Paper Title: Quantum advantage from measurement-induced entanglement in random shallow circuits.







^1 Department of Pure Mathematics, University of Waterloo
^2 Institute for Quantum Computing, University of Waterloo
^3 Department of Combinatorics and Optimization, University of Waterloo
^4 Perimeter Institute for Theoretical Physics, Waterloo
^5 California Institute of Technology
§ ABSTRACT 

We study random constant-depth quantum circuits in a two-dimensional architecture.
While these circuits only produce entanglement between nearby qubits on the lattice, long-range entanglement can be generated by measuring a subset of the qubits of the output state. It is conjectured that this long-range measurement-induced entanglement (MIE) proliferates when the circuit depth is at least a constant critical value d^*.
For circuits composed of Haar-random two-qubit gates, it is also believed that this coincides with a quantum advantage phase transition in the classical hardness of sampling from the output distribution.

Here we provide evidence for a quantum advantage phase transition in the setting of random Clifford circuits. 
Our work extends the scope of recent separations between the computational power of constant-depth quantum and classical circuits, demonstrating that this kind of advantage is present in canonical random circuit sampling tasks. In particular, we show that in any architecture of random shallow Clifford circuits, the presence of long-range MIE gives rise to an unconditional quantum advantage. In contrast, any depth-d 2D quantum circuit that satisfies a short-range MIE property can be classically simulated efficiently and with depth O(d). 
Finally, we introduce a two-dimensional, depth-2, “coarse-grained" circuit architecture, composed of random Clifford gates acting on O(log(n)) qubits, for which we prove the existence of long-range MIE and establish an unconditional quantum advantage.


Quantum advantage from measurement-induced entanglement 
 in random shallow circuits
    Mehdi Soleimanifar^5
    August 5, 2024
====================================================================================





§ INTRODUCTION

Identifying computational tasks where quantum computers yield an advantage compared to classical ones is a central goal of quantum information science. To make effective progress, one hopes to understand which families of quantum circuits admit efficient classical simulation algorithms, with a focus on quantum circuit architectures that are experimentally feasible in the near-term.

A key task involving quantum circuits is simulating measurement of their output state in the standard basis. In the worst case, classically sampling from this output distribution is intractable—even for the 2D brickwork architecture, and with circuit depth as small as d=3 <cit.>.
However, the classical hardness of this problem for circuit architectures composed of random local gates remains less well understood. Such random quantum circuits have diverse applications:  they underpin quantum supremacy experiments <cit.>, provide benchmarking schemes for near-term quantum devices <cit.>, and model typical quantum states arising from the dynamics or ground states of locally-interacting quantum systems <cit.>.
Random instances of quantum circuits can also help identify generic features that relate to the classical hardness of simulating quantum circuits.

Perhaps surprisingly, it has been conjectured that the complexity of sampling from the output distribution of random geometrically local quantum circuits in a two-dimensional architecture undergoes a quantum advantage phase transition: polynomial-time classical simulation is believed to be possible if and only if the circuit depth is 
below a constant critical value d^* <cit.>.
Recent works have proposed an intriguing physical origin for this computational transition, linking it to a change in the amount of measurement-induced entanglement (MIE) in random quantum circuits <cit.>.

Measurement-induced entanglement, which enables quantum protocols such as teleportation and entanglement swapping, occurs when spatially separated subsystems of a many-body state become entangled due to measurements on other subsystems. When combined with unitary dynamics, MIE gives rise to emergent phases with distinctive entanglement patterns <cit.>. 
In 2D constant-depth quantum circuits, where only nearby qubits may be entangled prior to measurements, MIE can lead to the formation of entanglement between more distant qubits. This in turn can affect the cost of classically simulating measurement of all qubits in the output state of the quantum circuit.  Based on numerical simulations and heuristic mappings to statistical models, previous works <cit.> have suggested that for circuit depth d < d^*, MIE results in short-range entanglement, making the circuit amenable to classical simulation techniques. 
However, for depths d ≥ d^*, MIE induces long-range entanglement, which poses an obstacle for the efficient classical simulation of random quantum circuits. <ref> illustrates the change in behaviour of MIE  at the conjectured critical circuit depth d^*=6 for two-dimensional random Clifford circuits. 
A similar transition in MIE has been recently observed in random 2D circuits composed of one- and two-qubit gates that form a universal gate set <cit.>.

