.>, this model has been used to describe a vacancy in a graphene crystal <cit.>. The Anderson model, which has been used to study the effect of adatoms on the graphene electron system <cit.>,  it includes the possibility of electron transfer from the host to some energy level that belongs to the adsorbed atom <cit.>.
Here, we take into account an extra magnetic term for both models to describe the exchange interaction between the local spin on the impurity site and the electrons that flow through the GJJ. 

We focus on a single impurity located in the middle of the normal phase region and analyze how it affects the LDOS within the superconductive gap |E|<Δ.
This quantity is experimentally detectable by scanning tunneling spectroscopy <cit.>. 
Here, the LDOS reveals that ABSs are sensitive to the type of scattering processes caused by the impurity.
In proximity to the impurity, the energy dependence of the LDOS is sensitive to the magnetic properties of the impurity, and it is different for the Anderson or Lifshitz model. 
We analyze the conditions for the occurrence of an impurity-induced subgap bound state. Finally, at the energy corresponding to this impurity-induced bound state, we analyze the spatial dependence in the normal phase region along the transverse direction and apply a Fourier analysis.

This paper is structured as follows. Section <ref> provides technical details on the Dirac-Bogoliubov-De Gennes model used to describe ABSs. Sections <ref> and <ref> present the Anderson and Lifshitz models, respectively, and we analyze the energy dependence of the LDOS near the impurity placed in the middle of the normal phase region. Section <ref> examines the spatial dependence of the LDOS along the normal phase region, and we analyze it in Fourier space. Finally, the conclusions are reported in Section <ref>.







§ MODEL



We analyze a GJJ in the ballistic regime, as depicted in Fig. <ref>, which is composed of two identical metallic superconductive electrodes on the sides and a graphene monolayer in the middle.
The entire device has a finite width W and is considered infinite along the x̂-direction, the normal graphene stripe has length L, and we assume that W≫ L.
The electron system is described by the following Dirac-Bogoliubov-de Gennes (D-BdG) Hamiltonian <cit.>


    ℋ̂_D-BdG=   ∑_ζ= ±∫ d^2rΨ̂_ζ^†(r) H_D-BdGΨ̂_ζ(r),
    
    H_D-BdG=   τ_z[U(r) 1_σ+ħ v_D/i(∂_xσ_x+∂_yσ_y)]
       +τ_x1_σReΔ(r)-τ_y1_σImΔ(r) ,


where ζ=± denotes valley indices.
Here, the four components spinors Ψ̂_±(r) are defined as

    Ψ̂_+(r)=[ψ̂_A, K, ↑^†(r), ψ̂_B, K, ↑^†(r), ψ̂_A, K^', ↓(r), ψ̂_B, K^', ↓(r)]^† ,
       Ψ̂_-(r)= [-ψ̂_B, K^', ↑^†(r), ψ̂_A, K^', ↑^†(r),-ψ̂_B, K, ↓(r), ψ̂_A, K, ↓(r)]^† ,


where v_ D∼ c/300 is the Fermi velocity in monolayer graphene, the set composed by the two-dimensional matrix identity _σ (_τ) and the Pauli matrices {σ_x,σ_y, σ_z } ({τ_x,τ_y, τ_z }) act on the two-dimensional sublattice A-B (electron-hole) subspace.
The superconducting order parameter Δ(r), and the scalar potential U(r) are defined across the junction as follows


    Δ(r)   =Θ(|x|-L / 2) Δ e^i ϕ_0(x), 
    ϕ_0(x)   =Θ(x) ϕ_R+Θ(-x) ϕ_L,


    U(r)=-μ_0Θ(L / 2-|x|)-U_0Θ(|x|-L / 2),


where Θ(x) is the Heaviside step function, and we assume that U_0 ≫ |μ_0| <cit.>.



The multiple coherent Andreev reflections at the interfaces between the normal phase region and the superconductive sides at x=± L/2 generate a set of ABSs <cit.>. These eigenstates of the D-BdG Hamiltonian have subgap eigenenergies, i.e. |E| <Δ. In the short junction limit L≪ξ, where ξ = ħ/(πΔ) is the coherence length, the eigenstates with energies above the gap, i.e. |E| >Δ, form the Andreev continuum. These states have eigenenergies that are independent of the phase difference ϕ=ϕ_ R-ϕ_ L, so ABSs are the only ones responsible for carrying the Josephson equilibrium supercurrent <cit.>.
In this study, we disregard the continuum and focus on the low-energy characteristics of the GJJ.


To obtain ABSs, we seek the eigenstates of the D-BdG Hamiltonian by exploiting the uniformity of the system along the y-direction


    φ_k, E,ζ(r)=e^i k y/√(W)φ̃_k, E(x),


where  k determines the wavenumber along the y direction, 
and E is the eigenenergy.
Since the D-BdG Hamiltonian in Eq. (<ref>) is independent of the valley index ζ, we omit it in φ̃_k, E(x), and it leads to double degeneracy in the ABS spectrum.

As shown in Fig. <ref>, there are three sharp partitioned sectors along the x direction for which we look for a solution to the stationary D-BdG problem of the following form


    φ̃_k, E(x) =  N_k,Eφ̃_k, E^(S-L)(x)     x≤ -L/2, 
     N_k,Eφ̃_k, E^(N)(x)     -L/2< x< L/2, 
     N_k,Eφ̃_k, E^(S-R)(x)     x ≥ L/2,


and afterward, we impose the continuity at the interfaces x=± L/2. The coefficient N_k,E denotes the normalization factor.
To search for eigenfunctions, we follow the approach used in Ref. pellegrino_commphys_2022 where the junction length was taken infinitesimal L/ξ→ 0. Here, we generalize to the case of a finite junction length L.

We start from the superconductive sides (x<-L/2 and x>L/2).
We write both φ̃_k, E^(S-L)(x) and φ̃_k, E^(S-R)(x) as a linear combination of terms of the form e^s η_λ xa_k, E, λ,s,  where s=±, λ=±. By imposing the exponential form, it solves the D-BdG equation in the superconductive sides for  


    η_λ= i √((U_0+λ√(E^2-Δ^2)/ħ v_D)^2-k^2) .


We exclude pairs of coefficients λ and s such that (λ,s)=(+,+) and (-,-) (where λ s=+) on the left side, and (λ,s)=(+,-) and (+,-) (where λ s=-) on the right side, since they correspond to 

wavefunctions that cannot be normalized.
Then, we obtain the following expressions


    φ_k, E^(S-L)(r)=   e^i k y[x^L_k,E e^-η_+(x-L/2)a_k, E,+,-. 
    
            +   .y^L_k,E e^η_-(x-L/2)a_k, E,-,+],


    φ_k, E^(S-R)(r)=   e^i k y[x^R_k,E e^η_+ (x-L/2)a_k, E,+,+. 
    
            +   . y^R_k,E e^-η_- (x-L/2)a_k, E,-,-] .


The four-dimensional vectors a_k, E, λ, s can be represented as


    a_k, E, λ, s= W_λ, sb_k, E, λ ,
 

where W_λ, s=_τ (Q_λ, sΛ_λ, s) is defined in terms of the matrices 


    Q_λ, s   =1/√(2)[σ_z+i(e^-s z_λσ_--e^s z_λσ_+)],
    Λ_λ, s   =1/√(2[1+e^s(z_λ+z_λ^*)])(1-σ_z)+
       +1/√(2[1+e^-s(z_λ+z_λ^*)])(1+σ_z),


which act on the sublattice space, where z_λ satisfies the identity e^s z_λ=k v_ D(k+s η_λ)/[U_0+λ√(E^2-Δ)], and 


    b_k, E, λ=1/√(2)[ e^i (ϕ_j + λ/2β(E) ), 0, e^-i λ/2β(E), 0]^T,


with β(E)= arccos(E / Δ) and j=R,L.

Assuming that the superconductive sides are in the large doping regime, i.e. U_0 ≫ |μ_0|, and Δ≪ U_0, then η_λ, defined in Eq. (<ref>), can be approximated <cit.> as


    η_λ≈  i U_0/ħ v_D- λ√(Δ^2-E^2)/ħ v_D ,


and z_λ≈ i π /2, which leads W_λ,s≈1/√(2)(σ_z+s σ_x). 

To find the solution of the stationary equation in the normal phase sector φ̃_k, E^(N)(x), we employ the transfer matrix approach <cit.> 
After algebraic manipulations, the stationary problem can be rewritten as the following ordinary differential equation in terms of the transfer matrix


    φ̃_k, E^(N)(x)=𝒯(k, E ; x) φ̃_k, E^(N)(-L / 2),



    d  T(k,E;x)/d x=[i τ_zσ_xE/ħ v_D+i σ_xμ_0/ħ v_D+σ_z k]  T(k,E;x) ,


with boundary condition 𝒯(k, E ; x=-L / 2)=1_τ1_σ.


The transfer matrix which solves the above problem is explicitly expressed as


    T(k,E;x) = _τ+τ_z/2𝕋(k,E;x) +  _τ-τ_z/2𝕋(k,-E;x) ,


where


    𝕋(k,± E;x)   = sin[q_± (x+L/2)] [iμ_0± E/ħ q_±σ_x+k/q_±σ_z ]
       + cos[q_± (x+L/2)]_σ ,


and q_± =√((μ_0± E/ħ)^2 -k^2).

Using the transfer matrix, we can impose the continuity condition on the wave functions at the N-S interfaces, and  we obtain the compact condition


    φ̃^ (S-R)_k,E(x=L/2)= T(k,E;L/2)φ̃^ (S-L)_k,E(x=-L/2) .


After a few algebraic manipulations, it can be rewritten as


    x^ R_k,Ea_k,E,+,+ +y^ R_k,Ea_k,E,-,-= 
       = T(k,E;L/2)[x^ L_k,Ea_k,E,+,-+y^ L_k,Ea_k,E,-,+] .


The expression above is a homogeneous system of four equations for the variables { x^ L_k,E,y^ L_k,E,x^ R_k,E,y^ R_k,E}. The solution of this system is not trivial only if the associated matrix is singular, i.e. its determinant D is zero. 
For each pair of values of the y component of the wave vector k and the phase difference ϕ, the energies E(j,k,ϕ) that nullify the determinant D are the eigenenergies of the ABSs, with j being the subband index.
The determinant D is an even function of energy, which leads to an even number of subbands E(j,k,ϕ), so the subband index runs over the range {± 1, ±2,…, ± n }, where 2n is the number of subbands, and the following property E(j,k,ϕ)=-E(-j,k,ϕ) is valid.
For each ABS, after finding the eigenenergy E(j,k,ϕ), we choose the nontrivial solution that guarantees the condition |x^ L_k,E(j,k,ϕ)|^2+|y^ L_k,E(j,k,ϕ)|^2=1.





Under the approximation in Eq. (<ref>), the normalization factor N_k,E(j,k,ϕ) is determinated for a given subband index j, y-component of the wavevector k, and phase difference ϕ, by enforcing the condition below


    ∫^W/2_-W/2 dy ∫^∞_-∞d x φ^†_k, E,ζ(r)  φ_k, E,ζ(r)|_E=E(j,k,ϕ) =1 ,


which gives


    N_k,E(j,k,ϕ)   =  1/√(1+B(j,k, ϕ))√(√(Δ^2-E(j,k,ϕ)^2)/ħ v_D) ,


where 


    B(j,k, ϕ)=   L/ξ√(1-(E(j,k,ϕ) / Δ)^2)
    ×   [x_La_k, E,+,-+y_La_k, E,-,+]^†𝒜(k, E(j,k,ϕ)) 
    ×   [x_La_k, E,+,-+y_La_k, E,-,+] ,


    𝒜(k, E)   =∫_-L / 2^L / 2d x/L𝒯^†(k, E, x) 𝒯(k, E, x) 
       = _τ+τ_z/2𝔸(k,E) +  _τ-τ_z/2𝔸(k,-E) ,



    𝔸(k,± E)   =[(μ_0± E)^2/ħ^2 ^2 q^2_±-k^2sin(2q_± L) /2q_± ^3 L]_σ
       
    -[k(μ_0± E)/ħ q_±-k(μ_0± E)sin(2q_± L)/2ħ q^3_± L]σ_y 
       +
    k sin^2(q_±L)/q^2_± Lσ_z ,


and x^ L_k,E and y^ L_k,E are two components of the nontrivial vector which solves the system in Eq. (<ref>) with E→ E(j,k,ϕ).


From now on, we focus on the short junction regime L/ξ≪ 1, where the subband index j can assume two values, namely {-1,1}, and we have E(j,k,ϕ)=jϵ(k,ϕ).
In particular, we will consider a finite value of L/ξ≪ 1, and note that in the limiting case L/ξ→ 0, we have B(j,k, ϕ)→ 0.

To describe the low-energy electronic properties, we project the full D-BdG Hamiltonian in Eq. (<ref>) onto the ABS subspace with the projector 𝒫̂_A <cit.>. 

The Andreev Hamiltonian ℋ_A=𝒫̂_A ℋ̂_D-BdG𝒫̂_A is expressed as


    ℋ̂_A=∑_ζ= ±∑_kϵ(k, ϕ) ( γ̂_ζ,+, k^†γ̂_ζ,+, k-γ̂_ζ, -,k^†γ̂_ζ,-,  k),


where  γ̂_j, ζ, k is the fermionic ABS operator. 

Each set of values of ζ,k determines a two-level system with energy splitting 2ϵ(k,ϕ).

Fig. <ref> shows the ABS spectrum as a function of the component k, with the phase difference set at ϕ=3π/4 and the chemical potential in the normal phase at μ_0 = 5 ħ /L (a) and μ_0 = 8 ħ /L (b). The solid black line is the spectrum obtained numerically for a finite short junction L/ξ=π/20, while the red dashed line is the universal analytical expression ± E(k,ϕ)=±Δ√(1-τ(k) sin^2(ϕ/2)), obtained in the limit L/ξ→ 0, where the k dependence is totally included in the transmission probability τ(k) which depends on the nature of the normal phase stripe <cit.>.
For the graphene electron gas <cit.>, the transmission probability of the normal state is τ(k)=(^2-k^2)/[^2-k^2 cos ^2 (L √(^2-k^2) )], with =μ_0/(ħ). 
From the comparison of the dispersion relations in Fig. <ref>, we see that the extrema are located at the same k, and in the limiting case L/ξ→ 0 they correspond to the extrema of the transmission probability <cit.>.
In Fig. <ref> one can see that there is an energy window, which we call mini-gap, in which excitations are prohibited <cit.>. The minigap δ(ϕ)=min_k[ϵ(k,ϕ)] depends on the phase difference ϕ.
In particular, in the limit L/ξ→ 0, we have δ(ϕ)= Δ |cos(ϕ/2)|.


We conclude this Section by introducing the particle density and the charge density operators, which are the quantities of interest.
Within the D-BdG formalism, the particle density and the charge density operators are expressed, in terms of the four component spinors defined in Eq. (<ref>), as


    ρ̂(r)=   ∑_ζΨ̂_ζ^†(r) _τ_σΨ̂_ζ(r),
    ρ̂_ C(r)=   ∑_ζΨ̂_ζ^†(r) τ_z _σΨ̂_ζ(r),


where in ρ̂_ C electrons (holes) have charge +1 (-1).


Applying the projector P̂_A on the density operators in Eq. (<ref>), we define the Andreev particle and charge densities operators as


    ρ̂_ A(r)=   P̂_Aρ̂(r)P̂_A=
            ∑_ζ∑_j,j'∑_k,k'φ^†_k, E(j,k,ϕ),ζ(r)
       ×_τ_σφ_k', E(j',k',ϕ),ζ(r)
            γ̂_ζ,j, k^†γ̂_ζ,j', k'
            ,
    ρ̂_ C,A(r)=   P̂_Aρ̂_ C(r)P̂_A=
            ∑_ζ∑_j,j'∑_k,k'φ^†_k, E(j,k,ϕ),ζ(r)
       ×τ_z _σφ_k', E(j',k',ϕ),ζ(r)
            γ̂_ζ,j, k^†γ̂_ζ,j', k'
            ,


whose support is the ABS subspace.

In the following sections, we study modifications of the spin-resolved local particle density of states (LDOS).
The spin-up electron and the spin-down hole LDOS are defined as


    ρ_ e↑(r, Ω)   = Tr[
         ρ̂_ A, e ↑(r)δ(Ω-Ĥ_ tot)] ,
    ρ̂_ A, e ↑(r)   =1/2[ρ̂_ A(r)+ρ̂_ C,A(r)] ,




    ρ_ h↓(r, Ω)   = Tr[
          ρ̂_ A, h ↓(r)δ(Ω-Ĥ_ tot)] ,
    ρ̂_ A, h ↓(r)   =1/2[ρ̂_ A(r)-ρ̂_ C,A(r)] ,


where we have Tr[·]=∑_ζ,j,k⟨∅|γ̂_ζ,j,k·γ̂^†_ζ,j,k| ∅⟩, where the reference state | ∅⟩ is the vacuum of all field operators γ̂_ζ,j,k.
The spin-down electron LDOS is given by ρ_ h↓(r, Ω)=ρ_ e↓(r, -Ω). 

So far we considered the bare GJJ in the short-junction regime. 
In the following, we include the presence of a localized impurity described in terms of the Anderson or Lifshitz model.
The total Hamiltonian Ĥ_ tot is composed of the Andreev Hamiltonian Ĥ_ A and a term Ĥ_ imp that takes into account the interaction of the ABSs with a single impurity.









§ SINGLE ANDERSON IMPURITY
 


In this Section, we discuss the Anderson model <cit.> which describes an impurity that causes inelastic scattering processes in a GJJ electronic system. Within this model, each impurity has two localized electronic states, distinguished by the spin projection along the z direction. For the spin-up and spin-down cases, the energies are E_d, ↑=μ_0+ϵ_d + ϵ_ Z and E_d,↓=μ_0+ϵ_d - ϵ_ Z, respectively. The
energy ϵ_d is the impurity energy with respect to the Fermi level μ_0, and the Zeeman energy term   breaks the time-reversal symmetry, and here any Coulomb repulsive term <cit.> is neglected.
Within the Bogoliubov-de Gennes approach, the Hamiltonian of the impurity can be expressed in a spinorial form as


    ℋ̂_ D   = Φ̂_d^† (ϵ_dτ_z+ϵ_Z_τ)Φ̂_d


where Φ̂_d^†=[c_d,↑^†, c_d,↓], and the fermionic operators c_d,↑^† (c_d,↓) creates (annihilates) an electron with spin-up (spin-down) bounded to the impurity. 


Here, we describe the interaction between a single impurity and the electron system in a graphene monolayer in terms of a tunneling Hamiltonian <cit.> expressed as


    V̂_ D   = V̂_d+V̂^†_d ,
    V̂_d   =∑_ζ= ±∫ d^2rΦ̂_d^† V_d, ζ(r) Ψ̂_ζ(r),


where the matrices V_d,ζ are defined as


    V_d,+(r)     =[[    v_A, d(r)    v_B, d(r)            0            0;            0            0 -v_A, d^*(r) -v_B, d^*(r) ]], 
    
            V_d,-(r)     =[[ -v_B, d^*(r)  v_A, d^*(r)            0            0;            0            0    v_B, d(r)   -v_A, d(r) ]].


We assume that the impurity is located at r_d in a carbon site and that the tunneling term acts on the electron system in graphene at the atomic scale.
Following Ref. pellegrino_commphys_2022, using a microscopic description based on a tight-binding model for the electron system in graphene, the matrix elements of the potential V̂_ D are expressed as


    v_α, d(r)=     t_0 √(A_c)[m_d  δ_α, A  + (1-m_d) δ_α, B] δ(r-r_d),



where A_c=3√(3)a^2/2 is the area of the unit cell <cit.>, with a=1.42 Å, t_0 is the tunneling amplitude which, without loss of generality, is taken real and positive. If the impurity is located at a carbon site belonging to the sublattice A (B) then one has m_d=1 (m_d=0). 

Here, we consider that the carbon site where the impurity is located belongs to sublattice A, and we have v_α, d(r)= t_0 √(A_ c)δ_α, Aδ(r-r_d).
The results remain unchanged regardless of the sublattice chosen, as a result of the sublattice symmetry.


The local interaction potential projected in the ABS subspace is explicitly written as


    𝒱̂_dP̂_ A = t_0∑_j ∑_ζ∑_k Φ̂_d^† w_d, ζ,j,kγ̂_ζ,j,k


where


    w_d,±,j,k   =g_d,±φ_k, E(j,k,ϕ),±  (r_d)
       =g_d,±e^i k y_d/√(W)φ̃_k, E(j,k,ϕ)(x_d),



    g_d, +   =  √(A_c)[  1  0  0  0;  0  0 -1  0 ] ,
    
     g_d, -   =  √(A_c)[  0  1  0  0;  0  0  0 -1 ] .


To study the local effect induced by an Anderson impurity on the electron system in a GJJ, we use the total Hamiltonian ℋ̂_ tot=ℋ̂_ A+ℋ̂_ imp, where

ℋ̂_ imp=Ĥ_ D+ 𝒱̂_d 𝒫̂_A+ 𝒫̂_A𝒱̂^†_d, which consists of the impurity Hamiltonian Ĥ_ D, and the tunneling terms which connect the ABSs and the Anderson impurity. 

To calculate LDOS, we use the following expressions


    ρ_ e ↑(r, Ω)   =-1/π Im Tr[  ρ̂_ A,e ↑(r) 𝒢̂_ tot(Ω+iη)],
    ρ_ h ↓(r, Ω)   =
    -1/π Im Tr[  ρ̂_ A,h ↓(r) 𝒢̂_ tot(Ω+iη)],

which involves the total Green function 


    𝒢̂_ tot(Ω)=(Ω-ℋ̂_ tot)^-1 ,


with η=0^+.
After algebraic manipulations, we write the projection of the total Green function onto the ABS subspace as


    𝒢̂(Ω)=   𝒫̂_A𝒢̂_ tot(Ω) 𝒫̂_A=(Ω-ℋ̂_eff)^-1,
    ℋ̂_eff=   ℋ̂_A+𝒫̂_A𝒱̂_d^†(Ω-ℋ̂_d)^-1𝒱̂_d𝒫̂_A.



Before calculating the LDOS in the presence of a single Anderson impurity, we consider the clean GJJ.
For the unperturbed GJJ system, the ABS Green function takes the simple form 


    𝒢̂_0(Ω)   =(Ω-ℋ̂_A)^-1
       =∑_j=±∑_ζ= ±∑_k1/Ω-j ϵ( k, ϕ)γ̂_j, ζ, k^†γ̂_j, ζ, k,


and by replacing it in Eq. (<ref>), we find the LDOS


    ρ_0,ℓ(r, Ω)=-1/π Im Tr[ρ̂_ A,ℓ(r) 𝒢̂_0(Ω+iη )].




It is useful to introduce the following 4×4 matrix Green function


    G̅_0(r,r',Ω)=∑_j ∑_k
    φ_k, E(j,k,ϕ),ζ (r) φ^†_k, E(j,k,ϕ),ζ (r')/Ω -jϵ(k,ϕ)
       =  ∑_k  e^ik(y-y')/W[ ∑_j φ̃_k, E(j,k,ϕ)(x) φ̃^†_k, E(j,k,ϕ)(x')/Ω -jϵ(k,ϕ)],


which acts on the direct product of the electron-hole subspace and sub-lattice subspace on which the spinors of the D-BdG formalism are built. It allows one to write a compact and intuitive form of the LDOSs. 
Although the valley index ζ is indicated in the initial line, the spinorial wavefunction does not depend on it, so it is omitted in G̅_0(r,r',Ω).
Furthermore, looking at the second line of Eq. (<ref>), each entry of G̅_0(r,r',Ω) has the form of a Fourier transform, and we calculate them using the Fast Fourier transform algorithm <cit.>, see details in Appendix <ref>. 
Starting from the definition in Eq. (<ref>), applying the cyclic property of the trace, and using Eq. (<ref>), the LDOS in the clear limit takes the following compact form


    ρ_0,ℓ(r, Ω)=-1/π Im{ tr[M_ℓG̅_0(r,r,Ω+i η)] },


where ℓ={ e ↑, h ↓}, M_ e ↑=1/2(_τ + τ_z) _σ, and M_ h ↓=1/2(_τ - τ_z) _σ, and tr[·] denotes the trace over the direct product of the electron-hole subspace and sub-lattice subspace.

To calculate LDOS in the presence of a single impurity, one has to find the ABS Green function 𝒢̂(Ω) expressed in Eq. (<ref>). 
For this aim, we rewrite Eq. (<ref>) in the form of a Dyson equation as


    𝒢̂(Ω)=𝒢̂_0(Ω)+𝒢̂_0(Ω)  Σ̂(Ω) 𝒢̂(Ω) ,


where  the self-energy is


    Σ̂(Ω)   =  𝒫̂_A𝒱̂_d^†(Ω-ℋ̂_ D)^-1𝒱̂_d𝒫̂_A.



The exact solution to the Dyson equation, as detailed in Appendix <ref>, is expressed in the following closed-form 
 

    𝒢̂(Ω)  =𝒢̂_0(Ω)+∑_ζ,ζ'∑_j, j^'∑_k, k^' F_ζ,j,k,ζ',j',k' (Ω)γ̂_ζ,j,  k^†γ̂_ζ^', j^', k^',


where the analytical expression of F_ζ,j,k,ζ',j',k'(Ω) is reported in Appendix <ref>, see Eq.(<ref>).

Based on the results in Eq.(<ref>), for both electrons and holes, the LDOS is written as the sum of two terms ρ_ℓ(r, Ω)=ρ_ℓ,0(r, Ω)+δρ_ℓ(r, Ω), where



    δρ_ℓ(r, Ω)=-1/π Im Tr{ρ̂_ A,ℓ(r)[𝒢̂(Ω+iη )- 𝒢̂_0(Ω+iη )]}.


Using Eq. (<ref>), we have the following


    δρ_ℓ(r, Ω)=-1/π Im∑_ζ∑_j,j'∑_k,k'φ^†_k, E(j,k,ϕ),ζ(r) M_ℓ
       ×φ_k', E(j',k',ϕ),ζ(r) F_ζ,j',k',ζ,j,k(Ω+i η) ,


where η=0^+.
Applying the cyclic property of the trace and using Eqs. (<ref>) and (<ref>), the expression above can be rewritten as



    δρ_ℓ(r, Ω)=-1/π Im tr{M_ℓG̅_0(r,r_d,Ω+iη)
    ∑_ζ g_d,ζ^† X(Ω + i η) [_τ - Y(Ω+iη)]^-1 g_d,ζG̅_0(r_d,r,Ω+iη)  } ,



where


    Y(Ω)= ∑_ζ g_d,ζG̅_0( r_d, r_d,Ω) g^†_d,ζX(Ω) ,


and the nature of the impurity is included in the 2×2 matrix X(Ω). For an Anderson impurity, one has 


    X(Ω) → t_0^2 G_d(Ω) ,


with


    G_d(Ω)    =  [(Ω-ϵ_Z)_τ -ϵ_dτ_z]^-1
       =
    [ 1/Ω-(ϵ_d+ϵ_ Z)              0;              0 1/Ω+(ϵ_d-ϵ_ Z) ] .


Fig. <ref> shows the LDOS for electrons with spin-up (solid black line) and holes with spin-down (red dashed line) as a function of energy Ω. The junction length, Fermi level and phase difference are set to L/ξ=π/20, μ_0=5 ħ/L, and ϕ=3π/4, respectively. To obtain the LDOS numerically, a finite value of η=10^-3Δ is used, which produces a fictitious finite linewidth of the energy levels within the mini-gap and a smoothing of the square root divergences that are expected at the extrema of the dispersion relations E(j,k).

Figs. <ref> (a) and (b) illustrate the case of clean GJJ, evaluated in the normal phase region at r_0=(0,0) and at the N-S interface r_0'=(-L/2,0), respectively. In both panels, the LDOS ρ_0, e ↑ and ρ_0, h ↓ are zero for energies within |Ω| < δ(ϕ). 


Figs. <ref> (c) and (d) illustrate LDOS in r_0=(0,0) and r'_0=(-L/2,0), respectively, when a non-magnetic Anderson impurity with parameters (ϵ_d,ϵ_ Z)=(0.2,0) Δ is present in r_d=r_0. Figs. <ref> (e) and (f) show the same LDOS when the impurity is magnetic, with parameters (ϵ_d,ϵ_ Z)=(0.2,0.1) Δ.
For both Anderson impurities, we fix the tunneling amplitude at t_0=√(Δħ W/(10 A_ c)  ).
In the absence of any spin-splitting term, such as the Zeeman term, the total Hamiltonian Ĥ_ tot is particle-hole symmetric. This symmetry leads to the identity ρ_ e,↑(r,Ω)=ρ_ h,↓(r,-Ω). The relation between the spin-down electron and the spin-down hole LDOS, which is independent of the Hamiltonian, is ρ_ e,↑(r,Ω)=ρ_ e,↓(r,Ω). Therefore, with a particle-hole symmetric Hamiltonian, the electron LDOS is independent of the spin z-projection, i.e. ρ_ e,↑(r,Ω)=ρ_ e,↓(r,Ω).
In the presence of a non-magnetic Anderson impurity where the particle-hole symmetry is present (see Fig. <ref> (c) and (d)), two bound states within the mini-gap, due to the hybridization of ABS and impurity, arise at opposite energies, with Ω_0=-0.185 Δ≲ -ϵ_d and Ω_0'=-Ω_0.
In Fig. <ref> (e) and (f), the effect of a magnetic Anderson impurity is taken into account, which breaks the particle-hole symmetry. This is evidenced by the emergence of two sharp peaks within the minigap at energies Ω_0=-0.098 Δ≳ -ϵ_d + ϵ_ Z and Ω_0'=0.267 Δ≲ϵ_d + ϵ_ Z, which are no longer opposite in energy, Ω_0'≠ -Ω_0.



















§ SINGLE LIFSHITZ IMPURITY



In this Section, we consider an impurity that acts as a source of elastic scattering processes for the electron system in a GJJ, and it is modeled by the Lifshitz model <cit.>. 
Similarly to Sect. <ref>, we consider a localized impurity that is placed at position r_d and at a carbon site that belongs to sublattice A, which generates the potential below


    Û_ D   = 
    ∑_κ'=K,K'∑_κ=K,K'∑_α=A,B∑_α'=A,B∑_s=↑,↓∫ d^2 r u_s
       ×δ_s,s'δ_α,Aδ_α',A A_ cδ(r-r_d) 
     ψ̂_α,κ, s^†(r)  ψ̂_α', κ', s'(r) ,


here u_s is the spin-resolved interaction energy, but spin-flip processes are excluded.
Due to the short-range potential of the impurity, there is a mixing between the valleys K and K'.
We focus on the low-energy electronic system and project the total Hamiltonian onto the ABS subspace. The ABS Green function can be expressed in the form of a Dyson equation, see Eq. (<ref>), where the self-energy is given by


    Σ̂(Ω)   =   P_ AÛ_ D P_ A
       =∑_ζ,ζ'∑_j,j'∑_k,k'w^†_d,ζ,j,k U_d w_d,ζ',j',k'γ̂_ζ,j,  k^†γ̂_ζ^', j^', k^' ,


with


    U_d=u_d τ_z + u_ Z_τ ,


u_d=(u_↑+u_↓)/2 and u_ Z=(u_↑-u_↓)/2.

Following the procedure used for a single Anderson impurity (details in Appendix <ref>), we find that the ABS Green function Ĝ(Ω) has the form shown in Eq. (<ref>), with



    X(Ω) → U_d.


Moreover, for both electrons and holes, the LDOS can be expressed as ρ_ℓ(r, Ω)=ρ_ℓ,0(r, Ω)+δρ_ℓ(r, Ω), where ρ_ℓ,0(r, Ω) is equal to Eq.(<ref>), while the modification induced by a single Lifshitz impurity is given by Eq. (<ref>) with the substitution in Eq. (<ref>).

Fig. <ref> shows the LDOS for electrons with spin-up (solid black line) and holes with spin-down (red dashed line) as a function of energy Ω, by selecting the junction length, Fermi level and phase difference used in Fig.  <ref>.


Fig. <ref> (a) [(c)] and (b) [(d)] show LDOS at r_0=(0,0) and r'_0=(-L/2,0), respectively, in the presence of a non-magnetic [magnetic] Lifshitz impurity located at r_d=r_0 and characterized by (u_d,u_Z)=(-1/8,0)  ħ W/A_ c [(u_d,u_Z)=(-1/8,0.1225) ħ W/A_ c]. 

A non-magnetic Lifshitz impurity acts as a source of elastic scattering that does not disrupt Cooper pairs <cit.>. Therefore, for any value of u_d and u_ z=0, the mini-gap is not affected by impurity scattering.
In fact, in Fig. <ref> (a) and (b), the LDOS ρ_0, e ↑ and ρ_0, h ↓ are exactly zero within the energy range Ω < δ(ϕ), despite the presence of a strongly attractive non-magnetic Lifshitz impurity. 
In Appendix <ref>, we verify semi-analytically that the emergence of a bound state within the mini-gap energy generated by the non-magnetic Lifshitz impurity is impossible.

The spin-independent strong attractive interaction considered produces a pair of symmetric peaks, Ω_0=-0.623 Δ and Ω_0'=-Ω_0, which appear outside the mini-gap energy window, δ(ϕ)<|Ω_0|<Δ, and preserves the particle-hole symmetry property ρ_ e ↑(r,Ω)=ρ_ h ↓(r,-Ω).

In contrast, a magnetic Lifshitz impurity,  which breaks the particle-hole symmetry, is a source of elastic scattering processes that can disrupt Cooper pairs <cit.>. In Fig. <ref> (c) and (d), we consider a spin-resolved potential, with u_↑=u_d+u_ Z=0.25 × 10^-2ħ W/A_ c and u_↓=u_d-u_ Z=-0.2475 ħ W/A_ c. This term u_ Z reduces the interaction with the spin-up channel and increases the attractive interaction with the spin-down channel. Two sharp peaks appear at Ω_0=-0.272 Δ and Ω_0'=0.786 Δ, and only the first one is within the mini-gap energy window.





§ ANALYSIS IN REAL SPACE



In this Section, we investigate the structure of the density fluctuations that are generated near the impurity, which can result in a sharp peak in the LDOS at an energy within the mini-gap energy δ(ϕ).

In the presence of a single impurity at r_d=(x_d,y_d)=(0,0), Figs. <ref> (a)-(b) and Figs. <ref> (a)-(b) [Figs. <ref> (a)-(b) and Figs. <ref> (a)-(b)] show the LDOS for spin-up electrons and spin-down holes, respectively, along the N-S interface r=(-L/2,y) and in the middle of the normal region along the transverse direction r=(x_d,y), with the Fermi level set at μ_0=5 ħ/L [μ_0=8 ħ/L].

In all panels, Ω_0 is the energy at which a sharp peak of the LDOS appears in the range of -δ(ϕ)<Ω_0<0. We have considered an Anderson non-magnetic impurity (solid black line), an Anderson magnetic impurity (green dashed line), and a Lifshitz magnetic impurity (blue dashed-dotted line).
For all the cases considered, the finite signals of LDOSs shown here are consequences of the hybridization between the impurity and ABSs. In fact, the LDOSs for a clean GJJ both ρ_ e ↑,0(r,Ω_0) and  ρ_ h ↓,0(r,Ω_0)  are exactly zero.
Moreover, regardless of the nature of the impurity, the LDOS exhibits a similar pattern of oscillations. This oscillatory behavior reminds us of Friedel oscillations that arise in an electron gas from localized perturbations <cit.>, where their characteristic wavevector 2 is a direct consequence of an important property of the perturbated system, namely the existence of a sharp Fermi surface in an electron gas <cit.>.
We now explore the Fourier analysis of the spatial pattern of the LDOS to determine what information about the ABSs in a GJJ can be obtained.
For this aim, we define the partial Fourier transform as


    ρ̃_ℓ(k;x,Ω)=
      
       ∫_-∞^∞ d y e^-i k yρ_ℓ(r,Ω) ,


where ℓ={ e ↑, h ↓}, and they have been calculated by using the Fast Fourier transform algorithm <cit.> (details in Appendix <ref>). 

Fig. <ref> (c)-(d) and Fig. <ref> (c)-(d)  [Fig. <ref> (c)-(d) and Fig. <ref> (c)  -(d)]  show respectively the partial Fourier transform ρ̃_ℓ(k;-L/2,Ω_0) and ρ̃_ℓ(k;x_d,Ω_0) (ℓ={ e ↑, h ↓}) of LDOS illustrated in  Fig. <ref> (a)-(b) and Fig. <ref> (a)-(b)   [Fig. <ref> (a)-(b) and Fig. <ref> (a)  -(b)], and the Fermi level is μ_0=5 ħ/L [μ_0=8 ħ/L].

Independently of the type of impurity and doping level, ρ̃_ℓ(k;-L/2,Ω_0) and ρ̃_ℓ(k;x_d,Ω_0) have a global maximum at k=0, indicating that the most effective scattering processes that lead to the bound state at Ω_0 have zero transferred momentum.

In addition to the peak at k=0, for both considered doping levels, ρ̃_ℓ(k;-L/2,Ω_0) and ρ̃_ℓ(k;x_d,Ω_0) show a pronunciated profile around finite wavenumbers. 

To understand the origin of these special transferred momenta, we recall that Fig. <ref> shows how the ABS dispersion relation ϵ(k,ϕ) and the analytical expression E(k,ϕ)= Δ√(1-τ(k) sin^2(ϕ/2)) have extreme points located at the same momenta <cit.>. 
In particular, for a given phase difference ϕ,  the minima of E(k,ϕ) occur when the transmission probability τ(k) reaches its maximum value τ(k)=1 (total transmission).
For the normal state of the graphene electron gas <cit.>, the total transmission occurs for k_0=0 (Klein tunneling <cit.>)  and for ± k_n=±√(μ_0^2/(ħ)^2-(n π)^2/L^2) (stationary wave condition), where n=1,…,⌊μ_0L /(πħ)⌋ (⌊⌋ is the integer part). 
Klein tunneling is independent of the doping level, whereas the number of momenta that satisfy the stationary wave condition is a function of the doping level.
For example, at a doping level μ_0=5ħ/L, only ± k_1 satisfy the stationary wave condition. When the doping level increases to μ_0=8ħ/L, ± k_1 and ± k_2 satisfy the stationary wave condition.

At special momenta k_0 and ± k_n, which correspond to highly transmissive channels of the normal graphene stripe, the ABS dispersion relation ϵ(k,ϕ) shows quasi-degenerate minima, and at these energy values, the density of states exhibits square root divergences <cit.>.

At the doping level μ_0=5ħ/L, regardless of the nature of the impurity, along the N-S interface x=-L/2, we observe in Fig. <ref> (a)-(b) an oscillating behavior of spin-up electrons and spin-down holes LDOSs with a characteristic wavenumber 2 k_1, as confirmed by the partial Fourier Transform in  Fig. <ref> (c)-(d), where pronounced peaks around ± 2 k_1 appear. 
Along the x=x_d line within the normal phase region, both the spin-up electron and spin-down hole LDOSs show a pattern in Fig. <ref> (a)-(b) that is reminiscent of beats. In fact, in the corresponding partial Fourier Transforms in Fig. <ref> (c)-(d) pronounced peaks appear around two wavenumbers k_1 and 2 k_1.

This result suggests that the scattering processes with exchanged momentum 2 k_1 and k_1 play a crucial role in the generation of bound states, and that the ABSs involved have momenta around the special ones k_0 and ± k_1.

At the doping level of μ_0=8ħ/L, the spin-up electron and spin-down hole LDOSs along the N-S interface at x=-L/2 display an oscillating behavior with a wavenumber of 2 k_2, as seen in Fig. <ref> (a)-(b). The partial Fourier Transforms in Fig. <ref> (c)-(d) provide further confirmation, with two distinct peaks at ± 2 k_2.
Along the x=x_d line in the normal phase region, both the spin-up electron and spin-down hole LDOSs in Fig. <ref> (a)-(b) show a pattern that resembles beats, and the partial Fourier Transforms in Fig. <ref> (c)-(d) are prominent around ±(k_1-k_2) and ±(k_1+k_2).
This result indicates that the momenta of ABSs more involved in the formation of the bound states are close to the specific values k_0, ± k_1, and ± k_2.
In conclusion, the Fourier analysis of LDOS reveals the momenta mainly involved in the scattering processes with ABSs associated with a high transmission probability τ(k), corresponding to the minima of ϵ(k,ϕ). 










§ CONCLUSIONS



In this work, we investigated the LDOS in the normal phase stripe of a ballistic short GJJ. In particular, we studied local effects in the vicinity of a short-range impurity and the generation of bound states resulting from hybridization with ABSs.
We examined two distinct approaches for describing a single short-range impurity, the Anderson model and the Lifshitz model. The Anderson model treats the impurity as a source of inelastic scattering, whereas in the Lifshitz model the impurity only causes elastic scattering.

We have found that in the absence of magnetic terms the Lifshitz impurity does not produce any bound state within the mini-gap. This result can be seen as a consequence of the Anderson theorem, which states that this kind of impurity cannot disrupt Cooper pairs. However, the Anderson impurity is capable of forming two bound states within the minigap, which are symmetric in energy with respect to the Fermi level.
By introducing a magnetic term, we have observed that the Lifshitz impurity disrupts Cooper pairs, and it can induce a bound state within the mini-gap. For the Anderson model, an additional magnetic term can break the energy symmetry of the two bound states within the mini-gap.
In summary, from the energy features of the LDOS within the mini-gap, one can distinguish the nature of an impurity coupled to the electron system of a short ballistic GJJ.
Moreover, we have studied the spatial structure of the density fluctuations in a bound state within the minigap. 
In these cases, we have observed that, apart from the Lifshitz non-magnetic model, given the phase difference and the Fermi level, LDOS exhibits a unique oscillating and decreasing trend as one moves away from the impurity, regardless of the nature of the impurity.
For each spatial profile, the corresponding Fourier analysis shows that the characteristic wavevectors of the density oscillations are connected to the momenta of the high-transmissive channels in ballistic graphene,  providing insight into normal-phase graphene.