Establishing this quantum advantage of random quantum circuits—and more broadly, a separation between efficient quantum and classical computation—has remained elusive. 
A series of recent works <cit.> have provided unconditional results addressing a more limited notion of quantum advantage. 
In these works, a computational problem is introduced that can be solved by geometrically-local constant-depth quantum circuits, but not by any classical probabilistic circuit whose depth grows sub-logarithmically with the input size. These works also provide distributions over instances such that classical shallow circuits cannot even solve an instance in the average case (see e.g., <cit.>).
Existing results along these lines make use of certain specially tailored families of shallow quantum circuits. This raises the question of whether generic shallow quantum circuits over a fixed architecture demonstrate a similar quantum advantage. 

In this paper, we answer this question in the affirmative. We describe a family of random quantum circuits whose measurement distributions cannot be simulated with shallow classical circuits. The key to our findings is establishing a connection between the emergence of long-range MIE at the critical depth d^* and the limitations of low-depth classical simulation methods. 

Let us now describe our results in more detail.





§ OVERVIEW OF RESULTS



 §.§ Measurement-induced entanglement

Consider n qubits at the vertices of a 2D square lattice. 
These qubits are partitioned into three sets A, B, and C where A is a single qubit and B is a square region that shields A from C, see <ref>.
All qubits are initialized in the |0⟩ state. We apply a unitary U to A∪ B∪ C and then measure qubits in B in the computational basis. For the moment we consider the case where U is a random depth-d circuit in the brickwork architecture, see <ref>.  We study the setting where the side length of B is much larger than 2d. By a lightcone argument, this ensures that qubit A is unentangled from qubits in C in the output state of the quantum circuit, and only may become entangled as a result of the measurement of qubits in B. 

In this setting, one can ask: What is the shallowest depth d^* at which there is measurement-induced entanglement between A and C?

For simplicity we can imagine that we have already taken the  limit of an infinite lattice n→∞. Then for depth d<d^* it is conjectured that  the expected purity 𝔼[(ρ^2_A)] of qubit A (averaged over the random choice of circuit and measurement outcomes on B) approaches 1 exponentially fast as |B| grows, while for d≥ d^⋆ this expected purity is at most 1-γ(d) where γ(d)>0 is a constant that may depend on depth d, but is independent of the size of the shielding region B <cit.>.

This scenario motivates the following definitions of short-range and long-range MIE for output states of 2D random quantum circuits. In the following definitions, a 2D, n-qubit, random quantum circuit is described by a probability distribution over quantum circuits that act on n qubits arranged at the vertices of a two-dimensional square grid. A family of 2D random quantum circuits is described by a sequence of such probability distributions, with an increasing number of qubits n_1,n_2,n_3,…. Note that here we do not restrict the gates of the circuit to act on nearest-neighbor qubits.

[Short-range MIE]
A family ℱ of random 2D quantum circuits satisfies the short-range MIE property if the following holds. Suppose ψ is a random pure state on a 2D lattice of n qubits that is generated according to ℱ. We measure all qubits of ψ in an L× L shielding region B centred at qubit A∈ [n], obtaining outcome x∈{0,1}^|B|. Let ρ_A=ρ_A(x) be the postmeasurement state of qubit A. Then

    𝔼[Trρ_A^2]≥ 1-e^-Ω(L).

Here the expectation is taken over the randomness in generating ψ as well as the distribution of random measurement outcomes on qubits in B.






[Long-range MIE] A family ℱ of random 2D quantum circuits satisfies the long-range MIE property if there is a constant c∈ (0,1) such that the following holds. Let ψ be a random pure state on a 2D lattice of n qubits that is generated according to ℱ.  We measure all qubits of ψ in an L× L shielding region B centred at qubit A∈ [n], obtaining outcome x∈{0,1}^|B|. Let ρ_A=ρ_A(x) be the postmeasurement state of qubit A. Here L is any positive integer such that |C|≠ 0. Then

    𝔼[Trρ_A^2]< c,

where the expectation is taken over the randomness in generating ψ as well as the distribution of random measurement outcomes on qubits in B.