The authors thank G. G. N. Angilella, J. Ankerhold, F. Bonasera, P. Hakonen, C. Padurariu, and V. Varrica for illuminating discussions and fruitful comments on various stages of this work. E.P. acknowledges support from the PNRR MUR project PE0000023-NQSTI. F.M.D.P acknowledges support from the project PRIN 2022 - 2022XK5CPX (PE3) SoS-QuBa - "Solid State Quantum Batteries: Characterization and Optimization". G.F. thanks for the financial support Università degli Studi di Catania, Piano di Incentivi per la Ricerca di Ateneo.
I.V. thanks the Institut für Komplexe Quantensysteme and IQST, Universität Ulm (Germany) for the great hospitality and stimulating environment.





figuresection



§ CALCULATION OF THE ABS'S GREEN FUNCTION WITH A SINGLE IMPURITY



In this Appendix, we show the details for the calculation of the ABS's Green function in the presence of a short-range impurity.
The approach described below is used for both the Anderson model (see Sect. <ref>) and the Lifshitz model (see Sect. <ref>).
In both cases, the ABS's Green function is involved in a Dyson series, see Eqs. (<ref>)-(<ref>), of the form


    Ĝ(Ω)=Ĝ_0(Ω)+Ĝ_0(Ω) Σ̂(Ω)Ĝ(Ω) ,


where Ĝ_0(Ω) is defined in Eq. (<ref>) and


    Σ̂(Ω)=∑_ζ,ζ'∑_j,j'∑_k,k'w^†_d,ζ,j,k X(Ω) w_d,ζ',j',k'γ̂_ζ,j,  k^†γ̂_ζ^', j^', k^' .


For the Anderson model, we have the following correspondence


    X(Ω) →[ t_0^2/Ω-(ϵ_d+ϵ_ Z)                  0;                  0 t_0^2/Ω+(ϵ_d-ϵ_ Z) ] ,


which is equal t_0^2 G_d(Ω), defined in Eq. (<ref>). Whereas, for the Lifshitz model, we have

    X(Ω) →[  u_d+u_z        0;        0 -u_d+u_z ] ,


which corresponds with U_d, defined in Eq. (<ref>).
By expanding the ABS's Green function as


    𝒢̂(Ω)=∑_n=0^∞𝒢̂_n (Ω),


where the n-th term


    𝒢̂_n(Ω)=𝒢̂_0(Ω)[Σ̂(Ω) 𝒢̂_0(Ω)]^n ,


for n>0 it can be expressed as



    Ĝ_n(Ω)   =   ∑_j,j^'∑_ζ,ζ^'∑_k,k^'1/Ω-j ϵ(k,ϕ)
      w^†_d, j,ζ,k
    X(Ω)
     
    Y^n-1(Ω)  w_d,j',ζ',k'  G_0(j^',k^',Ω)  γ̂^†_j,ζ,kγ̂_j^',ζ^',k^'



where


    Y(Ω)  =  ∑_ζ,j,k w_d,ζ,j,k    w^†_d,ζ,j,k/Ω-j ϵ(k,ϕ)     X(Ω) .








It is convenient to write the ABS's Green function as


    Ĝ(Ω)   =Ĝ_0(Ω) +∑_j,j^'∑_ζ,ζ^'∑_k,k^'1/Ω-j ϵ(k,ϕ)w^†_d,ζ,j,k
    X(Ω) 
       ×[ ∑^∞_n=0 Y^n(Ω)]  w_d,ζ',j',k'1/Ω-j' ϵ(k',ϕ)γ̂^†_j,ζ,kγ̂_j^',ζ^',k^',


in the square brackets, we have isolated the geometric series of ratio Y(Ω), which can be expressed in a closed form, and 𝒢̂(Ω) can be written in the form


    𝒢̂(Ω)  =𝒢̂_0(Ω)+∑_ζ,ζ'∑_j, j^'∑_k, k^' F_j,ζ,k,j',ζ',k' (Ω)γ̂_j, ζ, k^†γ̂_j^', ζ^', k^',


where


    F_ζ,j,k,ζ',j',k' (Ω) =1/Ω-j ϵ(k,ϕ) w^†_d,ζ,j,kX(Ω)
       ×[_τ-Y(Ω)]^-1 w_d,ζ',j',k'1/Ω-j' ϵ(k',ϕ).



The result expressed in Eq. (<ref>) is formally valid for both types of impurities, the differences from the Anderson model and the Lifshitz model are generated by the form of the corresponding matrix X(Ω), see Eqs. (<ref>) and (<ref>).



§ NUMERICAL FOURIER TRANSFORM



The discrete Fourier transform (DFT) is a mathematical operation that converts a finite sequence of discrete values { z_n } into another sequence {z̃_j }, according to the definition 


    z̃_j = ∑^N-1_n=0 e^-i 2π j n/N z_n,


and the inverse operation (IDFT) is defined as


    z_n = 1/N∑^N-1_j=0 e^i 2π j n/Nz̃_j.


Both of these can be calculated rapidly using the fast Fourier transform (FFT) <cit.>.

We demonstrate how IDFT has been used to calculate the 4×4 matrix G̅_0(r,r',Ω). According to Eq. (<ref>), each entry of G̅_0(r,r',Ω) is an inverse Fourier transform from the k domain to the y domain, i.e.  


    f(y)= ∑_k e^i k yf̃(k) .
  

In the case of a wide GJJ (W ≫ L,ξ), the summation over k can be replaced by an integration 


    f(y)=  W /2π∫^∞_-∞ dk e^i k yf̃(k) .
 

Now, we take f̃(k) in the range [-k_ max,k_max] and select the finite sequence f̃_j=f(k_j-k_ max), with k_j = 2 k_ max j/N and j=0,…,N-1. In terms of the definition of IDFT, we have the following


    f_n=1/N∑^N-1_j=0 e^ iy_n k_jf̃(k_j-k_ max) .


where y_n=π n /k_ max, where the resolution in the y-domain is Δ y = π /k_ max and the spatial range is [-π N/(2 k_ max),π N/(2 k_ max)].
When k_ max is fixed, the discrete values obtained by the IDFT approach the following expression in the limit of large N≫1:


    f_n= 1/2 k_ max∫^k_ max_-k_ max d k e^i k y_nf̃(k).


Comparing with Eq. (<ref>), we can see that


    f(y_n)= lim_k_ max→∞lim_N →∞ k_ max W/π f_n .


To calculate G̅_0(r,r',Ω), we have approximated the expression above with the finite values of N=23062 and k_ max=100/L.
According to this approximation, the function f(y) is calculated in the spatial range [-y_ max, y_ max] where y_ max=362.2 L. 

To calculate numerically the partial Fourier transform ρ̃_ℓ(k;x,Ω), defined in Eq. (<ref>), we have used an analogous approach.
For a given pair of x and Ω,  ρ̃_ℓ(k;x,Ω) has a form of the type below


    g̃(k)=
    
     ∫_-∞^∞ d y e^-i k y g(y) .


We consider the function g(y) in the interval [-y_ max,y_ max] and select a finite sequence g_n=g(y_n-y_ max), where y_n=2y_ max (n/N) and n=0,…,N-1.
Keeping fixed y_ max, in the limit N→∞ the discrete values obtained by the DFT of Eq. (<ref>) tend to be 


    g̃_j = N/2y_ max∫^y_ max_-y_ max dy e^-i k_j y g(y),


with k_j= π j/ y_ max, comparing it with Eq. (<ref>), we find g̃(k_j)= lim_ y_ max→∞lim_N →∞2 y_ max/Ng̃_j.


To calculate ρ̃_ℓ(k;x,Ω), we have used an approximation of the limit above, taking the finite values of N=23062 and y_ max=362.2 L. This enabled us to compute the partial Fourier transform in the range of [-100/L,100/L].






§ IMPOSSIBILITY OF A BOUND STATE GENERATED BY A NON-MAGNETIC LIFSHITZ IMPURITY








In this Appendix, we show that a single non-magnetic Lifshitz impurity cannot create a bound state in the subgap energy range |E| <Δ.
For a given phase difference ϕ, within the subgap energy window, the only region in which a generic impurity can create a bound state (with infinite lifetime) is within the minigap, δ(ϕ)=min_k[ϵ(k,ϕ)].
In particular, the occurrence of a pole of the function F_ζ,j,k,ζ',j',k' (Ω), see Eq. (<ref>), indicates the emergence of a bound state due to the hybridization of ABSs and the single impurity <cit.>. This is possible if there is an energy Ω that solves the following condition


    Det[_τ-Y(Ω)]=0 ,


where form of Y(Ω) depends on the specific nature of the impurity (see App. <ref>).
Here, we focus on the case of a single non-magnetic Lifshitz. Using Eqs. (<ref>) and (<ref>), after simple algebraic manipulations, below we report an equivalent equation to Eq. (<ref>) in terms of the 4×4 matrix G̅_0(r,r',Ω), defined in Eq. (<ref>), i.e.



    Det[  1/u_d A_ c- G̅_0(r_d,r_d,Ω)_11-G̅_0(r_d,r_d,Ω)_22            G̅_0(r_d,r_d,Ω)_13+G̅_0(r_d,r_d,Ω_0)_24;              G̅_0(r_d,r_d,Ω)_31+G̅_0(r_d,r_d,Ω)_42 -1/u_d A_ c- G̅_0(r_d,r_d,Ω)_33-G̅_0(r_d,r_d,Ω)_44 ]=0.
 

Conversely, we see that for any energy Ω within the mini-gap |Ω|<δ(ϕ), Eq. (<ref>) is an algebraic quadratic equation where the unknown variable is 1/(u_d A_ c). This algebraic equation has at least a real solution only if its associated discriminant D(Ω) is non negative, D(Ω)≥0, where


    D(Ω)=   [G̅_0(r_d,r_d,Ω)_11+G̅_0(r_d,r_d,Ω)_22+G̅_0(r_d,r_d,Ω)_33+G̅_0(r_d,r_d,Ω)_44]^2
       -4[G̅_0(r_d,r_d,Ω)_13+G̅_0(r_d,r_d,Ω)_24][G̅_0(r_d,r_d,Ω)_31+G̅_0(r_d,r_d,Ω)_42] .




Fig. <ref> illustrates the discriminant D(Ω) as a function of the energy Ω within the mini-gap δ(ϕ), for three different values of the phase difference: ϕ=3π/4 (solid lines), ϕ=0.1π (dashed lines) and ϕ=0.9π (dashed-dotted lines), and two values of the Fermi level: μ_0=5 ħ/L (black lines) and μ_0=8 ħ/L (red lines).
In all the cases presented, we have a finite short junction length L/ξ=π/20, and the impurity is placed in the middle of the normal phase stripe r_d=(0,0). Here, the discriminant D(Ω) is always negative, which confirms that a non-magnetic Lifshitz single impurity cannot generate a bound state within the mini-gap.







 Paper Title: End-to-End Protocol for High-Quality QAOA Parameters with Few Shots.


 


These authors contributed equally to this work.
Global Technology Applied Research, JPMorganChase, New York, NY 10017, USA


These authors contributed equally to this work.
Global Technology Applied Research, JPMorganChase, New York, NY 10017, USA


ruslan.shaydulin@jpmchase.com
Global Technology Applied Research, JPMorganChase, New York, NY 10017, USA
 
Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL 60439, USA

Global Technology Applied Research, JPMorganChase, New York, NY 10017, USA


§ ABSTRACT 
    The quantum approximate optimization algorithm (QAOA) is a quantum heuristic for combinatorial optimization that has been demonstrated to scale better than state-of-the-art classical solvers for some problems. For a given problem instance, QAOA performance depends crucially on the choice of the parameters. 
    While average-case optimal parameters are available in many cases, meaningful performance gains can be obtained by fine-tuning these parameters for a given instance. 
    This task is especially challenging, however, when the number of circuit executions (shots) is limited. 
    In this work, we develop an end-to-end protocol that combines multiple parameter settings and fine-tuning techniques.
    We use large-scale numerical experiments to optimize the protocol for the shot-limited setting and observe that optimizers with the simplest internal model (linear) perform best. 
    We implement the optimized pipeline on a trapped-ion processor using up to 32 qubits and 5 QAOA layers, and we demonstrate that the pipeline is robust to small amounts of hardware noise. 
    To the best of our knowledge, these are the largest demonstrations of QAOA  
    parameter tuning on a trapped-ion processor. 
    
    


End-to-End Protocol for High-Quality QAOA Parameters with Few Shots
    Marco Pistoia
    August 5, 2024
===================================================================





§ INTRODUCTION






Quantum computing has shown great promise in tackling computational problems that are difficult for classical computers. 
Among such problems, combinatorial optimization problems are of particular interest due to their ubiquity in fields including finance, logistics, and operations research and due to the existence of quantum algorithms offering speedups <cit.>. 
The quantum approximate optimization algorithm (QAOA) <cit.> 
is a prominent quantum heuristic that has been demonstrated to achieve better scaling than state-of-the-art classical solvers for certain combinatorial optimization problems, including maximum 8-satisfiability <cit.> and low autocorrelation binary sequence <cit.> problems. QAOA solves an optimization problem by preparing a parameterized quantum state such that upon measuring it, a high-quality solution is obtained with high probability.
To apply QAOA to a given problem, QAOA parameters must be set.


The performance of QAOA is highly sensitive to the choice of these parameters, and its parameter optimization has been widely studied in the community <cit.>. For many problem classes, optimal parameters have been derived in the infinite-size limit and empirically demonstrated to achieve good performance for finite-sized instances <cit.>. Even when rigorous theoretical analysis is out of reach, one fixed set of empirically obtained QAOA parameters can work well for most instances <cit.>. 
Nonetheless, 
there is still nontrivial variation in the optimal parameters between instances, which is often amplified by adding weights to the problem.
Thus, fine-tuning the average-case or infinite-size parameters for a given instance is necessary to fully exploit the algorithm's potential <cit.>.
Fine-tuning these parameters is challenging, however, especially when the number of circuit executions (shots) is limited, as is often the case with current quantum hardware.


Shots are the fundamental currency of near-term quantum computation. One “shot” represents an execution of a quantum circuit followed by a measurement. Each optimization iteration in QAOA requires hundreds or thousands of shots to minimize the sampling error of an expectation value evaluation <cit.>. 
The limitations of near-term quantum devices, such as their scarcity, frequent and time-consuming recalibration, and slow operation time, constrain quantum resource availability and, thus, the total number of shots available for an algorithm run. This constraint is particularly pronounced in atomic platforms such as trapped-ion and neutral atom quantum processors, where measurement time is on the same order of magnitude as gate time <cit.>.

The optimization of QAOA parameters presents a significant challenge if the number of shots is limited. This challenge is exacerbated by the fundamental limits that quantum mechanics imposes on the cost of computing gradients of quantum circuits <cit.>. The high cost of computing the gradient motivates the use of derivative-free optimization, which typically either assumes a deterministic objective <cit.> or requires a high number of shots to converge <cit.>. As a consequence of these challenges and despite the recent progress <cit.>, the problem of optimizing parameterized quantum circuits with a small number of shots remains open.












In this work, we propose and implement an end-to-end protocol for obtaining high-quality QAOA parameters with a small number of shots. Our protocol integrates multiple techniques to reduce the cost of parameter optimization, as shown in <Ref>. These techniques include previously studied ones such as initialization with instance-independent or “fixed” parameters <cit.> and rescaling of weighted problems <cit.>, as well as new components to carry out parameter fine-tuning in the shot-frugal setting, including optimizer selection, hyperparameter tuning, and shot budget allocation.
We use extensive numerical experiments to optimize all aspects of our pipeline. In doing so, we demonstrate that the optimizer with the simplest internal model
is the best option in shot-frugal scenarios. 
Our protocol performs well without further classical configuring when given a new problem instance.


We demonstrate the effectiveness of our protocol by deploying it on trapped-ion quantum devices applied to QAOA circuits with up to 32 qubits. Our protocol is optimized in noiseless simulation but is robust to small amounts of noise and achieves good performance on hardware. For example, in one instance of 20-qubit 3-regular MaxCut with p = 5, the parameter setting protocol achieves up to 56.61% relative 
approximation ratio
(AR) improvement in noiseless simulation while it holds 46.88% relative AR
improvement under hardware noise. We observe that as the circuit size grows, the noise becomes too strong, and the performance of the protocol deteriorates.



§ BACKGROUND


The quantum approximate optimization algorithm (QAOA) <cit.> solves combinatorial optimization problems by preparing a parameterized quantum circuit such that upon measuring it, high-quality solutions are obtained with high probability.
The circuit
is defined by two operators, problem Hamiltonian H_P and mixer Hamiltonian H_M, a hyperparameter p, and an initial state |ψ_0⟩:

    |ψ(γ,β)⟩ =e^-iβ_pH_Me^-iγ_pH_P…e^-iβ_1H_Me^-iγ_1H_P|ψ_0⟩ ,

where γ=[γ_0,⋯,γ_p] and β=[β_0,⋯,β_p] are free parameters. 
As the number of layers p approaches infinity, the QAOA circuit with appropriate parameters |ψ(γ, β)⟩ approaches the ground state of the problem Hamiltonian H_P, and the corresponding energy approaches the optimal value of the problem's objective function.
The parameters γ, β are typically obtained by using a classical optimizer that iteratively updates them based on the measurement outcomes:

    min_γ, β⟨ψ(γ, β)|H_P|ψ(γ, β)|,⟩

where ⟨ψ(γ, β) | H_P | ψ(γ, β)|$⟩ is the expectation value of the energy.

The optimization of variational parameters within the QAOA framework presents a significant challenge. On the one hand, the objective function in practical implementations is inherently stochastic. 
In each QAOA iteration, the expectation value⟨ψ(γ, β) | H_P | ψ(γ, β)|$⟩ is estimated and given to the optimizer as the objective by sampling numerous measurement results from |ψ(γ, β)⟩. Each measurement represents an execution of the entire circuit, and the number of circuit executions used to estimate a state is referred to as the number of shots. The most interesting and promising use cases involve limited shots of the stochastic objective. 


On the other hand, there is limited access to gradients of the QAOA objective, which means practical QAOA experiments have to rely on optimization techniques that do not utilize derivative information, a task that is naturally more complex than gradient-based optimization <cit.>. 
In the absence of gradients, quantum computing researchers have turned to derivative-free optimization (DFO) techniques as the classical optimization approaches in their QAOA work. DFO approaches can coarsely be categorized into direct-search and model-based methods. Direct-search methods evaluate the objective at a geometric pattern of points around a candidate point. If a better point is observed, the best point is updated; otherwise, the displacement is decreased in the geometric pattern. Model-based methods also evaluate points near a candidate point and use these evaluations to build various local or global models of the objective being optimized.

Applying DFO methods that are designed for deterministic objectives to stochastic objectives often leads to suboptimal performance. Rigorous convergence guarantees for stochastic DFO methods typically demand a substantial number of samples to accurately construct or adjust optimization models <cit.>; applying them to quantum optimization tasks will likely be difficult. There is ongoing research aimed at adapting deterministic DFO methods to better accommodate the inherent noise within stochastic objectives, striving for a balance between robustness and sample efficiency <cit.>.


In this paper, we apply QAOA to weighted maximum cut (MaxCut) and portfolio optimization (PO) problems. We now briefly discuss how QAOA is instantiated to be applied to these problems. 


Given an undirected graph G=(V, E) with an edge weight w_uv associated with each edge (u,v)∈ E, 
find s∈{-1,1}^|V|, that will 

    maximize  f(s) = ∑_(u,v)∈ Ew_uv/2 (1-s_us_v).



Mapping spin variables s_i onto the spectrum of Pauli Z matrices, we obtain the Hamiltonian that encodes the MaxCut problem on qubits:

    H_P = ∑_(u,v)∈ Ew_uv/2 (I - Z_uZ_v).


We use the Pauli X mixer Hamiltonian when applying QAOA to the MaxCut problem:

    H_M = ∑_iX_i.


The initial state |ψ_0⟩ is set to be the ground state of the mixer Hamiltonian, which for H_M in <Ref> is

    |ψ_0⟩ = |+⟩^⊗ n.



Given assets with expected returns μ∈ℝ^n
and covariance 
Σ∈ℝ^n× n, a risk factor q ∈ℝ, and a budget K ∈ℕ, find x∈{0,1}^n that will

    minimize        f(x) = q x^T Σx - μ^T x
        subject to       1^T x = K.



Mapping binary variables x_i to the Pauli Z matrices as x_i → (I - Z_i)/2, we get the Hamiltonian

    H_P=1/2q∑_i < jW_ijZ_iZ_j-1/2∑_i(q∑_jΣ_ij-μ_i)Z_i+c

encoding PO on qubits, where c=1/2∑_i(q∑_j = iW_ij-μ_i) is a constant. To preserve the Hamming weight of the state, we use an XY mixer with a 1-dimensional ring connectivity defined as

    H_M = ∑_i ∑_j=i+1X_iX_j + Y_iY_j.

The initial state is prepared as a Dicke state <cit.>, which is a superposition of all feasible (i.e., Hamming weight K) bitstrings with an equal probability. 


Given a solution s or x to the problem, we use approximation ratio (AR) to quantify the quality of the solution. For MaxCut, it is defined as

    AR(s) = f(s)-f_ min/f_ max-f_ min,

where f_ min and f_ max are the minimum and maximum  value of f(s), respectively; that is,

    f_ min = min_s f(s), 
    
        f_ max = max_s f(s).

For PO, we need to take constraints into consideration:

    AR(x) = 
      f(x)-f_ max/f_ min-f_ max,    ∑_i x_i = K, 
    
      0,    ∑_i x_i ≠ K,

where f_ min and f_ max are 

    f_ min = min_∑_i x_i = K f(x), 
    
        f_ max = max_∑_i x_i = K f(x).


We also use the metric of relative AR improvement, defined as 

    AR(x) - AR_ini/AR_opt - AR_ini,

where AR_ini and AR_opt are the approximations ratios corresponding to the collection of solutions produced by QAOA circuits with initial parameters γ_ini, β_ini and optimal parameters γ_opt, β_opt:

    AR_ini = 𝔼_x∼|ψ(γ_ini,β_ini)⟩ [AR(x)]
    AR_opt = 𝔼_x∼|ψ(γ_opt,β_opt)⟩ [AR(x)].

In practical evaluations, we obtain the optimal parameters by performing noiseless optimizations with unlimited shots.





§ RESULTS


We now present our results. We begin by summarizing our protocol in <Ref> and briefly introducing existing techniques our protocol uses. We proceed by describing the rest of its components, with optimizer selection in <Ref>, hyperparameter study in <Ref>, and budget allocation in <Ref>. We then present the performance of the protocol on trapped-ion hardware in <Ref>.




 §.§ End-to-end protocol for QAOA parameter optimization


<Ref> shows an overview of our protocol. Given a problem instance, we first follow <cit.> to rescale the weights. We divide the objective function 

    √(1/|E_2|∑_i,jw_ij^2 +1/|E_1|∑_iw_i^2 ),

where |E_2| is the number of quadratic terms in the objective function and |E_1| is the number of first-order terms. This rescaling rule can also be extended for problems with higher-degree terms. 

The difficulty of optimizing the parameters in QAOA heavily depends on the initial point selection. It has been shown for several problem settings that
the optimized parameters for different problem instances are approximately equal <cit.>.
Consequently, the averaged optimized parameters from several problem instances serve as a high-quality initial point. Thus, we use the parameters given in <cit.> for unweighted MaxCut with 3-regular graphs as our initial points for MaxCut and follow the empirical observation in Ref. <cit.> to 
use the averaged optimized parameters for the Sherrington-Kirkpatrick model <cit.> for PO. 
The values of γ and β are further rescaled so that they are on the same scale for the optimizer.

The rest of the components concern optimizer selection, hyperparameter tuning, and shot budget allocation, which we describe in detail in the following subsections.




 §.§ Optimizer choice




We conduct an evaluation of various optimization methods under the shot-frugal setting.
Specifically, we examine the performance of 
COBYLA <cit.>,
BOBYQA <cit.>, 
NELDER_MEAD <cit.>, 
ESCH <cit.>,
DIRECT_L <cit.>,
CRS2_LM <cit.>,
SPSA <cit.>,
GSLS <cit.>,
and
IMFIL <cit.>. These methods have been considered in the quantum
optimization context and are available in optimization packages
NLopt <cit.>, SciPy <cit.>, PDFO <cit.>, and
Scikit-Quant <cit.>. 
Some other methods we also tested but either work very similarly to one of the above methods or perform undesirably include
UOBYQA <cit.>, 
NEWUOA <cit.>,
LINCOA <cit.>, 
and 
SNOBFIT <cit.>.
We compute the energy landscapes of 60 random p=1 PO instances and efficiently test the optimization quality of
each method. 



Some of the tested methods have dozens of hyperparameters that can be adjusted
before running; other methods have only one or two. A complete study of the
performance of each method as its hyperparameters change is far beyond the scope
of this manuscript. Instead, we focus our numerical studies on the allocation of the total budget, an additional hyperparameter they all share that is crucial to our restricted setting. This is particularly important if the optimization method needs a few initial function evaluations before being able to make a prediction since more function evaluations lead to fewer shots per evaluation. In addition, the “initial step size” of model-based methods, which determines the spread of the initial pattern of points, is a crucial hyperparameter that impacts the performance significantly. We also vary it and choose the best-performing one for these methods in our benchmarking.
We test a total of 1,460 optimization configurations, and each configuration is evaluated 5 times with different sampling seeds on each of the instances. Please refer to <Ref> for the details of the efficient performance evaluation.


<Ref> shows the performance comparison of the tested methods using mean relative AR improvement (<ref>) as the metric.
We see that BOBYQA, COBYLA, and NELDER_MEAD  perform the best in this setting among all tested methods. We choose COBYLA over NELDER_MEAD since the former is considered as an improved version of the latter <cit.>. Between BOBYQA and COBYLA, we choose the latter, attributing to the fact that a simple model minimizes the number of initial function evaluations and maximizes the number of shots per evaluation. COBYLA assumes a linear model and needs only 2p+1 initial function evaluations to build the model for 2p parameters in a p-layer QAOA. BOBYQA follows almost the same strategy as COBYLA except that it assumes a quadratic model, which requires 1/2(2p+1)(2p+2) initial function evaluations to fully determine the model. 
This quadratic growth is less desirable than COBYLA's linear growth as we go to higher ps.
Similarly, other methods generally have more complicated assumptions and are thus more demanding in terms of the number of evaluations. The improved accuracy in the predictions does not compensate for the loss resulting from the large decrease in the number of shots per evaluation. 





 §.§ Hyperparameter selection


We now fix COBYLA as our optimization method and investigate its hyperparameter choice. COBYLA only has one hyperparameter to tune besides the budget allocation strategy, which is the initial step size (“rhobeg”). 
As previously mentioned, this hyperparameter determines the distance between the initial point and other initial evaluations, which is crucial for establishing an accurate linear model while using few evaluations. 
We run exact simulations of 100 random 12-qubit MaxCut instances and 60 random 12-qubit PO instances from p=1 to 5 and plot the optimization results under varying initial step sizes. 

In <Ref>, we plot the mean relative AR improvement with standard error over instances as a function of initial step size, where each curve represents the performance at the given number of function evaluations after the initial evaluations. The performance converges toward the maximum possible relative improvement as we increase the function evaluations, as expected. 
We observe that MaxCut, having a more accurate initialization strategy, requires a smaller initial step size than does PO.
With a given problem and initialization strategy, we notice that COBYLA is generally not sensitive to the initial step size, even when the number of evaluations is very low. The same trend can be observed across different ps, which we show in <Ref> and <Ref> in <Ref>.
For subsequent experiments, we use an initial step size of 0.1 for MaxCut and 0.5 for PO.





 §.§ Budget allocation


We now study the budget allocation strategy. We primarily focus on a per-instance total budget of 10,000 shots. This value of total budget was chosen to match the constraints of trapped-ion hardware. Prior QAOA experiments on the device used in this work used <5000 shots and p≤ 2 <cit.>.
For comparison, previous hardware demonstration on a superconducting processor used 25,000 shots per evaluation and 6 evaluations per optimizer iteration <cit.>. The number of shots per evaluation is inversely proportional to the number of total function evaluations:

    #shots/evaluation = ⌊shot budget=10,000/#evaluations allowed⌋.

We perform hyperparameter grid searches on the number of shots per function evaluation versus the maximum number of function evaluations on 1,000 random 12-qubit MaxCut instances from p=2 to 5.

<Ref> shows the mean relative AR improvement with standard error over instances as a function of the maximum number of function evaluations given to the optimizer, starting from the required number of initial function evaluations for building the linear model in COBYLA. We observe that in exchange for more function evaluations, the rapidly reduced number of shots per evaluation significantly impacts the measurement accuracy and optimizer behavior. With a good initial point, the optimizer can get close to the optimal point in a few iterations after the initial evaluations. Thus, the best strategy is to maximize the number of shots per evaluation so that the optimizer can rely on the initial evaluations to accurately predict candidate points. Empirically, we observe that 2 iterations after the initial evaluations work the best on the average case.

<Ref> shows a contour plot spanned by the number of additional evaluations and the number of shots per evaluation. The color represents mean relative AR improvement, and the three lines correspond to a total budget of 10k, 20k, and 30k, respectively. Focusing on the 10k budget line, we make the same observation as in <Ref>: smaller number of steps and higher per-evaluation shot budget
give the best results. Following the contours, We see that increasing the number of evaluations has little return, in contrast to the steady gain in increasing the number of shots per evaluation. If the number of shots is as low as 200, the optimizer will even find lower-quality parameters than the initial ones due to the extreme variance in sampling the objective function values. We expect the budget allocation strategy to be the same for similar budgets, whereas a substantially higher budget can allow more evaluations. We show results for p ∈{2, 3, 4} in <Ref> and <Ref> in <Ref>.












 §.§ Hardware demonstrations






We demonstrate the effectiveness of our parameter setting protocol with Quantinuum's noisy emulator of H1-1 (denoted as H1-1E), as well as the H1-1 <cit.> and H2-1 <cit.> quantum processors. Each optimization has a total budget of 10k shots across all evaluations for one problem instance.

We first validate the protocol on MaxCut problems. In <Ref>, we report the average performance over five n=12 graphs with QAOA depth up to 5. The Noiseless (∞) line optimizes the QAOA parameter under the noiseless backend and with an unlimited number of iterations. In other words, it represents the performance of the best achievable parameters. The Noiseless (2) line optimizes the QAOA parameter under the noiseless backend and with two additional iterations after the first 2p+1 initial evaluations, representing the results of the protocol in noiseless simulation. The H1-1E (2) line optimizes with two additional iterations with an emulator backend, which mimics the H1-1 quantum device, representing the deployment of the protocol in a practical scenario. The Initial line presents the performance under the fixed initial parameter. The results validate the effectiveness of the protocol, showing that under different p, the protocol can improve the parameter quality nontrivially. At p=1, all the fine-tuning results are close to the initial since the initial parameters are of very high quality. The gap between Noiseless (2) and H1-1E (2) quantifies the impact of the hardware noise. 

In <Ref>, we utilize all the qubits in the H1-1 processor and validate the protocol on three n=20 instances. The hardware results are shown on the H1-1 line. The proposed protocol performs well on quantum hardware, confirming the emulator results. In these selected instances, the initial ARs of p=5 are close to p=3 because the quality of initial parameters is relatively poor for these instances, highlighting the significance of the instance-level fine-tuning. The proposed protocol achieves up to 56.61% relative AR improvement in noiseless simulation while obtaining 46.88% relative AR improvement under hardware noise.

We use our protocol to optimize one n=32 MaxCut instance on the H2-1 processor. The ARs with the standard error over shots are shown in <Ref>. The “Hardware evaluation” column shows the results directly reported by the hardware
optimization. The “Exact evaluation” column shows the results evaluated with a noiseless state vector simulator using the parameters found by the hardware optimization. 
Since the circuit size is larger than in the previous experiments, the gap between the hardware and noiseless simulation also becomes larger. Nonetheless, for p=5, n=32 and a QAOA circuit with 240 two-qubit gates, the protocol is still able to optimize the parameter beyond the high-quality initial parameters. We plot the observed distributions with initial and optimized parameters in <Ref>.


We also use PO to verify our findings and numerically demonstrate the protocol's general applicability. PO is a harder problem setting because the initial parameter quality is generally poorer than MaxCut's. In addition, the constraint-preserving mixer has a larger overhead in the circuit.
We validate the protocol with H1-1E for five n=10 PO instances with p up to 7, as shown in <Ref>. The QAOA circuit has 97 two-qubit gates for initial state preparation and 65 two-qubit gates for one QAOA layer. We consistently observe the improved AR over the initial parameter setting. Meanwhile, as p becomes larger, the circuit encounters
an increasingly greater amount of noise, and the AR performance gap between the noisy and noiseless simulation becomes larger. 
For the p=7 experiments with 552 two-qubit gates, the relative AR improvement in the noisy simulation is 11.02% while it is 43.52% in the noiseless simulation. 
Parameter optimization in a highly noisy environment is challenging even under the fine-tuning setup, suggesting the necessity of deploying error suppression strategies and developing error correction techniques in larger-scale hardware experiments. 


 



§ DISCUSSION


QAOA has shown algorithmic speedup over the classical state of the art for some problems. High-quality parameters are necessary to realize its quantum speedup. However, although optimal parameters in the average cases are usually available, instance-level fine-tuning is necessary for maximizing the QAOA performance. 

We propose an end-to-end protocol for QAOA parameter setting. Focusing on weighted combinatorial optimization problems within the same family, we assume we have a fixed parameter initialization schedule for the unweighted problems from the same family. We first rescale the weighted problem instance such that the parameters for unweighted problems can be applicable. 
Then, we need to tune the hyperparameters of the classical optimizer. Under the shot-limited cases, we benchmarked 12 optimization algorithms and found that COBYLA and BOBYQA perform the best. Considering a realistic shot budget of 10k per problem instance, we use the simplest method, COBYLA. To tune the hyperparameters of COBYLA, we observe that a fixed initial step size works stably in QAOA with different layers for problems of different sizes. Then, we determine the maximum number of iterations of COBYLA to optimize the budget allocation. Under the setting of 10k shots, we assume an equally allocated budget and performe a grid search on the number of shots for one sample. We set the number of steps to be 2 and observe that it consistently works well. 

Our protocol can be applied to different problems. We test the protocol with MaxCut and PO instances. For MaxCut, we solve a 32-qubit problem with up to 5 layers of QAOA and performed hardware demonstration with both H2-1 and H1-1 devices. For PO, we solve 10-qubit problems with up to 7 layers of QAOA and demonstrate in the emulator of the H1-1 device. The protocol is effective and robust to noise. For example, in one instance of N20 MaxCut with p=5, the fine-tuning strategy achieves up to 56.61% relative AR improvement in noiseless simulation while it holds 46.88% relative AR improvement under hardware noise.

This protocol provides an end-to-end solution for QAOA parameter settings on a quantum device. 
The protocol is also generally applicable when we have many more shots, 
where we could run more fine-tuning iterations. For that scenario, some open and interesting directions still exist. For example, we can allocate the shots dynamically, allowing each iteration to have different shots. We can also assign the shot budget to the problem instances that perform relatively poorer under the fixed initial parameters. 





§ METHODS

We now present our approach to test our protocol through both numerical simulations and quantum demonstrations. We detail our methods for noiseless and noisy simulations, hardware experiments, and the specific combinatorial optimization problems we considered. 



 §.§ Evaluation and data generation

We perform both numerical simulations and quantum demonstrations to derive, tune, and validate our protocol. For noiseless simulations, we employ an optimized MPI-enabled QAOA simulation package, QOKit <cit.>, to simulate up to 32 qubits on 2 Nvidia A100-80GB GPUs. For noisy simulations, we use the Quantinuum H1 emulator <cit.>.
For hardware experiments, we obtain the results on the Quantinuum H1-1 and H2-1 quantum processors. H1-1 and H2-1 are trapped-ion devices that have full connectivity and high gate fidelity. For detailed specifications of H1-1 and H2-1, please refer to <cit.> and <cit.>.
Whenever the experiment setting is not exact (by exact, we mean noiseless with unlimited shots), we first obtain optimized parameters under the experiment settings and then report the results by exactly evaluating these parameters. This measure is to present the quality of the optimized parameters truthfully and fairly.

We consider two combinatorial optimization problems: weighted maximum cut (MaxCut) and portfolio optimization (PO). For MaxCut, we generate random 3-regular graphs with random edge weights sampled from a mixture of three Gaussian distributions {(𝒩(0, 1), 0.5), (𝒩(5, 2), 0.3), (𝒩(10, 1), 0.2)}, where 0.5, 0.3, and 0.2 are the probabilities of sampling from the corresponding Gaussian distribution. 
For PO, we use historical stock market data from Yahoo! Finance as the source for generating the expected return vectors and the covariance matrices. The 60 instances we use for each n are from the first 28 days of each month from January 2015 to December 2019 with n selected stocks from S&P 500 companies that have complete data during that period. 







 §.§ Low-cost optimizer benchmarking



We employ the quantum optimization helper package OSCAR <cit.> to conduct a computationally tractable evaluation of the various optimizers under different budget allocation strategies. For each QAOA instance, OSCAR computes a discrete energy landscape and then interpolates discrete points on the landscape to answer subsequent function evaluations without actually simulating the circuit. This enables us to quickly evaluate numerous optimization configurations with manageable costs.

Nonetheless, the discrete landscape computation is a grid search, which means its complexity is exponential in the number of points along each dimension. Although OSCAR supports reconstructing a landscape with a small number of sampled points, we choose to evaluate each point on the landscape with exact simulation to avoid approximations. Thus, this approach is viable only for low-depth QAOA landscapes. For our optimizer benchmark experiments, we use p=1 instances and a resolution of 128 by 128 over the π/4 by π/4 region where the center is the initial point we use in our experiments. <Ref> shows an example energy landscape we use. Notice that the local optimum (white star in the figure) is very close to the initial point (center of the figure), showing the effectiveness of our initialization strategy.

To simulate the energy with finite sampling errors, in addition to the energy landscapes, we compute the standard deviation landscapes to realize fast energy evaluation with an arbitrary shot. Based on the central limit theorem <cit.>, the deviation of a shotted estimation and exact QAOA energy will converge to a Gaussian distribution 𝒩(0, σ^2_exact/#shots), where σ_exact is the standard derivation of the energy evaluated by an exact QAOA state.

<Ref> shows an example standard deviation landscape we use. <Ref> and <Ref> show two landscapes sampled from the energy and the standard deviation landscapes with 5,000 and 500 shots per evaluation, respectively. We see that the finite sampling adds a noticeable salt-like noise to the landscapes, and fewer shots lead to heavier noise. To demonstrate the impact on optimization, we overlay COBYLA's evaluations on top of the landscape. Note that for demonstration purposes, we start from a lower left point instead of the center, which is the initial point we use in our experiments. We observe that the optimized point (yellow mark in the figure) is seriously affected by the sampling noise.






§ DATA AVAILABILITY


The data for reproducing figures used in this paper is available at <https://doi.org/10.5281/zenodo.12209739>.



§ CODE AVAILABILITY


The code for reproducing the data and figures used in this paper is available at <https://github.com/jpmorganchase/End-to-End_Protocol_for_High-Quality_QAOA_Parameters>.



§ ACKNOWLEDGMENTS

This material is based upon work supported in part by the U.S. Department of Energy, Office of Science, under contract number DE-AC02-06CH11357
and the Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research for Quantum Computing program.

The authors thank Danylo Lykov for his help with numerical experiments. TH, ZH, RS, and MP  thank their colleagues at Global Technology Applied Research of JPMorganChase for their support and helpful discussions.









§ DISCLAIMER

This paper was prepared for informational purposes with contributions from the Global Technology Applied Research center of JPMorganChase. This paper is not a product of the Research Department of JPMorganChase or its affiliates. Neither JPMorganChase nor any of its affiliates makes any explicit or implied representation or warranty and none of them accept any liability in connection with this position paper, including, without limitation, with respect to the completeness, accuracy, or reliability of the information contained herein and the potential legal, compliance, tax, or accounting effects thereof. This document is not intended as investment research or investment advice, or as a recommendation, offer, or solicitation for the purchase or sale of any security, financial instrument, financial product or service, or to be used in any way for evaluating the merits of participating in any transaction.

The submitted manuscript includes contributions from UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.  The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan <http://energy.gov/downloads/doe-public-access-plan>.




§ ADDITIONAL RESULTS

Here, we show additional results we have obtained, including the initial step size study for p=1,…,4 (<Ref> and <Ref>) and the budget allocation study for p∈{2, 3, 4} (<Ref> and <Ref>).












 Paper Title: Thermalization propagation front and robustness against avalanches in localized systems.






Scuola Superiore Meridionale, Largo San Marcellino 10, I-80138 Napoli, Italy
 

Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Complesso di Monte S. Angelo, via Cinthia, I-80126 Napoli, Italy


SISSA, Via Bonomea 265, I-34136 Trieste, Italy
INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy


Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Complesso di Monte S. Angelo, via Cinthia, I-80126 Napoli, Italy


Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Complesso di Monte S. Angelo, via Cinthia, I-80126 Napoli, Italy

§ ABSTRACT 




We investigate the robustness of the many-body localized (MBL) phase to the quantum-avalanche instability by studying the dynamics of a localized spin chain coupled to a T=∞ thermal bath through its leftmost site. By analyzing local magnetizations and quantum mutual information, we estimate the size of the thermalized sector of the chain and find that it increases logarithmically slowly in time. This logarithmically slow propagation of the thermalization front allows us to lower bound the slowest thermalization time, and find a broad parameter range where it scales fast enough with the system size that MBL is robust against thermalization induced by avalanches. The further finding that the imbalance – a global quantity measuring localization – thermalizes over a time scale exponential both in disorder strength and system size is in agreement with these results.






Thermalization propagation front and robustness against avalanches in localized systems
    Angelo Russomanno
    
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§  INTRODUCTION


Thermalization in quantum systems occurs in a way remarkably different than in classical systems, by a mechanism called eigenstate thermalization hypothesis (ETH) <cit.>. In a thermalizing quantum system eigenstates are locally equivalent to thermal density matrices, and this gives rise to long-time thermalization of local observables <cit.>. Generic isolated quantum systems are expected to thermalize and obey ETH <cit.>. It is therefore of particular interest to find systems that avoid thermalization: In this case, quantum information encoded in the initial state can persist for long times, with relevance for technological applications, as quantum memories <cit.>. 