As noted above, we expect 2D brickwork random circuits to have the short-range MIE property for depth d<d^* and to satisfy the long-range MIE property for d≥ d^*. In fact, in the latter case we expect that measurement-induced entanglement is “everywhere"—that is, present with respect to more general tripartitions [n]=ABC of the qubits (not just the case discussed above where B is a square shielding region centred at A). We also expect that this proliferation of MIE gives rise to genuine multipartite entanglement. We shall use the following notion of long-range tripartite MIE, which is phrased in more general terms and can be applied to random circuits in other architectures. 

[Long-range tripartite MIE]
A family ℱ of random quantum circuits satisfies the long-range tripartite MIE property if there are constants c_1,c_2∈ (0,1) such that the following holds. Suppose ψ is a random n-qubit pure state that is generated according to ℱ. Let Ω={h,i,j}⊆ [n] be a uniformly random triple of qubits. Let ρ_Ω=ρ_Ω(x) be the state of qubits in Ω after measuring all other qubits of ψ in the computational basis and obtaining outcome x∈{0,1}^n-3. Let ρ_h, ρ_i, ρ_j be its one-qubit reduced density matrices. Then, with probability at least c_1>0 we have

    Tr[ρ_h^2]<c_2   and  Tr[ρ_i^2]<c_2   and  Tr[ρ_j^2]<c_2.

Here the probability is over the random choice of ψ, the choice of triple {h,i,j}, and the random measurement outcomes on qubits in [n]∖{h,i,j}.



Rigorously establishing the conjectures described above, and the existence of a phase transition in measurement-induced entanglement for 2D brickwork circuits at a constant depth d^*, is a challenging open problem. Part of the challenge is that the shallow circuit depth does not allow the quantum system to fully randomize in most senses of interest.  On the other hand, for larger circuit depths d growing with n, random circuits can be easier to analyze as they begin to inherit properties of the Haar-random unitary ensemble, such as the unitary t-design property and anticoncentration.

Indeed, we show that long-range MIE exists whenever there is  subsystem anticoncentration–a feature of random quantum circuits which is expected to hold for depth Ω(log(n)) <cit.>. Let P_AB(0)=⟨ 0^ABC|U^† |0⟩⟨ 0|_AB U|0^ABC⟩ be the probability of measuring all-zeros on qubits in AB. Define a measure of subsystem anticoncentration

    χ≡ 4^|AB|𝔼_U[P_AB(0)^2]-1.

Here χ depends on the tripartition ABC as well as the random circuit ensemble.  Note that if U is a Haar-random n-qubit unitary then 

    χ_Haar=2^|AB|-1/2^|ABC|+1

which is exponentially small in |C|. We say that the subsystem AB anticoncentrates with respect to a random circuit ensemble if χ→ 0 as |C| grows.

The following result shows that subsystem anticoncentration implies long-range MIE. 


For any tripartition of the qubits ABC where A is a single qubit, we have

    𝔼[Trρ_A^2]≤1/2+3/2√(χ).



We will use this bound to establish long-range MIE in a certain “coarse-grained" circuit architecture which is introduced below; this describes depth-2 circuits with gates acting on O(log(n)) qubits at a time. However, we expect long-range MIE to be present even in constant-depth 2D random quantum circuits composed of two-local gates and with depth above the critical value. Such circuits do not satisfy the subsystem anticoncentration condition considered above. Indeed, for |C|<n, the χ value for depth-d random quantum circuits composed of two-local gates over an arbitrary architecture admits the following lower bound

    χ   =∑_s∈{0,1}^|AB|𝔼_U[⟨0^n|U^† (Z(s)_AB⊗ I_C)|0^n⟩^2]-1
       ≥𝔼_U[⟨0^n|U^† Z_A U|0^n⟩^2]≥1/3(2/5)^d

where the last inequality follows from Lemma 2.12 in Ref. <cit.>; see also <cit.>. Thus, χ≥Ω(1) for d=O(1), demonstrating that a different proof technique is needed to establish long-range MIE in this setting.

Next we describe how MIE is related to classical simulation and quantum advantage.