In an ergodic system, thermalization occurs because its various parts can exchange particles and energy, so a possible way for a system to avoid thermalization is to exhibit insulating behavior, an example of which is given by Anderson localization <cit.>, occurring in non-interacting disordered systems. In the presence of small enough interactions, the system is still space localized and non thermalizing, a phenomenon called many-body localization (MBL) <cit.>. Following the works <cit.>, this topic has been tremendously explored over the past few years, from a theoretical <cit.>, experimental <cit.>, and numerical <cit.> point of view, particularly focusing on one-dimensional systems.
MBL systems display many interesting features such as the emergence of a complete set of quasi-localized integrals of motion <cit.>, area law entanglement in all many-body eigenstates <cit.>, logarithmic growth of entanglement entropy with time <cit.>. This last slow-entanglement-growth property is especially relevant, due the role played by entanglement in giving rise to ETH of local observables <cit.> (a thermal local density matrix is maximally mixed, and so any subsystem thermalizes becoming maximally entangled with the rest). The properties of MBL systems have been extensively described in different reviews <cit.>. 

However, the stability of this phase has been put into question in the thermodynamic limit, as it was pointed out that under certain circumstances many-body localized systems may be unstable towards rare regions of small disorder by a mechanism dubbed “quantum avalanches” <cit.>. In different works, they have attempted to tackle the matter of one- or higher-dimensional systems both theoretically <cit.> and experimentally <cit.>, studying the spectral properties or the dynamics of a MBL system in contact with an ergodic inclusion <cit.>. 
In some important cases the effect of the ergodic inclusion was studied using a Lindbladian acting on an end of the system <cit.>, as we better clarify below. <cit.>  

The search for evidence of the avalanche mechanism in standard MBL models is still very active.
One way is through many-body resonances, which allow globally different spin configurations to interact <cit.>. They are negligible in the MBL phase but there is a crossover regime where they become more and more relevant with increasing system size leading eventually to avalanches <cit.>, that spread just thanks to strong many-body resonances between rare near-resonant eigenstates <cit.>. Avalanches are researched also from the experimental point of view, where one monitors their spread by measuring the site-resolved entropy over time <cit.>. 

The way we are interested in here relies on the idea that an ergodic inclusion (that occurs almost certainly if the system is large enough) can thermalize the whole chain if the thermalization time of any subchain is short enough <cit.>. This time is numerically estimated by taking a subchain and coupling it to a thermal bath by one of its ends (that simulates the part of the chain that has already thermalized). If the slowest thermalization time, corresponding to the thermalization time of the spin farthest from the thermal bath, scales slowly enough with the size of the subchain (fast thermalization) then the avalanche propagates and the system thermalizes <cit.>.

In this work we do something quite similar, and we take an MBL system, couple it to a T=∞ thermal bath by one of its extremities, and study how the slowest thermalization time scales with the system size, in order to understand if MBL is robust to avalanches. We use the Lindbladian bath considered in Ref. <cit.> restricting it to just the leftmost site of the system. This bath bath is less general than the one considered in <cit.>, but such that it can be easily studied with an approach of Hamiltonian quantum trajectories, and able to provide a lower bound to the slowest thermalization time.

An advantage of this choice of bath is that it allows in some context a scaling to large system sizes. We see that because we study not only the case an MBL system, but also the one of an Anderson localized system. The latter provides useful information also for the MBL case, because the coupling to the bath makes the system interacting <cit.>. Although the chain coupled to the bath is nonintegrable, we can reach quite large system sizes using the Hamiltonian quantum-trajectory approach. One can show that the dynamics is equivalent to an average over many realizations (quantum trajectories) of a unitary Schrödinger evolution with noisy Hamiltonian (unitary unraveling <cit.>). Considering the Anderson-localized chain, one finds that along each stochastic trajectory the system is integrable, and can be described by a Gaussian state, so that a numerical analysis up to large system sizes is possible.

We focus on the scaling of the slowest thermalization time with the system size and we do that looking at the propagation of the thermalization front through the system, an approach that, to the best of our knowledge, has not yet been used in this context. Heat propagates through the system, from the bath at the leftmost site, and at any time there is a subchain on the left side that has already thermalized [see the cartoon in Fig. <ref>]. We estimate the length of this thermalized subchain by defining two length scales, one based on the behavior of local magnetizations, and another on the behavior of two-site mutual information.

We find that in a localized system (both Anderson and MBL) these length scales increase logarithmically in time. We can lower bound the slowest thermalization time by using this logarithmic increase to estimate the time when the thermalization front reaches the other end of the chain. We find that this time scale exponentially increases with the system size, and there is a regime of parameters where this exponential increase is fast enough that localization is robust against thermalization induced by avalanches. We study the slowest thermalization time also looking at the thermalization behavior of the imbalance and get results in agreement. 
























The paper is organized as follows. In Sec. <ref> we define our model, and in Sec. <ref> we discuss the numerical methods we use in the case of many-body localization. In Sec. <ref> we numerically show that the imbalance thermalization time is exponential both in the system size and in the disorder strength, and this allows us to show that there is a critical strength beyond which MBL is robust against the avalanche instability. In Sec. <ref> we define a thermalization length scale using the local magnetizations and show that it logarithmically increases with time. This implies that the thermalization time of the imbalance exponentially increases with system size. In Sec. <ref> we consider the mutual information and use it to define another thermalization length scale that also increases logarithmically with time. In Sec. <ref> we draw our conclusions and sketch perspectives of future work. In the Appendixes we discuss some important aspects that would have broken the main discussion. In particular, in Appendix <ref> we review how the Hamiltonian quantum-trajectory approach works, in Appendix <ref> we discuss how the Gaussian nature of the state in the case of the Anderson model simplifies the numerical analysis, and in Appendix <ref> we review the derivation of the threshold above which the scaling of the slowest thermalization time is slow enough to guarantee robustness against avalanches. 





§  MODEL


We consider the 1D spin-1/2 XXZ Heisenberg model in a random magnetic field 


    Ĥ_0 = ∑_j=1^L h_j Ŝ_j^z +  J ∑_j=1^L-1 ( Ŝ^x_j Ŝ^x_j+1 + Ŝ^y_j Ŝ^y_j+1 + ΔŜ^z_j Ŝ^z_j+1 ),


where L is the system size, Ŝ^α_j ≡σ̂_j^α/2 are the on-site magnetizations (α∈{ x, y, z }) and J = 1 sets the energy scale. The on-site magnetic field h_j are chosen randomly and uniformly in the interval [-W, W], and the system conserves the total magnetization S^z_ tot=∑_j=1^LŜ^z_j in the z-direction. We consider open boundary conditions. 

This model was widely studied as a paradigmatic model displaying MBL behavior <cit.> for several reasons. It exhibits a transition from the ergodic to the MBL phase at a critical disorder strength W_c, and it is believed to capture all essential properties of the MBL phase and of the localization transition. At Δ = 1 various estimates of W_c have been made collocating 3.5 ≲ W_c ≲ 5.4 <cit.>, although the debate is still open. Furthermore, the Jordan-Wigner transformation <cit.> allows to map this model to a system of interacting spinless fermions with tunneling matrix element J and nearest-neighbor interaction strength Δ (see Appendix <ref> for details), connecting the model and the experiments on quasirandom optical lattices <cit.>. 

We couple one end of this system to a thermal bath that induces T=∞ thermalization. This evolution is described by the Lindbladian 


    ρ̇̂̇(t) = - i [ Ĥ_0, ρ̂(t) ] - γ^2 [Ŝ^z_1 ,[Ŝ^z_1, ρ̂(t)]] ,


where γ^2 is the coupling to the bath. Restricting to any S^z_ tot subspace, one can see that the identity  is the only steady state of this Lindblad dynamics, implying the T=∞ thermalization inside that subspace. A similar Lindbladian, acting on all the sites and not just on the first one, has been studied in <cit.>. 


In order to study this Lindbladian, we use a quantum trajectory approach. More specifically we rely on the so-called unitary unraveling <cit.> where the dynamics of Eq. (<ref>) is described by an average over many realizations of unitary Schrödinger evolutions with a noisy Hamiltonian. So one should evolve with the kicked Hamiltonian


    Ĥ_γ(t) = Ĥ_0 + γξ(t)Ŝ_1^z ,


where Ĥ_0 is defined in Eq. (<ref>) and ξ(t) is an uncorrelated Gaussian noise, for which ⟨ξ(t)|=⟩ 0, ⟨ξ(t)ξ(t')|=⟩δ(t-t') and all the cumulants are vanishing (angular brakets denote the average over noise realizations). In order to implement it numerically, we must discretize it over time intervals τ≪ 1/J and Trotterize it, so that the n-th evolution step is given by the action of the operator


    Û_n = ^-iη_nγŜ_1^z^-iτĤ_0 ,


where η_n are Gaussian random variables such that η_n = ∫_(n-1)τ^nτξ(t) dt, so that ⟨η_n|=⟩0 and ⟨η_nη_n'|=⟩τδ_n,n'. In the following we fix τ = 0.05 (choosing a shorter τ does not affect the results). Averaging over random realizations, in the limit τ→ 0 one recovers the Lindblad equation, Eq. (<ref>), as we show in Appendix <ref>, and in the Anderson-model case Δ = 0 the dynamics along each trajectory is given by Gaussian states, allowing thereby to numerically address large system sizes , as we discuss in detail in Appendix <ref>.






§ METHODS


For the dynamics, it is customary to evolve the system starting from the antiferromagnetic Néel state |ψ⟩ = |↑↓…↑↓⟩, which has total magnetization S_ tot^z = 0 and is included in the smaller sector of the Hilbert space of dimension 𝒩 = LL/2. 
In order to detect the effect of the kick on the localized chain, we compare two different evolutions, one with the Hamiltonian, Eq. (<ref>), without noise (γ=0)


    |ψ_0(t)⟩ = e^- i Ĥ_0 t|↑↓↑↓ ...⟩ ,


and one with the fully noisy Hamiltonian Ĥ_γ [Eq. (<ref>)], fixing the same disorder realization


    |ψ_γ(t)⟩ = ∏_n=1^[t/τ]Û_n |↑↓↑↓ ...⟩ ,


ending both evolutions at some final time t_f. The state |ψ_0(t)⟩ is unaffected by noise and will be considered as a reference. We average each of the quantities over N_r different disorder/noise realizations. When γ=0 in each realization we take a different random choice of the onsite fields h_j, for j∈{1, …, L} in Eq. (<ref>). When γ=1 we take in each realization a different choice of the h_j and also a different choice of the random sequence η_n, with n∈{1,…,[t_f/τ]} providing the noise in Eq. (<ref>). We indicate the average over the N_r disorder/noise realizations with an overline (…). We evaluate the errorbar on this average as the root mean square deviation divided by √(N_r), performing error propagation where appropriate.

In the case of the Anderson model we take N_r = 1200 while in the case of MBL, in which the simulation times are longer, the number of realizations will be going from N_r = 500 for the smallest system L = 8, to N_r = 100 for the biggest one L = 16. When we compare γ=1 and γ = 0 we average over the same set of disorder realization.  To simulate the time evolution we use exact diagonalization and Krylov subspace projection methods <cit.>. Exact diagonalization is much more efficient in the case of the Anderson model (Δ=0) thanks to the mapping of each trajectory to a free fermion model (see Sec. <ref>). In  the following text, whenever we consider an interacting case, we fix Δ=1, and when we consider coupling to the thermal bath we take γ=1.






§ IMBALANCE AND LOCAL MAGNETIZATIONS






 §.§ Imbalance behavior and robustness against avalanche instability


The imbalance ℐ between the even and odd sites in the spin representation is defined as 


    ℐ(t) = 1/L∑_j=1^L (-1)^j ⟨ψ(t)|Ŝ_j^z|ψ(t) ⟩.


It is a global feature of the system which can be computed from local quantities, and can be experimentally measured <cit.>. 
The normalization ensures ℐ(t=0) = 1. 
The long-time stationary value of the imbalance effectively serves as an order parameter of the MBL phase, which is why it has been widely used in the literature <cit.>. For disorder strength W < W_c, a power-law decay ℐ(t) ∝ t^-β has been observed <cit.>, while, in localized ones for W > W_c, one numerically sees convergence to a constant value at long times <cit.>, indicating the existence of a set of conserved quantities preserving local information about its initial state. (See <cit.> for the challenges related to numerically observing the latter regime.)

The goal of our analysis is to understand how the imbalance applying the kick ℐ_γ = 1 differs from the unbiased case, ℐ_γ = 0 , scaling the size of the system L and the strength of the disorder W. 
We show some examples of the behavior of the imbalance in Fig. <ref> for L=16 and two different values of W [W=2 in panel (a) and W = 8 in panel (b)]. 
We see that the coupling to the bath γ causes ℐ_γ(t) to deviate from ℐ_0(t) starting at a time t ∼ 1, eventually decaying to zero. 

In order to perform a size scaling we define a relative imbalance ℐ_r as


    ℐ_r(t) = I_0(t) - I_γ(t)/I_0(t)  .


This quantity increases from the initial value ℐ_r(t=0)=0 [due to ℐ_0(t=0) = ℐ_γ(t=0)] to the asymptotic value ℐ_r(t→∞)=1 [due to ℐ_γ(t→∞)=0], as we see in Fig. <ref>. We can study the time scale over which this happens and we do that computing the minimum time t it takes for the relative imbalance ℐ_r to grow above a fixed threshold, within the statistical error. Given the computational limitations, we fixed the common threshold to be r_ th=0.17. <cit.>



We show results in Fig. <ref>. From one side we see that t displays a behavior consistent with an exponential increase with the disorder strength W, as the fits in Fig. <ref>(a) shows. This is in agreement with the exponential dependence on the disorder strength of the thermalization time, seen when one end of a MBL chain is time-periodically coupled to an ergodic one <cit.>. Furthermore, the exponentially increase with disorder strength is also displayed by the Thouless time t_Th, usually defined as the longest physically relevant relaxation time in the system <cit.>.
From the other side we find that t also exponentially increases with the size L of the system, as we show in Fig. <ref>(b).








So we conclude that, at least for the parameters considered in Fig. <ref>, there is some α > 0 such that the thermalization time of the imbalance has the behavior


    t∼exp(α LW) .


This numerical finding has interesting implications for the scaling of the slowest thermalization time, a point very important for the robustness of MBL under the effect of avalanches. As shown for instance in Ref. <cit.>, MBL is robust if the slowest thermalization time t_s (the one of the spin farthest from the bath) scales faster than 4^L (see Appendix <ref> for more details). If we have that t in Eq. (<ref>) scales faster than 4^L, then we expect the same happening for t_s, since by definition t_s> t. This condition leads to 


    W > W = 2ln 2/α .


Indeed, we have numerically verified that for sufficiently large values of W, many-body localization (MBL) is robust against avalanche instability. For W < W, MBL might still be stable because the time scale t_s might still grow faster than 4^L. In fact, if the scaling behavior observed in Fig. <ref> holds over a sufficiently wide range of parameters, there exists a disorder strength beyond which MBL remains robust.

To gain a deeper understanding of the imbalance behavior, we will focus on the properties of the local magnetizations in the next section. This will allow us to observe the propagation of the thermalization front.














 §.§ Thermalization front and local magnetizations


In order to better understand these behaviors of the imbalance, let us consider the local magnetizations S_j^z_γ(t) = ⟨ψ_t|Ŝ_j^z|ψ_t|⟩ for the evolution with a given γ. Let us set 


    δ S_j (t) = |S_j^z_γ=0(t)-S_j^z_γ=1(t)| ,


and use them to see how thermalization propagates through the chain. 

In order to do that we define a length scale as


    h(t) = ∑_j=1^L(j-1) δ S_j(t)/∑_j=1^Lδ S_j (t) .


We plot h(t) versus t for the Anderson-model case in Fig. <ref>(a) and in the interacting case in Fig. <ref>(b). In the Anderson-model case, whatever the disorder strength W, we see that h(t) logarithmically increases with time for t large enough, as h(t)∼ Aln(t), up to a constant, and the same occurs in the interacting case, when W > W_c and one is inside the MBL regime. In Fig. <ref>(b) we also show the behavior for W = 2 in which h(t) seems to be growing faster than logarithmically and saturating. We can obtain the slope A with a linear fit of h(t) versus ln t. (We apply the fit for times large enough that the linear in ln t regime has already set in). We plot the 1/A resulting from this fit versus W in the insets of Fig. <ref>. In the Anderson-model case we see that 1/A irregularly increases with W, while in the interacting case the increase is regular and slightly above a straight line.



So we have a logarithmically propagating thermalization front. This gives rise to some interesting predictions on the behavior of ℐ_r(t). Because at time t only a fraction f(t)∼ Aln(t)/L of the chain has thermalized, we can predict a behavior 


    ℐ_r(t)∼ Aln(t)/L .


We have obtained this estimate as follows. Due to extensivity, let us write the long-time value of the noiseless imbalance I_γ(t)=qL for some q∈(0,1) and t≫ 1/J. If only a fraction of f(t) sites has thermalized, we can roughly approximate I_γ(t)∼ L q[1-f(t)], and so we apply Eq. (<ref>) and get Eq. (<ref>), valid for t≫ 1/J.

We have numerically checked that this prediction is obeyed for large L in the Anderson case as we show in Fig. <ref>(a), and for smaller system sizes in the interacting case as we show in Fig. <ref>(b). In both panels we fix W and plot Lℐ_r(t) versus t for different values of L, with a logarithmic scale on the horizontal axis. We see that the rescaled curves tend to a limit curve, meaning that the scaling with 1/L holds for large system sizes. Furthermore, the limit curve linearly increases with time in logarithmic scale, consistently with Eq. (<ref>). This finding implies an exponential scaling with the system size of the thermalization time of the imbalance, as the one shown in Fig. <ref>(b).



As we can see in the inset of Fig. <ref>(b), also the behavior of 1/A is consistent with a linear increase in W. We can therefore write 1/A ∼β W for some β>0. Substituting in Eq. (<ref>) and imposing ℐ_r(t)=r_ th(=0.17) 
we get


    t∼exp(r_ thβ L W) ,


that is consistent with Eq. (<ref>).
 


    












Let us see what the results above imply for robustness of localization against avalanches. From one side, in the interacting case we have derived Eq. (<ref>), that implies the robustness of MBL for W large enough, as we have see in Sec. <ref>. From the other let us assume to average over infinite disorder/randomness realization and observe that Eq. (<ref>) implies


    δ S_j(∞)=lim_t→∞δ S_j (t) = lim_t→∞|S_j^z_γ=0(t)| ,


because lim_t→∞S_j^z_γ=1(t)=0 ∀ j=1, …, L due to thermalization. If we had periodic boundary conditions (PBC), due to the symmetries of the problem averaged over disorder, we would get that the |S_j^z_γ=0(t)| are independent of j, and also the δ S_j(∞) are independent of j (let us write δ S_j(∞)≡δ S). So we get


    lim_t→∞h(t) = δ S∑_j=1^L(j-1)/δ S∑_j=1^L 1 = L-1/2   for PBC .


Taking open boundary conditions, as we are doing in the numerical analysis above, affects the behavior of the |S_j^z_γ=0(t)|, and then of the δ S_j(∞), for a finite length l_b near the boundaries. This length l_b is finite and does not scale with L because we are assuming the system with γ=0 to be localized (Anderson or MBL). So we can write in our case


    lim_t→∞h(t) = L-1/2+𝒪(l_b) .





 



 


In order to find the thermalization time t^* we can say that all the local magnetizations have thermalized when h(t^*) = lim_t→∞h(t), and using that, for large enough t, h(t)∼ Aln(t), we get with some simple algebra
 
 
    t^*∼exp(L/2A) ,

 
with some multiplicative constants in front that are irrelevant for the scaling with L. In order to have robustness against avalanches, we must have a scaling faster than 4^L. We meet this condition if exp(1/2A)>4 or 


    1/A > 4ln 2 .


 

In the inset of Fig. <ref>(a), we observe that the Anderson model consistently satisfies this requirement within the parameter range we investigate. The Anderson model is inherently integrable and unable to generate ergodic inclusions. However, if an ergodic inclusion is introduced into the model, it remains resistant to thermalization induced by avalanches within this parameter range.

In contrast, the MBL model satisfies Eq. (<ref>) provided that W exceeds a threshold value W^* ≳ 7, as shown in the inset of Fig. <ref>(b). For W > W^*, MBL remains robust. For W < W^*, we cannot make definitive claims about stability because t^* in Eq. (<ref>) only offers a lower bound on the slowest thermalization time t_s.


 
 
 




§ QUANTUM MUTUAL INFORMATION


The quantum mutual information (QMI) for two spatial subsets of the chain 𝒜, ℬ⊂ [1,L] is defined as


    I_m (𝒜, ℬ) = S (𝒜) + S(ℬ) - S (𝒜∪ℬ),


where S (𝒜) is the Von Neumann entropy S(𝒜) = - Tr (ρ̂_A lnρ̂_A), ρ̂_A(B)=_A(B)[|ψ⟩⟨ψ|] is the reduced density matrix of the subsystem 𝒜(ℬ).
The QMI can be used to study the spread of quantum correlations between two separated spatial subsets at distance j. More specifically, if 𝒜≡ [1] is the subsystem only made by the spin in position 1, and ℬ≡ [1+j] is the subsystem only made by the spin at position 1+j, we define


    I_mj≡ I_m([1],[1+j])  .


It was shown in Ref. <cit.> that QMI can be used as a probe to detect MBL.


In the closed system case, γ = 0, I_mj(t) asymptotically tends to a constant, that depends on the distance j and the strength of the disorder W, and is in general smaller than the infinite temperature value.
If we apply the kick, the system thermalizes to infinite temperature and the mutual information between any pair of sites becomes a constant independent of the the disorder strength and of the choice of the sites. 

In Fig. <ref>(a) we show an example of I_mj(t) versus t for L = 14, γ = 1, Δ = 1, W = 1, for every possible value of j, together with the infinite temperature value T = ∞. We see that, independently of the distance j, at long times all the I_mj(t) converge to the same limit, that cannot be distinguished on this scale from the T=∞ one. We obtain the T=∞ value with the construction introduced in <cit.>: We average I_mj over random superpositions of spin configurations, each of them being


    |ψ⟩=1/√(𝒩)∑_{s}^-iφ_{s}|{s}⟩


where {s} are the spin configurations consistent with vanishing S^z_ tot, 𝒩=LL/2 is the dimension of the Hilbert space, and φ_{s} are random numbers uniformly distributed in the interval [0,2π].  In Fig. <ref>(b) we see see that the agreement between the asymptotic value and the T=∞ value
improves for increasing system size, in agreement with the fact that fluctuations around the thermal value decrease for increasing system size. We consider here the case W = 1 because the thermalization time is short, but eventually occurs for generic W, due to the thermalization to T=∞ induced by the bath. 






In order to track the spreading of the QMI in the system we define the following lengthscale


    ℓ(t) = ∑_j=2^L (j-1)I_m_j(t)/∑_j=2^LI_m_j(t) .


The fact that I_mj(t →∞) →const implies that lim_t→∞ℓ(t) = L/2,  as this quantity independently of the disorder strength, as one can easily check substituting in Eq. (<ref>) a constant value to I_m_j(t). We plot ℓ(t) versus t in Fig. <ref>(a) for the Anderson model (Δ = 0) and in Fig. <ref>(b) for the interacting model (Δ = 1). In the former case we see a logarithmic increase of this quantity in time, whatever the disorder amplitude W, while in the latter this logarithmic increase occurs when W>W_c and one is inside the MBL regime. We also show an example of behavior for W < W_c (W = 2) in which the growth is faster then logarithmic. In Fig. <ref>(a) we plot for comparison also a case without thermal bath, where ℓ(t) suddenly reaches an asymptotic value much smaller than L/2 and does not increase logarithmically in time. In the presence of the thermal bath, we find the slope of the logarithmic increase ℓ(t) ∼ Bln t linearly fitting ℓ(t) versus ln t. (We apply the fit for times large enough that the linear in ln t regime has already set in). We find that the inverse slope saturates with in W, exactly as the slope of h(t) versus ln t (see inset of Fig. <ref>).


Also here, as in Sec. <ref>, we can provide a lower bound to the slowest thermalization time. As we have noticed above, the system asymptotically thermalizes, so in the t→∞ limit the I_m_j(t→∞) become uniform, and we see that asymptotically ℓ(t→∞) = L/2. One gets thermalization of the mutual information at the time t^*_ i when this asymptotic behavior is saturated, that is ℓ(t_ i^*) =  ℓ(t→∞). Because for t large enough we have ℓ(t) = B ln (t/t_0,  i) for some  constant t_0,  i (see main panel of Fig. <ref>), this condition is equivalent to Bln (t^*_ i/t_0,  i) = L/2, that is to say


    t^*_ i = t_0,  iexp(L/2B) .


If we impose a scaling faster than 4^L we find 1/B> 4ln 2, and – as we can see in the insets of Fig. <ref> – this condition is verified for all the parameters we are considering, also in the interacting case [inset of Fig. <ref>(b)]. So, the analysis of the QMI provides an interval of robustness of MBL larger than the one provided by the analysis of the local magnetizations in Sec. <ref>. We have nevertheless to be cautious with this result, because we are modeling the environment as an average over trajectories. In the case of the local magnetizations it is perfectly equivalent using the Lindbladian or the average over the stochastic Schrödinger equation, because these quantities are linear in the quantum state. By contrast, the QMI is a nonlinear function of the quantum state, and so here we are specifically addressing the properties of the stochastic Schrödinger equation, and that might induce some unknown bias.






































































§ CONCLUSION AND PERSPECTIVES


In conclusion we have studied a many-body or Anderson localized model coupled to a T=∞ thermal bath by one end. The coupling is described by a Lindbladian, and to numerically study it we have used a quantum-trajectory approach. More specifically, we have used a unitary unraveling such that the dynamics is an average over many realizations of a unitary Schrödinger evolution with noise. In the case of Anderson localization this approach allows to reach quite large system sizes, due to the Gaussian form of the state along each quantum trajectory.

We have first numerically studied the dynamics of the imbalance, a global quantity widely used to assess the presence of localization. We have defined its thermalization time as the time beyond which the normalized difference of the imbalance with and without thermal bath goes beyond a given threshold, and we see that in the MBL case this thermalization time exponentially increases with the strength of the disorder and the size of the system. We have shown that this result implies that the MBL is robust against the avalanche instability when the disorder strength goes beyond a given threshold. 







We have then focused on the heat propagation through the system considering how a thermalization front propagated. We have estimated the extension of the already thermalized part of the chain, defining two length scales using the onsite magnetizations and the quantum mutual information. We have found that they both logarithmically increase with time. This is true both for the MBL case and we could use this fact to lower bound the slowest thermalization time with a quantity exponentially scaling with the system size. We have found that for strong enough disorder strong-disorder regime this scaling is fast enough that the system is robust to avalanches.

The thermalization front propagating logarithmically slowly suggests that the localized integrals of motion might play a role also in case of coupling to a bath, and slow down the heat propagation well below the ballistic trend predicted by the Lieb-Robinson bound <cit.>. The reason for this idea is that the localized integrals of motion play a strictly similar role in slowing-down the propagation of quantum correlations in an isolated MBL system, that leads to a logarithmic increase of entanglement entropy. Further studies are needed to clarify this point.





Perspectives of future work are exciting. First of all we can focus our attention to the density-density correlator <cit.>, a quantity whose propagation properties are known in MBL systems, and see how its behavior is changed by the presence of a propagating thermalization front. Second we can perform a similar analysis on the OTOC (that for MBL systems has been considered in <cit.>) to see how the thermalization front affects the scrambling properties of the system. Finally, we can deeper study the relation between strength of localization of the integrals of motion, logarithmic thermalization front, scaling of the slowest thermalization time, and robustness of MBL to avalanches.


We acknowledge interesting discussions with R. Bachman and useful comments on the manuscript by D. Huse and D. Luitz. G. P., P. L. and A. R. acknowledge financial support from PNRR MUR Project PE0000023-
NQSTI and 
computational resources from MUR, PON “Ricerca e Innovazione 2014-2020”, under Grant No. PIR01_00011
- (I.Bi.S.Co.).
G. P. acknowledges computational resources
from the CINECA award under the ISCRA initiative. This work was supported by PNRR
MUR project PE0000023 - NQSTI, by the European
Union’s Horizon 2020 research and innovation programme under Grant Agreement No 101017733, by
the MUR project CN_00000013-ICSC (P. L.), and by
the QuantERA II Programme STAQS project that
has received funding from the European Union’s Horizon 2020 research and innovation program.















































§ DERIVATION OF THE LINDBLAD EQUATION FORM THE QUANTUM-TRAJECTORY SCHEME


Let us start with the Schrödinger equation with the Hamiltonian Eq. (<ref>)


    i/ t|ψ(t)⟩ = [Ĥ_0 + γξ(t)Ŝ_1^z]|ψ(t)⟩ .


We can write it as


    |ψ(t+Δ t)⟩⟨ψ(t+Δ t)|  =|ψ(t)⟩⟨ψ(t)|
         -iΔ t[Ĥ_0,|ψ(t)⟩⟨ψ(t)|] 
       -i Δ W_t γ [Ŝ_1^z,|ψ(t)⟩⟨ψ(t)|] ,


where ⟨Δ W_t|=⟩Δ t and ⟨Δ W_tΔ W_t'|=⟩δ_t, t'Δ t are Gaussian uncorrelated variables. Let us also write


    |ψ(t)⟩⟨ψ(t)|   = |ψ(t-Δ t)⟩⟨ψ(t-Δ t)|
       -iΔ t[Ĥ_0,|ψ(t-Δ t)⟩⟨ψ(t-Δ t)|] 
       -i Δ W_t γ [σ̂_1^z,|ψ(t-Δ t)⟩⟨ψ(t-Δ t)|] ,


and substitute it in the last term of Eq. (<ref>). We get many terms. Averaging over randomness and defining ρ̂_t=⟨ϱ(t)⟩ with ϱ(t)≡|ψ(t)⟩⟨ψ(t)|, we get


    ρ̂_t+Δ t   = ρ̂_t - i Δ t [Ĥ_0,ρ̂_t]  
         -(Δ W_tΔ W_t)γ^2 [Ŝ_1^z,[Ŝ_1^z,ρ̂_t-Δ t]]
       + o(Δ t) .


Using ⟨Δ W_tΔ W_t|=⟩Δ t and going in the limit Δ t → 0 we get


    / tρ̂_t =- i [Ĥ_0,ρ̂_t] - γ^2 [Ŝ_1^z,[Ŝ_1^z,ρ̂_t]] .






§ CASE OF THE ANDERSON MODEL


Applying the Jordan-Wigner transformation, the model in Eq. (<ref>) becomes the spinless-fermion model


    Ĥ_0 = ∑_j=1^L h_j n̂_j +  J ∑_j=1^L-1[1/2( ĉ_j^†ĉ_j+1 +  H. c.) + Δn̂_j n̂_j+1]


where ĉ^(†)_j are anticommuting fermionic operators acting
on the j-th site. The number of fermions N≡∑_j=1^Ln̂_j is conserved. In the Anderson-model case we have Δ = 0 and we get a quadratic fermionic Hamiltonian, so that the time evolved state |ψ(t)⟩ along each trajectory can be cast in the form of a generic Gaussian state (Slater determinant).
The full information of such state is contained in a L × N matrix W(t), defined by

    |ψ(t)⟩ = ∏_k=1^N [  ∑_j=1^L [ W(t) ]_j,k ĉ_j^†] |Ω⟩ ,


where |Ω⟩ is the vacuum of the fermionic operators ĉ_j. Because the number of fermions is conserved and we initialize with the Néel state we have N=L/2. At initial time, the matrix L × N matrix defining the Néel state is given by [W(0)]_k,l = ∑_j=1^L/2δ_2j,lδ_j,k. The discrete evolution step (see Eq. (<ref>))


    |ψ(t+τ)⟩ =  ^-iη_nγŜ_1^z^-iτĤ_0|ψ(t)⟩


is simply translated in the dynamics of the matrix W(t) as


    W(t+τ) = K_n^-iτ Q W(t) ,


where the noisy step is implemented through the L× L diagonal matrix K_n with matrix element [K_n]_i,j=δ_i,j + δ_1,iδ_1,j(^-iη_nγ-1), while the L× L matrix Q implementing the action of the Anderson Hamiltonian has matrix elements [Q]_i,j = δ_i,j h_j + J/2(δ_i,j+1 + δ_i,j-1). Thanks to the validity of Wick's theorem, one can use the matrix W(t) to obtain the expectation of any observable, and even the entanglement entropy, as clarified in <cit.>. In this way one can numerically reach quite large system sizes, up to L∼𝒪(10^2). 




§ SCALING OF THE SLOWEST THERMALIZATION TIME AND ROBUSTNESS OF LOCALIZATION AGAINST AVALANCHES


In this section we explain why, in order for localization to be robust against avalanches, the slowest thermalization time should scale faster than 4^L when one end of the system is coupled to a thermal bath. The thing has already been clearly explained in <cit.>, and we add this section just for completeness. Let us consider Fig. <ref> depicting an MBL chain. We have an ergodic inclusion of size r_0 (red region), and a part that has already thermalize (yellow region) made by two sectors, with the same size r (we assume the avalanches to act symmetrically). The rest of the system is already localized (blue region) and we want to see if thermalization can further propagate there by avalanches.




With this aim, we focus on a part of the localized region in contact with the ergodic region, made by two sectors of length Δ r (dark blue region). We ask ourselves if this region thermalizes due to the contact with the yellow region, that's to say if the avalanche propagates. This can happen if each sector in the dark-blue region (let's say the right one – they are equal) thermalizes fast enough. So the slowest thermalization time t_s of each dark-blue sector must be much shorter than the inverse gap 1/δ of the region obtained joining the red, the yellow and the dark blue regions. The point is that thermalization in each dark-blue sector must be faster than the time needed to the part of the chain involved in the thermalizing dynamics (red+yellow+dark-blue) to express finite-size revivals and other quantum dynamical effects connected with the discreteness of the spectrum that hinder thermalization.
On general grounds <cit.> (and we numerically verify it also in our work) one has t_s∼κ^Δ r for some κ>0. Being this a spin 1/2 system one has δ=2^-(r_0+2r+2Δ r). Imposing t_s ≪ 1/δ, and asking that this condition should be verified also in the limit of Δ r ≫ r, one finds


    κ < 4 .


By contrast, if κ > 4 MBL is robust to avalanches. So it is very important to know κ. In order to find it, we have done as biologists do when move a portion of neural tissue from in vivo to in vitro <cit.>: We have cut away the right blue region from the system in Fig. <ref>, we have connected its left site to a thermal bath simulating the yellow thermalized region in Fig. <ref>, and we have studied the scaling of the slowest thermalization time with Δ r. (We have renamed Δ r as L for merely aesthetic reasons.)

































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 Paper Title: Energy Transport Among Highly-Polarized Atoms.














Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010.


Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010.


Department of Physics and Astronomy, Ursinus College, Collegeville, PA 19426.


Department of Physics and Astronomy, Ursinus College, Collegeville, PA 19426.


Department of Physics and Astronomy, Ursinus College, Collegeville, PA 19426.


Department of Physics and Astronomy, Ursinus College, Collegeville, PA 19426.


Department of Physics and Astronomy, Ursinus College, Collegeville, PA 19426.


Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010.


Department of Physics and Astronomy, Ursinus College, Collegeville, PA 19426.


Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010.




























§ ABSTRACT 

A static electric field of a few V/cm shifts the energy levels of ultracold Rydberg atoms in a magneto-optical trap. For a given principle quantum number, most of the energy levels are nearly degenerate at zero field and fan out with increasing field to form a manifold. We excite Rydberg atoms to energy levels near the center of the manifold, where the spacing is nearly harmonic, and allow them to exchange energy via resonant dipole-dipole interactions. We measure the time evolution as energy spreads away from the center of the manifold, which reveals that the system fails to thermalize for long interaction times. A computational model that includes only a few essential features of the system qualitatively agrees with this result. 







Energy Transport Among Highly-Polarized Atoms
    Michael W. Noel
    August 5, 2024
=============================================






§ OUTLINEISH




  * Idea - explore energy transport among states in a nearly harmonic ladder coupled by dipole-dipole interactions.


  * modeling

Harmonic - energy spreads rapidly to uniformly populate ladder - thermalization.

Bands of states with small anharmonicity - energy transport is suppressed.


  * Stark effect - Hydrogen vs. Rubidium

nearly harmonic

bands of low m states


  * experiment

MOT

Rydberg excitation 

Stark state excitation

Field ionization

MW spectroscopy

Timing

Calibration


  * discussion

Rapid energy exchange

At steady state, much of the population remains in the initial state.




 
Harmonic ladders of energy levels are ubiquitous in nature. Beyond the paradigmatic quantum harmonic oscillator, energy ladders are the basis for many phenomena in atomic, chemical, and condensed matter systems. They emerge from vibrations in molecules and lattices <cit.>, quantum well states in thin films <cit.>, or as Hofstadter's butterfly via field-induced electron motion in solids <cit.>. These energy structures lead to interesting studies of nonequilibrium dynamics, topological phases, and energy transport. Recent applications include memories for quantum photonic networks <cit.> and control of molecular disassociation through vibrational ladder climbing <cit.>. Manipulation of quantum confinement in quantum dots and other nanostructures can tune electronic and optical properties necessary for lasing and band gap engineering <cit.>. Engineered higher-dimensional Hamiltonians are also possible by encoding energy levels as an additional degree of freedom, or synthetic dimension <cit.>. Recently, ladders of ns and np Rydberg levels have been mapped to a synthetic dimension to probe band structure <cit.>.

The high tunability of strong interactions among ultracold atoms make them excellent platforms for quantum simulation and studies of energy transport <cit.>. Recent work has explored transport in the context of quantum many-body phase transitions <cit.>, such as the degree of correlation between entanglement growth and diffusion <cit.>. Energy transfer through spatial redistribution of electronic states has also been realized with direct imaging <cit.>. On the contrary, slowed transport has been observed in the presence of quantum many-body scar states <cit.>, or in dynamically localized systems such as the quantum kicked rotor <cit.>.
































In this Letter, we measure the evolution of energy transport in nearly harmonic ladders of energy levels in ultracold Rydberg atoms. The energy exchange is mediated by resonant dipole-dipole interactions among the amorphous cloud of rubidium Rydberg atoms in a magneto-optical trap (MOT). The sensitivity of Rydberg atoms to external electric and magnetic fields offers some control over the parameters of the ladder, making this a promising model system.

When an atom is immersed in an electric field, the degeneracy among different angular momentum states is lifted.  In non-relativistic hydrogen, the state splittings are dominated by the linear Stark effect, which leads to a harmonic spacing.  While the ℓ-degeneracy is lifted, the m-degeneracy is not.  Thus, each rung of the ladder consists of a set of even or odd m-states with the number of degenerate states increasing at the center of the manifold.  There are however some deviations from this in alkali atoms due to the complexity of the ion core.  The quantum defects lift the ℓ degeneracy at zero field leading to a quadratic Stark shift at low field.  Above ℓ=3, the quantum defects are small so this quadratic shift quickly returns to a nearly linear shift as the field increases.  

Including fine structure and considering only the low m_j states in rubidium, we can think about the manifold as clusters of six states each with |m_j| = 1/2, 3/2, 5/2. For an initial cluster near the center of the manifold, dipole moments between clusters vary between 0 and 60 ea_0 whereas dipole moments within clusters can be much stronger, around 250 ea_0. Clusters are separated by about 0.8 GHz and the initial cluster is about 14 MHz wide. The closest pair of states within the initial cluster are separated by about 0.4 MHz.

For a sample of atoms initially excited to a state near the middle of the Stark manifold, we might expect dipole-dipole interactions to drive the system into thermal equilibrium, with population spread uniformly across the manifold.  Instead, we find that a significant fraction of the population remains in the initial state, even at long interaction times. A simple model of this system confirms this behavior, pointing to the interplay between interactions within a cluster and across rungs of the ladder as the cause of the non-thermal dynamics.



Our experiment makes use of ^85Rb atoms that are cooled and confined in a magneto-optical trap.  With the trapping laser driving the 5s → 5p cycling transition, we use two additional lasers to excite Rydberg atoms.  One laser operates at 776 nm to drive the 5p → 5d transition and the other excites from 5d to states with nf character at 1265 nm.

Figure <ref>(a) shows a Stark map for the region of interest.  We must first find the desired Stark state.  This is done by locking the frequency of the 1265 nm laser so that it can excite states at an energy just below the zero-field high-ℓ n=34 manifold.  We then measure the number of excited Rydberg atoms as the field is increased as shown in Fig. <ref>(b).  For this experiment we chose the state that tunes into resonance at 7.15 V/cm for this excitation energy.  For this initial state, neighboring clusters are nearly harmonically spaced and the spacings of states within each cluster is small.  Once the desired state is found, the static electric field remains fixed at this value.  

Upon excitation to the Stark state, atoms can immediately begin to exchange energy through dipole-dipole interactions, which spreads population to neighboring states.  To study this energy transport, we must measure the distribution of population across the Stark manifold.  Selective field ionization cannot resolve the signal from these closely spaced Stark states, but it can resolve the 36d state from those in the n=34 manifold.  We therefore use a pulsed microwave field as shown in Fig. <ref>(a) to drive population from the manifold states to the d-state.  As the microwave frequency is scanned we measure the d state population to reveal the spectrum of excited Stark states as shown in Fig. <ref>.



Each panel in Fig. <ref> shows the microwave spectrum for a particular time delay between the laser excitation pulse and the microwave probe pulse, which is the time during which atoms can interact and exchange energy.  Both pulses have a width of 50 ns and a delay of 0 ns marks the time at which the trailing edge of the laser pulse is aligned with the rising edge of the microwave pulse.  At 7.15 V/cm, the 36d state is split into five states; j=5/2, |m_j|=1/2, 3/2, 5/2, and j=3/2, |m_j|=1/2, 3/2.  The range of energy between these five states is slightly smaller that the separation between neighboring manifold states.  We therefore expect the microwave spectrum to show five peaks for a single Stark state cluster.  The colored bands in Figs. <ref> and <ref> mark the range of microwave resonances associated with a single cluster of Stark states.  To find the population in each Stark cluster, we integrate the signal in each band.  The resulting evolution of energy transport due to dipole-dipole interaction is shown in Fig. <ref> where the color and dashing of each curve matches that of Figs. <ref> and <ref>.