 §.§ Shallow and efficient classical simulation



We obtain a simple, efficient, and shallow classical simulation of 2D quantum circuits 
under the condition that they only generate short-range MIE. We expect that this condition is satisfied by 2D brickwork random circuits in  the low-depth regime d < d^* where MIE is expected to be short-range. 

Our classical simulation algorithm is a modified version of the recently introduced “gate-by-gate" method <cit.>.

    Consider a depth-d quantum circuit U=U_tU_t-1… U_1 acting on n qubits arranged at the vertices of a 2D grid. Suppose that each gate in the circuit is either a CNOT gate between nearest-neighbor qubits on the grid, or a single-qubit gate. Suppose that the output state of any subcircuit of U satisfies the short-range MIE condition <ref>. Then there is an efficient algorithm which samples from the output distribution of U. Moreover, this probabilistic classical algorithm can be parallelized to depth O(d) using gates that act on O(log^2(n)) bits at a time. 



The classical circuit in the above can be parallelized to depth O(d) even in a “programmable" setting where it takes as input the specification of the individual gates U_1,U_2,…, U_t in the circuit and outputs a binary string approximately sampled from the output distribution. 

Our algorithm complements an existing technique from Ref. <cit.> for classically simulating low-depth 2D circuits which is efficient under a different criterion for short-range MIE. The `patching algorithm' described in  <cit.> proceeds by first sampling from small disjoint subregions of the 2D lattice and then patching them together to obtain an overall sample from the output distribution. 
This algorithm succeeds when the short-range MIE property holds for subregions A of size O(log^2(n)) and when X = A ∪ B ∪ C corresponds to a subregion X ⊂ [n] obtained by tracing out disconnected squares of side O(log(n)) from the lattice.  
In contrast, our simulation algorithm for the quantum circuit U in <ref> relies on any subcircuit of U exhibiting short-range MIE when A is a single qubit and  A ∪ B ∪ C = [n] is a tripartition of the entire lattice. 






 §.§ Quantum advantage with random shallow Clifford circuits

Next we specialize to shallow Clifford circuits. These circuits are capable of outperforming classical constant-depth circuits at certain tasks—the kind of quantum advantage described in Refs. <cit.>. More precisely, these works use classically controlled (or “programmable") Clifford circuits: the quantum circuit takes as input a binary string x and then samples from the output distribution of a Clifford circuit C_x that depends on the input. 

To study quantum advantage in random Clifford circuits we view them as programmable Clifford circuits with random inputs. That is, for any depth-d circuit architecture with gates acting on k-qubits at a time, we consider a controlled Clifford circuit that takes as input a specification of the individual gates in the circuit. Note that a k-qubit Clifford gate can be specified by O(k^2) bits. So the programmable Clifford circuit is also depth-d but has gates that act on O(k^2) qubits at a time.

Can this programmable Clifford circuit, with random input, be simulated by a shallow classical circuit?  For a given choice of gates (input) the classical simulator succeeds if its output distribution is close in total variation distance to the correct output distribution. We say that a classical probabilistic circuit simulates random Clifford circuits in this architecture if it succeeds with high probability over the choice of random gates. 


The following result shows that shallow random Clifford circuit families with k=O(1)-qubit gates that generate long-range tripartite MIE cannot be simulated by constant-depth classical circuits using gates that act on K=O(1) bits at a time.


 Consider a depth-d circuit architecture with random Clifford gates. Suppose that the family of random Clifford circuits in this architecture satisfies the long-range tripartite MIE property (<ref>). Suppose n is sufficiently large. Then a probabilistic classical circuit composed of gates with fan-in K and circuit depth D which simulates n-qubit random Clifford circuits in this architecture satisfies D ≥1/4log(n)/log(K).


Since it is conjectured that shallow Clifford circuits in a 2D brickwork architecture with depth d≥ d^* satisfy the long-range tripartite MIE property, this result provides evidence that such random shallow Clifford circuits attain a quantum advantage over classical shallow circuits.

The proof of <ref> extends the classical lower bound from Ref. <cit.> which can be viewed as exploiting the fact that tripartite MIE in 2D grid graph states is hard for a classical shallow circuit to simulate. While that proof made use of the structure of 2D grid graph states, here we show that long-range tripartite MIE is all that is needed for quantum advantage.