At an interaction time of -100 ns there should be no signal in the d-state since the microwave probe pulse turns off before the Rydberg excitation pulse turns on.  In fact, we see signal in the 36d-state due to the finite turnoff time of the microwave pulse.  As the delay is increased to 0 ns, we see a rise in the overall signal level as well as population in neighboring Stark states due to the dipole-dipole coupling.  Beyond 0 ns, the interaction continues to spread population to neighboring manifold clusters although this spread is rather slow, leaving the bulk of the population in the initial Stark state.  In Fig. <ref>(a) we also see an asymmetry between clusters above (dashed lines) and below (solid lines) the initial state.  This is largely due to the variation in the microwave coupling strength between the manifold states and the 36d-state.

To measure the microwave coupling strengths, we switched to exciting the 36d-state <cit.>.  We then measured, using selective field ionization, the population in the n=34 manifold as we scanned the microwave frequency.  By separately exciting the five 36d-states, we produced a map of the coupling strength across the manifold for each 36d-state, which were weighted to match data shown for the initially excited state in Fig. <ref>.  Using the same integration bins as shown in Fig. <ref>, we found the overall weighting factors, which were used to scale the data in Fig. <ref>(a).  The result of this scaling, shown in Fig. <ref>(b), reveals a more symmetric population of clusters above and below the initially excited Stark state.

Ideally, we would quench the system into resonance after excitation to the initial state. However, since the dipole-dipole interactions are resonant at such a wide range of electric fields, we cannot excite at a non-resonant field. Thus, dipole-dipole interactions occur during the 50 ns laser excitation pulse and some of the energy transport visible before 0 ns in Fig. <ref> is possibly due to direct excitation of pair states <cit.>. However, after 0 ns the population of the initial state continues to decrease and the population of other manifold states continues to increase, a clear signature of dipole-dipole interactions. In fact, Seiler et al. determined that resonant Förster interactions among manifold states were required to model the decay of Rydberg atom population in an electrostatic trap <cit.>.

In order to understand the dynamics of energy transport in the manifold, it is important to have a good estimate of the Rydberg density. We measure our density using the simple set of few-body Förster interactions studied in <cit.>. We tune our 1265 nm laser to excite Rydberg atoms to the 36p_3/2,|m_j|=1/2 state in an electric field with no resonant interactions. We then quench into resonance, and the 36s state is populated via a two-body interaction at 3.29 V/cm or a three-body interaction at 3.52 V/cm. We measure the population fraction in the 36s state as a function of interaction time over 1 μs in 10 ns increments, while interleaving the two resonant electric fields. The diameter of the excitation beam is measured using a knife-edge.

We then simulate the time evolution of the 36s state fraction by numerically solving the Schrödinger equation in a cylindrical volume with the measured diameter. The only free parameters in the simulation are the density of the Rydberg atoms and an overall scale factor. The simulations are compared to the data and the simulated density with the smallest chi-squared value selected as the best fit. Since the two- and three-body Förster interactions scale differently with density, the fit is tightly constrained and yields a close match only for a small range of densities. Our measured Rydberg density is (4± 2)× 10^9 cm^-3, or an average interatomic spacing of about 6 μm. We have only performed this procedure for the density used in this experiment, yielding a single-point calibration. We are currently developing a full calibration along these lines, using a wider range of densities.



We use the calibrated density in our model of the manifold system. We numerically solve the Schrödinger equation using the two-body Hamiltonian

    ∑_a∑_iσ̂_aa^i E_a 
        +  ∑_a  b∑_c  d∑_i  j(σ̂_ab^i σ̂_cd^j +H.c.)μ_abμ_cd/R_ij^3,







where σ̂^i_ab is an operator that takes the i^th atom from Stark level a to Stark level b and R_ij is the distance between the i^th and j^th atoms. The transition dipole moments μ_ab, which connect Stark level a to b, are calculated using the Numerov method <cit.>. The first term gives the energy of each level and the second term represents represents the two-body Förster interactions. When c=a and d=b, the second term yields hopping interactions.

In our simulations, we assume that the atoms are essentially motionless and we ignore the angular dependence of the interactions. We randomly place dozens of atoms in a spherical volume at our measured density, while using the blockade radius to estimate a minimum distance between atoms. Including enough energy levels to faithfully represent the manifold necessarily limits the number of atoms we can include in the Schrödinger equation. We iterate over each atom in the volume and choose its three nearest neighbors and then run the simulation with those four atoms so that we include the possibility of “secondary” interactions. For example, two pairs of atoms in the initial cluster could exchange energy, resulting in a pair of atoms in higher energy clusters that are now able to interact. We repeat this process for multiple volumes and average so that the final results include a few hundred simulations.

With only four atoms in the model, the simulated dynamics will be slower than in the experiment <cit.>. Since we also do not attempt to model direct excitation of pair states, our simulated time evolution is not suited for quantitative comparison to the experiment. However, we can look at longer times in the simulation, when even for four atoms the populations of each energy level have plateaued.



As a starting point for understanding the spread of energy in our system, we consider a perfectly harmonic ladder of nine energy levels with four atoms initially in the middle level. We set the separation between energy levels to be 0.8 GHz, the mean separation between clusters in the actual experiment. The μ_ab are randomly drawn from a box distribution centered on the mean between-cluster dipole moments. Figure <ref> shows the distribution of energy, averaged over the last five microseconds of a 100 microsecond simulation. The system has thermalized, with a relatively even energy distribution that extends over all nine energy levels. When we run this model with five or six atoms and more energy levels, we find that the system still thermalizes.

The predictions of the harmonic ladder disagree strongly with the experimental result. We can attempt to improve the model by adding some anharmonicity. Moving outward from the initial cluster used in the experiment, the mean energy of each cluster shifts by about 0.1% of the energy spacing. With this small anharmonicity, we find that the results do not significantly change and the system still thermalizes.

We further attempt to make the model more realistic by adding clusters of energy levels. We include nine clusters of four | m_j|=1/2 and 3/2 energy levels, which are taken from the calculated Stark energy levels from Fig. <ref> along with the calculated μ_ab.  We include 
 only four-atom states whose detuning from the initial state is less than the typical energy spacing between clusters, resulting in about 125 000 states. 

While this model still ignores many energy levels with | m_j| > 3/2 , the simulation results shown in Fig. <ref> are in qualitative agreement with the data shown in Fig. <ref>. The model shows a similar degree of symmetric energy spreading to more distant clusters, while the bulk of the population remains in the initial cluster. The dynamics are strikingly simple given the complex structure of the energy levels and the couplings μ_ab.

Some of this simplicity may be attributed to our choice of initial cluster. As seen in Fig. <ref>, the initial cluster is so tightly spaced that the individual energy levels are not resolved. The bandwidth of our excitation laser is broad enough to excite a superposition of these energy levels. With so many possible initial pair states, there are dipole-dipole interactions with small detuning to every final pair of clusters. Interference among the many pathways from initial state to final state may play a role in the failure of this system to thermalize.

Our results indicate that the Rydberg Stark manifold could offer a promising platform for studying quantum dynamics with a rich and complex energy level structure. Experiments studying dipolar energy exchange in atomic systems usually start with a simpler set of energy levels. The many closely-spaced energy levels in the manifold are more similar to solid-state systems, offering prospects for quantum emulation. While the energy levels discussed in this experiment are arranged into nearly harmonic clusters, the addition of a static magnetic field could either broaden those clusters or cause them to overlap. For |m_j| > 5/2, there are also nearly harmonic ladders that fall between the ladder studied here.

It could be interesting to repeat this experiment for different initial clusters, or perhaps even excite individual levels within a cluster. For example, the dipole moments connecting energy levels within the cluster above our initial cluster are two orders of magnitude smaller. Exciting to this cluster would allow use to explore the importance of dipole-dipole interactions within the initial cluster.

Interactions beyond two-body energy exchange are also possible in the manifold. In fact, nearly every two-body interaction studied here is somewhat detuned from resonance. Since there are many energy-level spacings available within the clusters, the addition of a third atom can bring the interaction closer to resonance. While these three-body interactions scale as R^-6, with their smaller detuning we expect that they contribute significantly at higher densities. A study of the density dependence of energy transport in the manifold may be able to detect their effect.



This work was supported by the National Science Foundation under Grants No. 2011583 and No. 2011610, and S.E.S. is supported by the National Science Foundation Graduate Research Fellowship under Grant No. 2334429.

This work used the Delta system at the National Center for Supercomputing Applications through allocation PHY230142 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.













 Paper Title: Quantum Simulation via Stochastic Combination of Unitaries.


APS/123-QED



 peetz@ucla.edu
Department of Physics and Astronomy, University of California, Los Angeles, California 90095


College of Letters and Science, University of California, Los Angeles, California 90095


prineha@ucla.edu
College of Letters and Science, University of California, Los Angeles, California 90095



§ ABSTRACT 
Quantum simulation algorithms often require numerous ancilla qubits and deep circuits, prohibitive for near-term hardware. We introduce a framework for simulating quantum channels using ensembles of low-depth circuits in place of many-qubit dilations. This naturally enables simulations of open systems, which we demonstrate by preparing damped many-qubit GHZ states on ibm_hanoi. The technique further inspires two Hamiltonian simulation algorithms with asymptotic independence of the spectral precision, reducing resource requirements by several orders of magnitude for a benchmark system.


Quantum Simulation via Stochastic Combination of Unitaries
    Prineha Narang
    August 5, 2024
==========================================================




Introduction: Linear combination of unitaries (LCU) has proven to be an essential primitive in the design of quantum algorithms <cit.>, enabling the simulation of general operators within a larger, dilated Hilbert space <cit.>. While powerful as a theoretical tool, this ancilla-based dilation is often expensive in practice, generating extensive sequences of unitary operators controlled on many qubits. Combined with its need for post-selective measurements <cit.>, in total LCU can require many repetitions of deep quantum circuits, often infeasible for near-term quantum computers.

In this Letter, we propose a simple stochastic framework for mapping general quantum channels onto single- or no-ancilla circuits, avoiding the expensive many-qubit dilations and post-selection costs of LCU. The stochastic combination of unitaries (SCU) method decomposes a channel into a convex combination of simple transformations, which can then be efficiently sampled and run as independent circuits. This results in low-depth circuits at the cost of additional measurement overhead, invaluable for NISQ-era simulations. Because this technique naturally applies to the simulation of open quantum systems <cit.>, we demonstrate its practical advantages by simulating a noisy, eight-qubit entanglement generation process with high accuracy on ibm_hanoi.

Building on this framework, we then propose two stochastic algorithms for Hamiltonian simulation. As one of the most promising applications of quantum technology, Hamiltonian simulation is crucial not only for quantum dynamics <cit.> but also as a subroutine in ground state estimation <cit.>, material characterization <cit.>, and combinatorial optimization <cit.>. Despite substantial research into Suzuki-Trotter product formulas <cit.>, post-Trotter methods <cit.>, and randomized algorithms <cit.>, Hamiltonian simulation remains out of reach for near-term quantum computers. To alleviate these issues, we introduce two simple strategies; one samples a novel, Pauli-based decomposition of the Taylor series, and the other stochastically supplements the Suzuki-Trotter product formulas with higher-order corrections. Remarkably, both algorithms' circuit depths are asymptotically independent of the target precision, an algorithmic first that enables ultra-precise simulations. We provide resource estimates for simulating the transverse field Ising model, reducing gate counts by up to several orders of magnitude compared to existing methods.

Theory: Quantum channels are completely positive, trace-preserving maps and can be represented using an operator-sum representation 

    ℰ(ρ)=∑_i K_i ρ K_i^†,

where the Kraus operators K_i satisfy ∑_i K_i^† K_i = 1 <cit.>. We first decompose the Kraus operators into a unitary basis {U_j} with coefficients {c_j}. By appropriately normalizing the coefficients, we can then express the channel as a convex combination times a normalization factor λ := ∑_jk |c_j c_k^*|:

    ℰ(ρ)=λ(∑_j p_jj U_j^ρ U_j^†+∑_k<jp_j k/2(e^i α_j k U_j^ρ U_k^†+h.c.)),

where p_jk := λ^-1 |c_j c_k^*|. The channel consists of random-unitary terms (j=k) and coherence-preserving cross terms (j ≠ k).

To simulate the channel, we can sample the terms {U_j^ρ U_j^†} and {1/2 (e^i α_j k U_j^ρ U_k^†+ h.c.)} with their respective probabilities. The random-unitary terms map directly onto circuits consisting of U_j acting on the input state <cit.>, and the cross terms map onto circuits of the form shown in Figure <ref>. In Eq. (<ref>), each cross term and its Hermitian conjugate can be accessed simultaneously via a single X̂ measurement of the ancilla qubit <cit.>.

This stochastic combination of unitaries (SCU) thus generates an ensemble of circuits, with considerably simpler structure and depth compared to linear combination of unitaries (LCU) <cit.>. Repeatedly sampling this ensemble for N samples yields an unbiased estimator of the quantum channel <cit.>, with variance scaling as 𝒪(λ^2/N). In the special case of random-unitary channels, Eq. <ref> has unit norm (λ=1) and excellent scalability <cit.>. For general channels, we can mitigate the increased variance via additional measurements, scaling up the number of samples N by λ^2. For depth-limited devices, this trade-off between low-depth circuits and additional measurement overhead can enable otherwise infeasible simulations.




Damped GHZ Simulation: SCU naturally applies to the simulation of open quantum systems, which we demonstrate by simulating an eight-qubit GHZ state subject to CNOT-induced amplitude damping on ibm_hanoi. The GHZ state is a common example of multipartite entanglement, often analyzed in the context of quantum networks <cit.>. Realistic networks are subject to noise, such as photon loss and decoherence <cit.>, and our simulation serves as a guide that could be readily adapted to hardware. 

By applying a Hadamard gate and a chain of CNOTs, one prepares the n-qubit GHZ state,

    |GHZ⟩ = 1/√(2)(|0^n⟩ + |1^n⟩).

Because two-qubit gates are often error-prone in practice, we model a noisy GHZ state with a CNOT-induced amplitude damping channel, with Kraus operators K_0 = 1/2(1+√(1-p)) I + 1/2(1-√(1-p)) Z and K_1 =  √(p)/2 X + i √(p)/2 Y. That is, after each CNOT in the preparation of |GHZ⟩, we sample the convex decomposition of the quantum channel as in Eq. (<ref>); we then apply the sampled operators to the target qubit according to SCU. In the Pauli basis, the L_1 norms of K_0 ρ K_0^† and K_1 ρ K_1^† are 1 and p respectively, making the normalization constant of the channel 1 + p. For the full n-qubit GHZ preparation channel, assuming each CNOT has equal damping strength, the overall normalization constant is thus (1+p)^n-1.

We then measure the fidelity of the prepared noisy state ρ relative to an ideal state ρ^GHZ via a modified version of multiple quantum coherences (MQC) <cit.>, detailed in the Supplemental Materials. Following the approaches of <cit.>, we expand the fidelity as

    F(ρ, ρ^GHZ)=1/2(ρ_0^n, 0^n+ρ_1^n, 1^n+ρ_0^n, 1^n+ρ_1^n, 0^n).

In this sum, the all-zero and all-one populations are directly observable in the standard basis. The third and fourth terms are coherences, accessible via MQC. Specifically, the MQC circuit shown in <cit.> gives the signal S_θ = ρ_0^n, 0^n(θ), and we obtain the coherences via Fourier transform of this signal <cit.>. As discussed in the Supplementary Materials, the inversion stage of our MQC circuit exactly mirrors the preparation phase, including any sampled damping operators. We note that this technique generalizes the use of MQC to arbitrary channels, expanding beyond its current scope <cit.>.



For a damped GHZ state, the analytically predicted signal is the ideal GHZ sinusoid together with a damping prefactor,

    S_θ = 1/2 (1-p)^n-1 (1+cos (n θ)).

In Figure <ref>, we plot the MQC signal for different damping strengths p, comparing experimental results from ibm_hanoi with analytically predicted signals. This simulation gives excellent theoretical agreement in spite of the additional variance induced by SCU.

Finally, we Fourier transform these signals to compute the fidelity F(ρ, ρ^GHZ). For damping strengths p ∈{0,0.05,0.15}, we respectively measure the fidelities {0.932 ± 0.007,0.76 ± 0.01,0.539 ± 0.008} over five runs. These closely match the ideal predictions of {1,0.842,0.613}, up to a slight offset due to device noise.

Compared to a direct LCU approach, SCU results in much simpler circuits, with reasonable sampling overhead for small damping strengths. Notably, amplitude damping also has tailor-made circuits which outperform direct LCU. For example, the approach in <cit.> compiles to four CNOT gates for each instance of damping, compared to 2p/1+p CNOT gates on average with SCU. Concretely, in the eight-qubit GHZ simulation, the base MQC circuit with no damping requires 14 CNOTs. To include damping at p=15%, the implementation in <cit.> would require 56 additional CNOTs compared to only 3.7 on average with SCU, a massive improvement that enables an otherwise prohibitive simulation. We provide further resource analysis in the Supplemental Materials.


Hamiltonian Simulation: Building on this stochastic framework, we introduce two algorithms for Hamiltonian simulation: 1) convex Taylor sampling (CTS) and 2) stochastically enhanced product formulas. In each algorithm, we expand time evolution as a probabilistic combination of quantum operations,

    e^-iHtρ(0) e^iHt = λ(∑_j p_j U_j) ρ(0) (∑_k p_k U_k^†).

From this form, we approximate the channel via randomized sampling of {U_j} and {U_k^†}, obtaining left/right gate pairs for each time step. These transformations are then simulated via SCU with measurement overhead λ^2. While many such decompositions exist, we investigate expansions with minimal spectral error ϵ and normalization constant λ. Both of the algorithms have gate complexities which are asymptotically independent of the target precision, thus enabling simulations with low spectral error requirements.

Convex Taylor Sampling: The following theorem (proven in the Supplementary Materials) gives a novel decomposition of the Taylor series of a time evolution operator, with inspiration from Ref. <cit.>. We use this decomposition to construct a simulation algorithm, with step-by-step instructions in Figure <ref>. 





Theorem 1: For a Hermitian operator H = ∑_i c_i P_i and real parameter t,

    e^-iHt = L_c ∑_j p_j P'_j + √(1+L_s^2)∑_k p_k e^i θ P'_k,

where ∑_i p_i = 1, θ = (√(1+L_s^2)), and P'_i ∈±{I,X,Y,Z}^⊗ n represent Pauli strings up to a negative sign. Further, L_c and L_s are bounded as L_c ≤cosh(t |H|_1) - 1 and L_s ≤sinh(t |H|_1), where |H|_1 := ∑_i |c_i|. 
 


Theorem 1 expresses the Taylor series as a convex combination of unitary operators up to an overall L_1 norm. This form lends itself to convex Taylor sampling (CTS), an algorithm that leverages SCU to independently sample the left and right propagators, e^-iHt and e^iHt. Specifically, for a time step t, we truncate the expansion in Theorem 1 to order M and normalize by the overall L_1 norm μ(t) = L_c+√(1+L_s^2). We then sample the resulting convex combination

    𝒞_T^(M)(t) = ∑_j p'_j P'_j + ∑_k p'_k e^i θ P'_k,

where the coefficients p'_j := L_c/μ(t) p_j and p'_k := √(1+L_s^2)/μ(t) p_k form a probability distribution. Independently, we sample 𝒞_T^(M)(-t) to obtain an overall transformation of the form V_1 ρ V_2^†, as implemented in Figure <ref>. To minimize simulation error, we discretize into r time steps as follows:

    e^-iHt = √(λ)(𝒞_T^(M)(t/r))^r + 𝒪((|H|_1 t)^M+1/r^M),

where λ = μ(t/r)^2r is the normalization constant for the full channel. We see that for a tolerated error ϵ, CTS requires r ∼𝒪((|H|_1 t)^1+1/Mϵ^-1/M) time steps.



To mitigate the additional variance induced by the normalization constant λ, we budget a maximum measurement overhead, creating a second constraint on the number of time steps. Because the leading order contribution to λ is quadratic, we get

    λ∼ 1 + 𝒪((|H|_1 t)^2/r).

Thus, to limit the measurement overhead to λ^2, we require r ∼𝒪((|H|_1 t)^2(λ-1)^-1) time steps. For M > 1, this overhead constraint is asymptotically dominant over the constraint on the spectral precision.

The CTS algorithm is closely related to qDRIFT, a highly influential stochastic simulation algorithm <cit.>. qDRIFT approximates the time propagator to first order in time as a random-unitary channel, requiring r ∼𝒪((|H|_1 t)^2 ϵ^-1) time steps <cit.>. While both algorithms' gate complexities scale quadratically in |H|_1 and t, CTS is asymptotically independent of the spectral precision ϵ and instead scales with the maximum tolerated normalization constant λ. This difference is crucial because λ and ϵ typically differ by several orders of magnitude.