Note that <ref> applies to more general circuit architectures beyond the 2D brickwork architecture. Can we rigorously establish long-range MIE, long-range tripartite MIE,  and quantum advantage in some architecture of random 2D circuits?


To this end, we introduce a “coarse-grained” architecture  shown in <ref>. The qubits are arranged on a √(n)×√(n) two-dimensional lattice and the circuit applies only two layers of gates.
Each gate acts on a square τ×τ subgrid of qubits. Note that here the depth of the quantum circuit is 2 and τ is a parameter that describes the locality of gates. This parameter plays the same role as circuit depth in the 2D brickwork architecture—it measures the linear size of the lightcone of a single qubit [In particular, if U is a coarse-grained circuit and O_j is an operator acting on qubit j, then the support of UO_jU^† is contained within a square region of size O(τ^2) centered at j. ].

  
Random coarse-grained Clifford circuits with τ = O(√(log(n))) and with n sufficiently large, satisfy both the long-range MIE property, and the long-range tripartite MIE property.



To prove <ref> we first show that the coarse-grained architecture with the stated choice of τ satisfies a strong form of subsystem anticoncentration in which the parameter χ from <ref> decays exponentially with the size of C. This subsystem anticoncentration is established using a (first moment) probabilistic method. We then show that for Clifford circuits this strong form of subsystem anticoncentration is enough to guarantee both long-range and long-range tripartite MIE.

As a direct corollary of <ref> we obtain an unconditional quantum advantage with random Clifford circuits in the coarse-grained architecture:

Random coarse-grained Clifford circuits with τ = O(√(log(n))) and with n sufficiently large, cannot be simulated by any probabilistic classical circuit of depth D < 1/4log(n)/log(K) and fan-in K.



The family of random coarse-grained Clifford circuits with τ = O(√(log(n))) can be implemented (in a programmable sense) via a depth-2 quantum circuit with gates acting on k=O(log(n)^2) qubits. In contrast, <ref> states that any classical probabilistic circuit composed of gates with fan-in K=polylog(n) that simulates the random circuit family must have depth D>Ω(log(n)/loglog(n)).



 §.§ Discussion

We have shown that an unconditional quantum advantage with random shallow Clifford circuits follows from the presence of long-range tripartite MIE. We also exhibited a random circuit architecture with this property.

Moreover, <ref> provide evidence that 2D random shallow Clifford circuits in the standard brickwork architecture undergo a quantum advantage phase transition  that coincides with the emergence of long-range (and long-range tripartite) MIE at the critical depth d^*=6. This provides a scaled-down Clifford counterpart to the conjectured phase transition in the classical hardness of simulating random circuits composed of two-qubit Haar-random gates in the same architecture. 

There are several open questions related to this work. Can we prove the existence of a phase transition in MIE at a constant critical circuit depth d^* for 2D brickwork random shallow circuits? Can we show that the quantum advantage with shallow circuits follows from the long-range MIE property (rather than the tripartite version)? Can we establish a quantum advantage in circuit depth for random circuits composed of Haar-random gates (instead of Clifford gates)? 

In the remaining sections of the paper we prove Theorems 1-4, respectively.





§ SUBSYSTEM ANTICONCENTRATION IMPLIES LONG-RANGE MIE


Here we show that random circuits that anticoncentrate also exhibit long-range measurement-induced entanglement. In particular, we prove <ref>.  

Suppose U=U_mU_m-1… U_1 is an n-qubit quantum circuit where each gate U_j is a Haar random k-qubit gate acting on the qubits in its support.
Here our results apply to general quantum circuits—i.e.,  we do not specify an architecture or require the circuit to act on a system of qubits in a 2D geometry. In the following we shall relate properties of U to those of a random Clifford circuit D=D_mD_m-1… D_1 where each D_i is a random Clifford unitary that acts on the qubits in the support of U_i. 