Stochastically Enhanced Product Formulas: The Suzuki-Trotter product formulas are perhaps the most well-known algorithms for Hamiltonian simulation <cit.>. For the Hamiltonian H = ∑_j c_j P_j, the first-order product formula approximates its time evolution to 𝒪(t^2) as

    𝒮_1(t) = ∏_j=1^m e^-i c_j P_j t.

More generally, 𝒮_p(t) denotes the p-th order product formula. Higher-order product formulas correct asymmetries from the operators P_j not commuting.

We can enhance these formulas by stochastically implementing their remainders as higher-order error corrections. For example, the leading error of the first-order product formula is a sum of commutators, t^2/2∑_i<j[P_i, P_j]; more generally, we can compute the remainders of a p-th order product formula up to a truncation order M. This yields a convex combination up to the normalization constant μ(t):

    𝒞_𝒮_p^(M)(t)  = p_0 S_p(t) + ∑_k p_k P_k',

where the Pauli strings are computed from the remainder expressions. For a desired precision ϵ, this stochastically enhanced product formula requires r ∼𝒪(t^1+1/Mϵ^-1/M) time steps. 

Once again, the overall normalization constant λ = μ(t/r)^2r determines our sampling overhead, with asymptotic scaling determined by the leading-order remainder. For the enhanced p-th order product formula,

    λ_𝒮_p∼ 1 + 𝒪((α^(p)_commt)^p+1/r^p),

where α^(p)_comm is the L_1 norm of the nested commutator expression in Ref. <cit.>. Thus, we require r ∼𝒪((α^(p)_commt)^1+1/p(λ-1)^-p) time steps to bound the measurement overhead to λ^2. This is asymptotically dominant over the constraint on spectral precision.

Transverse Field Ising Simulation: To highlight the performance of these approaches, we compare gate costs for simulating the transverse field Ising model (TFIM). The TFIM Hamiltonian is H = -J ∑_i Z^(i) Z^(i+1) - h ∑_j X^(j), where J is the exchange interaction parameter and h quantifies the strength of the transverse magnetic field.

We compute the number of CNOT gates required to simulate the TFIM for system size n, evolution time t=n, and model parameters J=1 and h=1, following Ref. <cit.>. To compare dependencies on spectral precision ϵ and normalization constant λ, we consider the parameter ranges 10^-6≤ϵ≤  10^-3 and 2 ≤λ≤ 2000. Our resource estimates for all algorithms are combined in Figure <ref>. For a broad range of sizes, our algorithms reduce the CNOT counts by several orders of magnitude. This difference is particularly pronounced for simulations with low spectral error, i.e. ϵ≤ 10^-6, due to our algorithms' asymptotic independence of spectral precision. Importantly, for many parameter choices, the reductions in depth exceed the measurement overhead of λ^2, improving the overall runtime.

To bound the sampling overhead, we evaluate each algorithms' series expansion to truncation order M, using Pauli string algebra rather than a matrix representation. For CTS and the stochastically enhanced first-order product formula, we compute the respective expansions to order M=3 up to 200 qubits. For the enhanced second-order product formula, we compute corrections to order M=4 and use Markov sampling for orders 5 ≤ M ≤ 7 up to 50 qubits, as detailed in the Supplementary Materials.

By properly including cross terms, our proposed algorithms model time evolution coherently, enabling empirical resource estimates with the spectral norm <cit.>. In contrast, qDRIFT estimates require the diamond norm, which is challenging to compute for more than a few qubits. Following Campbell's paper <cit.>, we instead analytically bound the number of gates as N ≤ 2 λ_H^2 t^2 / ϵ_♢, where λ_H is the L_1 norm of the Hamiltonian and ϵ_♢ is the diamond norm error bound.  To enable comparisons in Figure <ref>, we apply a bound on the diamond norm described in Refs. <cit.>. For a target unitary channel 𝒰(ρ)=U ρ U^† and a probabilistic ensemble of unitary channels 𝒱_k(ρ)=V_k ρ V_k^†,

    𝒰-∑_k p_k 𝒱_k_♢≤ 2U-∑_k p_k V_k.


Discussion and Conclusion: We first demonstrated SCU in the context of open quantum systems, simulating an eight-qubit, noisy GHZ state with high accuracy on ibm_hanoi. SCU significantly reduced circuit depths while maintaining reasonable measurement costs, enabling an otherwise prohibitive simulation. Beyond this example, one could similarly model time-dependent bath interactions via sequential sampling of time-dependent unitary decompositions. More broadly, this framework can be applied to any transformation, opening the door to applications beyond quantum simulation.

In applying SCU to the problem of Hamiltonian simulation, we showed stochastically executing high-order series operators can significantly reduce CNOT costs, with interesting parallels to composite simulation algorithms <cit.>. These ideas can similarly apply to higher-order product formulas as well as different systems of interest, such as fermionic and power law Hamiltonians. The algorithms' asymptotic independence of target precision is particularly attractive for simulations requiring low spectral error. 

As seen in the Pauli decompositions and as further discussed in the Supplementary Materials, the measurement overhead of SCU can be substantially impacted by the choice of unitary bases and the resultant L_1 norm of the unitary decomposition. Selecting a proper unitary basis or utilizing Hamiltonian decomposition techniques <cit.> could further lower the L_1 norm and thus the measurement overhead.

The SCU approach provides a general framework for realizing quantum channels as ensembles of simple quantum circuits. SCU repeatedly samples a sub-normalized channel's convex unitary decomposition, yielding unitary gates for the random-unitary terms and single-ancilla dilations for the cross terms. This approach mitigates the need for the large ancilla networks and multi-controlled gates required in techniques like linear combination of unitaries <cit.>, quantum signal processing <cit.>, and qubitization <cit.>. Instead, SCU generates low-depth circuits at the cost of additional measurements, a valuable trade-off for near-term quantum devices.


Acknowledgements: This work is supported by an NSF CAREER Award under Grant No. NSF-ECCS-1944085 and the NSF CNS program under Grant No. 2247007. The authors acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.











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§ SUPPLEMENTARY MATERIALS
 



 §.§ Damped GHZ: Generalized MQC
 

In this section, we provide implementation details for the simulation of the damped GHZ state, building on the multiple quantum coherences (MQC) formalism used in previous works <cit.>. A general density matrix can be decomposed in terms of its coherences ρ̂_m as ρ = ∑_m,qρ_q, q-m|q⟩⟨ q-m| = ∑_m ρ̂_m, and MQC provides a scalable technique to access these quantities experimentally. Here, we leverage MQC to compute the fidelity of a damped GHZ state ρ relative to an ideal GHZ state ρ^GHZ. Eq. (<ref>) expresses this fidelity as a sum over the populations P̂ := ρ_0^n, 0^n+ρ_1^n, 1^n and the coherences Ĉ := ρ_0^n, 1^n+ρ_1^n, 0^n. The populations are directly observable by simply preparing the damped GHZ state and measuring in the standard basis. We compute the coherences as C = 2 √(I_n), where I_m(ρ̂) = Tr[ρ̂_m^†ρ̂_m] is an experimentally accessible metric called the multiple-quantum intensity.

As explained in <cit.>, the derivation of MQC assumes unitary evolution. Gärttner et. al. show that MQC also works for certain non-unitary channels, but amplitude damping as described here is not included due to its asymmetric decay to |0⟩. In the following derivation, however, we show how the linearity of the SCU decomposition allows us to access the quantum coherences of the damped GHZ state through a generalized MQC scheme.

Denote ℰ_n as the channel which prepares the damped n-qubit GHZ state from |0^n⟩, combing all CNOTs and single-qubit damping channels into a single operator. We decompose this channel as follows:

    ℰ_n(ρ_0)     = λ(∑_j p_jj U_j^ρ_0 U_j^† +∑_j, k<j p_j k(e^i α_j k U_j^ρ_0 U_k^†+e^-i α_j k U_k^ρ_0 U_j^†)) 
        = λ(∑_j p_jjρ_j +∑_j, k<j p_j k(e^i α_j kρ_jk + e^-i α_j kρ_jk^†)),

where ρ_j := U_j^ρ_0 U_j^† and ρ_jk := U_j^ρ_0 U_k^†. For the random-unitary terms, we use the standard MQC technique <cit.> as depicted in Figure <ref>, while absorbing any stochastically sampled gates into U_j. After preparing ρ_j, we apply the rotation operator R(θ) = (P(θ) X)^⊗ n, invert the preparation via U_j^†, and then measure the all-zero population ρ̂_̂0̂. For the following derivation, observe that R(θ)ρ̂_j,m R(θ)^† = e^-imθρ̂_j,m. We thus measure the observable

    Ŝ_θ,j    = Tr[ρ_0 U_j^† R(θ) U_j^ρ_0 U_j^† R(θ)^† U_j^]  
        = Tr[(U_j^ρ_0 U_j^†) R(θ) (U_j^ρ_0 U_j^†) R(θ)^†] 
        = Tr[ρ_j R(θ) ρ_j R(θ)^†] 
        = ∑_m',mTr[ρ_j,m' e^-imθρ_j,m] 
        = ∑_m e^-imθTr[ρ_j,-mρ_j,m] 
        = ∑_m e^-imθ I_m(ρ_j).


For the cross terms, the derivation is similar. In order to properly invert the preparation, we exactly mirror the preparation gates by re-applying any sampled damping gates and using the same ancilla qubits. Overall, this yields

    Ŝ_θ,jk    = Tr[ρ_0 e^-i α_j k U_j^† R(θ) e^i α_j k U_j^ρ_0 U_k^† R(θ)^† U_k^] + Tr[ρ_0 e^i α_j k U_k^† R(θ) e^-i α_j k U_k^ρ_0 U_j^† R(θ)^† U_j^]  
        = Tr[(U_k^ρ_0 U_j^†) R(θ) (U_j^ρ_0 U_k^†) R(θ)^†] + Tr[(U_j^ρ_0 U_k^†) R(θ) (U_k^ρ_0 U_j^†) R(θ)^†] 
        = Tr[ρ_jk^† R(θ) ρ_jk R(θ)^†] + Tr[ρ_jk R(θ) ρ_jk^† R(θ)^†] 
        = ∑_m,m' e^-imθ(Tr[ρ_jk,m'^†ρ_jk,m] + Tr[ρ_jk,m'ρ_jk,m^†]) 
        = ∑_m e^-imθ I_m(ρ_jk + ρ_jk^†) 
        = ∑_m e^-imθ I_m(e^i α_j kρ_jk + e^-i α_j kρ_jk^†)


Combining these two schemes in the framework of SCU, in total we measure the all-zero population signal Ŝ_θ as

    Ŝ_θ    = λ(∑_j p_j Ŝ_θ,j +∑_j, k<j p_j kŜ_θ,jk) 
        = λ(∑_j p_j ∑_m e^-imθ I_m(ρ_j) +∑_j, k<j p_j k∑_m e^-imθ I_m(e^i α_j kρ_jk + e^-i α_j kρ_jk^†) ) 
        = ∑_m e^-imθI_m(λ∑_j p_j ρ_j + λ∑_j, k<j p_j k (e^i α_j kρ_jk + e^-i α_j kρ_jk^†) ) 
        = ∑_m e^-imθI_m(ℰ_n(ρ_0))


Finally, by Fourier transforming the signals S_θ shown in Figure <ref>, we obtain the multiple-quantum intensity I_m(ℰ_n(ρ_0)). The coherences are then C = 2 √(I_n), which we use to compute the fidelity relative to the ideal GHZ state.

In the Letter, we compare the experimentally measured fidelities with those of an ideal damped GHZ state, i.e. one hypothetically prepared on a noise-free quantum device. For these values, we use the following analytic calculations. The n-qubit GHZ state preparation involves n-1 instances of CNOT-induced amplitude damping. As a result, the all-one population is reduced by a factor of (1-p)^n-1, and the coherences are reduced by (1-p)^(n-1)/2. Using these damped values in Eq. (<ref>) gives an analytically predicted fidelity of F = 1/4(1+(1-p)^(n-1) / 2)^2.






 §.§ Damped GHZ: Resource Analysis
 
In this section, we compare the resource costs of our SCU-based approach to the direct circuit implementation shown in Figure <ref>. Compared to previously proposed circuits for amplitude damping <cit.>, we designed this circuit to be CNOT-efficient to make a fairer comparison with our stochastic approach. In Table <ref>, we show that SCU requires a sampling overhead of (1+p)^2 per instance but has a reduced average CNOT cost of 2p/1+p.  SCU thus offers significant reductions in circuit depth, especially pronounced for weak damping parameters p. As an example, consider our eight-qubit GHZ simulation with p=5%. For this case, the SCU approach requires a sampling overhead of λ^2 =  1.98.  However, including damping in the eight-qubit MQC circuit uses 28 CNOT gates with the direct approach compared to an average of 1.27 CNOT gates with SCU. Thus, with minimal sampling overhead, our SCU-based approach reduces these gate costs by over an order of magnitude.


For strong damping parameters p, the average CNOT cost with SCU is at most 1, still an improvement over the direct circuit. However, as the sampling overhead per instance increases, the scalability of this method suffers and eventually becomes impractical. In general, we recommend researchers balance this trade-off between depth reductions and sampling overhead for their specific applications.








 §.§ Markov Chain Sampling
 
In order for our SCU-based algorithms to be useful in early demonstrations of quantum advantage, we need to classically compute stochastic correction operators for reasonably large system sizes n, i.e. approaching 100 qubits. With this goal in mind, here we limit our truncation order to at most M = 4; this is merely a heuristic choice with which we can quickly compute corrections up to n ≈ 50 qubits on our 2023 personal computers. However, by borrowing ideas from Markov chains, we can use even higher-order operators without the need to fully expand and simplify them. 

As an example, product formula remainder expressions can contain operators of the form A = ∑_i^O(n) c_i P_i and B = ∑_j^O(n) c_j P_j. For reasonably large n, we can use Pauli algebra to fully compute second-order terms like AB or even fourth-order terms like A^2 B A. In contrast, seventh-order terms such as A^4 B^3 quickly become too costly to expand in the same way. We propose partitioning such terms into multiple layers of sampling, which in aggregate approximate the full operator. For this case, we 1) separately expand A^4 = ∑_l c_l P_l and B^3 = ∑_m c_m P_m, 2) sample each as independent ensembles {P_l} and {P_m}, and 3) approximate A^4 B^3 as the product of sampled operators {P_l P_m}. This yields an unbiased estimator but generally increases the L_1 norm compared to full expansion.

For the enhanced second-order product formula, this structure-agnostic approach allows us to achieve a spectral error corresponding to truncation order M=7 while still managing to empirically compute the overhead bounds up to 50 qubits. The downside is that we miss out on potential simplifications between the various remainder terms, ultimately increasing the sampling overhead. For example, applying this scheme to all remainders of the first-order product formula would effectively ignore the commutator scaling of the second-order correction, causing the normalization constant λ to scale worse with respect to system size.



 §.§ Transverse Field Ising Model: Resource Analysis

After computing the number of time steps r, we convert to CNOT cost. For the standard product formulas, we simply multiply the number of two-qubit Pauli exponentials by a factor of two. For qDRIFT, we multiply the total number of gates by the probability of sampling a two-qubit Pauli exponential, again rescaling this by a factor of two. For the enhanced product formulas, the dominantly sampled operators are the standard product formulas 𝒮_p. For a time step t/r, the probability of sampling 𝒮_p is p_0 = 1/μ(t) = λ^-1/2r, and the probability of sampling first 𝒮_p for the left operator and then 𝒮_p^† for the right operator is p_0^2 = λ^-1/r. The probability of any other event is thus p_1 = 1 - λ^-1/r. Over all time steps r, the expected number of sampled terms N_1 other than the standard product formula 𝒮_p ρ𝒮_p^† is simply

    N_1 = r p_1 = r (1-λ^-1/r).

Asymptotically, for large r, we obtain

    N_1∼ln(λ).

We conclude that the gate cost per time step is negligibly increased compared to the standard product formula approach.



For CTS, the dominantly sampled gates will be Pauli exponentials from I - iHt, and the sampled left and right gates will typically differ. For the TFIM with J=h=1, approximately half of the sampled gates will be e^-i θ Z Z  and the other half will be  e^-i θ X. A controlled single-qubit Pauli exponential is locally equivalent to C(R_z(θ)), which compiles to two CNOT gates. As shown in Figure <ref>, a controlled two-qubit Pauli exponential can be implemented with four CNOT gates. Thus on average, CTS uses approximately six CNOT gates per time step.

To minimize the need for ancilla qubits, one can use mid-circuit measurements, repeatedly resetting and reusing the same physical qubits. In principle, this approach requires only a single ancilla qubit for the entire simulation. Depending on the device capabilities, however, such measurements could create a bottleneck; a practical solution is to use a pool of ancilla qubits, thus lowering the frequency of measurements per individual qubit.



 §.§ Proof of Theorem 1
 
First, we Taylor expand e^-iHt and separate the terms into real and imaginary subsets.

    e^-iHt = ∑_l=0^∞1/l!(-itH)^l 
        = ∑_m=1^∞(-1)^m/(2m)! (tH)^2m + I + i ∑_m'=0^∞(-1)^m'+1/(2m'+1)! (tH)^2m'+1.

Now substitute the following Pauli decompositions of each sum:

    ∑_m=1^∞(-1)^m/(2m)! (tH)^2m    = ∑_j c_j P_j = L_c ∑_j p_j P'_j 
    ∑_m'=0^∞(-1)^m'+1/(2m'+1)! (tH)^2m'+1    = ∑_k c_k P_k = L_s ∑_k p_k P'_k.

Because all powers of H are Hermitian, the coefficients c_i are strictly real. We absorb any negative signs into the operators P'_i := sign(c_i) P_i, and normalize by the L_1 norms L_c := ∑_j |c_j| and L_s := ∑_k |c_k|. This ensures that ∑_i p_i = 1, thus creating convex sums of unitaries. Continuing, we get:

    e^-iHt    = L_c ∑_j p_j P'_j + I + i L_s ∑_k p_k P'_k 
        = L_c ∑_j p_j P'_j + ∑_k p_k (I + i L_s P'_k) 
        = L_c ∑_j p_j P'_j + √(1+L_s^2)∑_k p_k e^i θ P'_k.

In the final step, we use the substitution I + i L_s P'_k = √(1+L_s^2) e^i θ P'_k, where θ := (√(1+L_s^2)). This step was inspired by the excellent work of Wan et. al. <cit.>. Finally, to prove the upper bounds on the L_1 norms, notice that ∑_m=1^∞1/(2m)! (tH)^2m = cosh(tH) - 1 and ∑_m'=0^∞1/(2m'+1)! (tH)^2m'+1 = sinh(tH). Because |H^n|_1 ≤ |H|_1^n, we conclude that L_c ≤cosh(t|H|_1) - 1 and L_s ≤sinh(t|H|_1).





 §.§ Worst-Case Complexity Analysis
 
Consider a general quantum channel ℰ, with the following Kraus decomposition: 

    ℰ(ρ)=∑_i^κ K_i ρ K_i^†

Any Kraus operator K_i can be written as a linear combination of unitary matrices, K_i = ∑_j^M_i c_ij U_ij. For instance, the Pauli strings form an orthonormal basis of operators. Then, we obtain:

    ℰ(ρ)     =∑_ijk c_ij c_ik^* U_ij^ρ U_ik^†


Theorem 2: Suppose that each of the κ Kraus operators K_i decomposes into M_i unitaries U_ij, and that ∑_i^κ M_i = N. Then, the L_1 sampling norm of the coefficients in Equation <ref> is at most M^* := max({M_i}). 


Proof: First, we directly compute the L_1 norm of the coefficients in Equation <ref>:

    |ℰ|_1     = ∑_ijk |c_ij c_ik^*| 
        = ∑_i (∑_j^M_i |c_ij| ∑_k^M_i |c_ik|) 
        = ∑_i (∑_j^M_i |c_ij|)^2

Now, use the property that ℰ is trace-preserving, i.e. ∑_i K_i^† K_i = I:

    I     = ∑_ijk c_ij^* c_ik U_ij^† U_ik^
        = ∑_ij |c_ij|^2 I + ∑_ij,k ≠ j c_ij^* c_ik U_ij^† U_ik^

Thus, the second sum must be 0, but more importantly,

    ∑_ij |c_ij|^2 = 1

In other words, the L_2 norm of the set {|c_ij|} is 1. According to the Cauchy-Schwarz inequality, for complex numbers u_i and v_i,

    |∑_i=1^n u_i v_i^* |^2 ≤∑_j=1^n |u_j|^2 ∑_k=1^n |v_k|^2

Setting v_i = 1  ∀ i, we obtain the following identity:

    |∑_i=1^n u_i |^2 ≤ n ∑_j=1^n |u_j|^2

This is exactly what we need – by knowing the L_2 norm, we can set an upper limit on the L_1 norm. For our application, we set u_i = |c_ij| to obtain the following inequality, valid for all i:

    (∑_j^M_i |c_ij|)^2 ≤ M_i ∑_j^M_i |c_ij|^2

Thus,

    |ℰ|_1 ≤∑_i M_i ∑_j^M_i |c_ij|^2 ≤ M^* ∑_ij |c_ij|^2

Because the L_2 norm is 1, we get our final result:

    |ℰ|_1 ≤ M^*.