For any s∈{0,1}^n, let Z(s)=∏_j=1^n Z_j^s_j. Here Z_j is the Pauli Z operator acting on the jth qubit. Let [n]=ABC be a tripartition of the qubits such that |A|=1. For the output state of the Clifford circuit D, the presence or absence of post-measurement entanglement between A and C after measuring B is determined by a set S defined as follows (this set indexes “entanglement-killing" stabilizers of the state D|0^n⟩). For P=X,Y,Z, let

    S_P≡   {s∈{0,1}^n: DZ(s)D^†=(-1)^α P_A⊗ Z(r)_B⊗  I_C 
        for some  r∈{0,1}^|B|, α∈{0,1}}

and let 

    S=S_X∪ S_Y∪ S_Z.
 



Suppose |ϕ⟩=D|0^n⟩ where D is a Clifford unitary. Let [n]=ABC be a tripartition of the qubits where |A|=1. There is post-measurement entanglement between A and C after measuring all qubits of B in the state ϕ, if and only if S=∅. 


Let G={DZ(s)D^†: s∈{0,1}^n} be the stabilizer group of ϕ.

First suppose S is nonempty. Then (-1)^α P_a⊗ Z(r)_B⊗ I_C is a stabilizer of ϕ, for some P∈{X,Y,Z}, r∈{0,1}^n, α∈{0,1}. After measuring all qubits in B in the computational basis, the postmeasurement state is stabilized by ± P_A for some choice of sign ± 1 determined by the measurement outcomes. In other words, the postmeasurement state is a pure 1-qubit stabilizer state.

Next suppose we measure all qubits in B , obtain measurement outcome x∈{0,1}^|B|, and the postmeasurement state ρ_A on qubit A is pure. The stabilizer group of this state is generated by

    G'={(-1)^x_j Z_j: j∈ B}∪ G”

where

    G”={P∈ G: P  is Z-type on B}.

(We say that an n-qubit Pauli operator is Z-type if it is a tensor product of I and Z Pauli operators.) Since ρ_A is pure, it must have a stabilizer ± P_A⊗ I_BC for some P∈{X,Y,Z}. This stabilizer can be expressed as a product of elements from the set of generators above. That is, for some T⊆ G” and R⊆ B we have

    P_A=∏_Q∈ TQ ∏_j∈ R(-1)^x_jZ_j.

From the above we conclude that 

    (-1)^∑_j∈ R x_j P_A⊗ Z(R)_B⊗ I_C=∏_Q∈ TQ =DZ(s)D^†

for some binary string s∈{0,1}^n. Therefore s∈ S and S≠∅.



<ref> shows that emptiness of S determines postmeasurement entanglement for Clifford circuits. We now show how this criterion can be extended to our random non-Clifford circuit U of interest.
 
Define Π(x)=|x⟩⟨ x|_B⊗ I_AC for x∈{0,1}^|B|. If we measure all qubits in B, we obtain x with probability

    q(x)≡⟨ 0^n|U^†Π(x) U|0^n⟩.

The postmeasurement state on A is then

    ρ(x)=1/q(x)Tr_BC(Π(x) U|0^n⟩⟨ 0^n|U^†Π(x)).


The expected purity of the postmeasurement state on A is upper bounded as

    𝔼_U 𝔼_x∼ qTr(ρ(x)^2)≤1/2+1/2√(3𝔼_D[|S|]).



Note that the LHS describes the purity of the postmeasurement state in the universal circuit U, while the RHS only involves an average over the Clifford circuit D. Moreover, this average is upper bounded as

    1/3𝔼_D[|S|]=𝔼_D[|S_Z|]
       ≤𝔼_D[∑_s∈{0,1}^AB, s≠ 0^|AB|⟨ 0^n|D^† Z(s)_AB⊗ I_C D|0^n⟩^2]
       =4^|AB|𝔼_D[|⟨ 0^n|D^†(|0⟩⟨ 0|_AB⊗ I_C) D |0^n⟩|^2]-1
       =4^|AB|𝔼_U[|⟨ 0^n|U^†(|0⟩⟨ 0|_AB⊗ I_C) U |0^n⟩|^2]-1
 
where in the last line we used the two-design property of the Clifford group. Here <ref> is the quantity denoted χ in the Introduction, which measures anticoncentration of subsystem AB.


Write

    α_P(x)≡1/2⟨ 0^n|U^†Π(x)·(P_A⊗ I_BC) U|0^n⟩,

so that

    ρ(x)=1/2α_I(x)∑_P∈{I,X,Y,Z}α_P(x) P.

Using the fact that ∑_xΠ(x)⊗Π(x)=2^-|B|∑_s∈{0,1}^|B| Z(s)⊗ Z(s) we get

    ∑_x∈{0,1}^|B| |α_P(x)|^2
       =1/4· 2^|B|∑_s∈{0,1}^|B| |⟨ 0^n|U^† (P_A⊗ Z(s)_B) U|0^n⟩|^2.

Then

    𝔼_U 𝔼_x∼ q(|α_P(x)|/2α_I(x))
    =2^|B|𝔼_U 𝔼_x∼unif |α_P(x)|
       ≤ 2^|B|(𝔼_U 𝔼_x∼unif |α_P(x)|^2)^1/2
       =(1/4∑_s∈{0,1}^|B|𝔼_U  |⟨ 0^n|U^† (P_A⊗ Z(s)_B) U|0^n⟩|^2)^1/2.

Now we use the fact that every gate in U is drawn from the Haar measure, and the fact that the Clifford group is a two-design. This allows us to replace the average over U with the average over the random Clifford circuit D in the above. Then for P∈{X,Y,Z} we have

    𝔼_U 𝔼_x∼ q(|α_P(x)|/2α_I(x))
       ≤(1/4∑_s∈{0,1}^|B|𝔼_D  |⟨ 0^n|D^† (P_A⊗ Z(s)_B) D|0^n⟩|^2)^1/2
       =1/2√(𝔼_D |S_P|)
       =1/2√(3)√(𝔼_D |S|).

Now the expected purity of the postmeasurement state is upper bounded as

    𝔼_U 𝔼_x∼ qTr(ρ(x)^2)   =𝔼_U 𝔼_x∼ q∑_P∈{I,X,Y,Z}2|α_P(x)|^2/4|α_I(x)|^2
       ≤𝔼_U 𝔼_x∼ q∑_P∈{I,X,Y,Z}|α_P(x)|/2|α_I(x)|
       =1/2+3𝔼_U 𝔼_x∼ q|α_Z(x)|/2|α_I(x)|

where we used the fact that |α_P(x)|≤α_I(x) for all x∈{0,1}^|B|. Plugging <ref> into the above completes the proof.





§ EFFICIENT AND SHALLOW CLASSICAL SIMULATION OF 2D CIRCUITS WITH SHORT-RANGE MIE


In this section we prove <ref>. In particular, we consider the output state |ψ⟩=U|0^n⟩ of a 2D, depth d, quantum circuit and describe an efficient classical simulation method that approximately samples from its output distribution, assuming that the state after each gate in the circuit has short-range MIE. We will also show that the algorithm can be parallelized to depth O(d). The simulation algorithm is a variant of the gate-by-gate method proposed in Ref. <cit.>.

Here we assume the circuit is expressed as U=U_mU_m-1… U_1|0^n⟩ where each U_t is either a CNOT gate or a single-qubit gate. 

In addition to the output state ψ and its output distribution P(z)=|⟨ z|ψ⟩|^2, it will also be convenient to consider states and output distributions associated with subcircuits. That is, define

    |ψ_t⟩=U_tU_t-1… U_1|0^n⟩  and   P_t(z)=|⟨ z|ψ_t⟩|^2,

for 0≤ t≤ m and z∈{0,1}^n. 

Now imagine that we measure all qubits in BC of the state ψ_t and obtain outcomes ω∈{0,1}^|BC|. The probability of this measurement outcome is P_t,BC(ω)≡∑_y_BC=ω_BC P_t(y). The postmeasurement state on A is

    ρ_t(ω)=1/P_t,BC(ω)⟨ω|_BC|ψ_t⟩⟨ψ _t|ω⟩_BC.


If instead we only measure the qubits in B of the state ψ_t, obtaining outcome ω_B, then the postmeasurement state of A is


    ρ̃_t(ω_B)≡1/P_t,B(ω_B)⟨ω_B |Tr_C(|ψ_t⟩⟨ψ_t|)| ω_B⟩

where P_t,B(ω_B)=∑_y: y_B=ω_B P_t(y). The following claim shows that the expected fidelity between ρ_t(ω) and ρ̃_t(ω_B) is equal to the average purity of the latter state.


    𝔼_z∼ P_t,BC[ F(ρ_t(z), ρ̃_t(z_B))]=𝔼_z_B∼ P_t,B[Tr( ρ̃_t(z_B)^2)]




Using the definitions (<ref>) we have

    𝔼_z∼ P_t,BC[ F(ρ_t(z), ρ̃_t(z_B))]
       =∑_z∈{0,1}^|BC|⟨ψ_t| (| z_B⟩⟨ z_B|⊗ |z_C⟩⟨ z_C| )ρ̃_t(z_B)|ψ_t⟩
       =∑_z_B∈{0,1}^|B|⟨ψ_t | (|z_B⟩⟨ z_B|⊗ I_C) ρ̃_t(z_B)|ψ_t⟩
       =∑_z_B∈{0,1}^|B|P_t,B(z_B)Tr( ρ̃_t(z_B)^2).


Define conditional probability distributions

    q_t(x_A|z_Bz_C)≡⟨ x_A|ρ_t(z_Bz_C)| x_A⟩

and

    q̃_t(x_A|z_B)≡⟨ x_A|ρ̃_t(z_B)| x_A⟩.


In the above discussion we fixed a partition [n]=ABC but in our classical simulation algorithm—<ref>—we allow this partition to depend on t in a simple way. In particular, let Γ⊆ [m] be such that U_t is a single-qubit gate if and only if t∈Γ (it is a CNOT otherwise).

 Then for t∈Γ we let A(t) be the qubit on which U_t acts, B(t) be all qubits within a square subgrid centred at A(t) with side length L, and C(t)=[n]∖(A(t)∪ B(t)). We write Square_L(A) for the set of all qubits that are a horizontal or vertical distance at most L/2 from qubit A, so that B(t)=Square_L(A(t)).

Let P̃ be the output distribution of <ref>. Recall that Γ⊆ [m] are the indices of one qubit gates in the circuit, and P_t(x)=|⟨ x|U_tU_t-1… U_1|ψ⟩|^2. Then

    P̃-P_m_1≤ 2∑_t∈Γ(1-_z∼ P_t,B(t)[Tr(ρ̃_t(z)^2)])^1/2.




The proof of <ref> is given in the Appendix. It is obtained by combining <ref> with a robustness property of the gate-by-gate algorithm established in Ref.<cit.>, which we adapt to our setting.


The left-hand side of <ref> is the approximation error of our algorithm: the total variation distance between the distribution sampled by our classical simulator, and the true output distribution of the quantum circuit.
We are successful in classically simulating the quantum circuit if we can make the right-hand-side of <ref> small— at most ϵ=O(n^-c) for a given constant c≥ 1. Let us assume that MIE in the circuit is short-ranged in the sense that, at each time 1≤ t≤ m in the circuit, the output state |ψ_t⟩ has the short-range MIE property (<ref>). In particular, for each t∈Γ this gives

    _z∼ P_t,B(t)[Tr(ρ̃_t(z)^2)]=1-e^-Ω(L).

Plugging this into <ref> we see that the RHS is O(me^-L). For polynomial-sized circuits (i.e., m=O(poly(n))) we can make this approximation error at most ϵ=O(n^-c) by choosing L=O(log(n)). 

Let us now show that <ref> has O(poly(n)) runtime with this choice of L.  The runtime of <ref> is O(mT(L)) where T(L) is the cost of classically computing the conditional probability q̃_t(y|x_B(t)) in line 9 of the algorithm. Using (<ref>) we see that this conditional probability is a ratio of the two marginal probabilities 

    ⟨ y_A x_B|Tr_C(|ψ_t⟩⟨ψ|_t)| y_A x_B⟩

and P_t,B(x_B). Since we are tracing out subsystem C, each of these marginals can be computed by removing all gates of U_tU_t-1… U_1 that lie outside of the so-called lightcone of AB. This gives a depth d circuit V acting on qubits of a subgrid with side length L+2d. Each of these marginal probabilities can be computed using a classical tensor network algorithm from Ref. <cit.> (see Thm 4.3 of that work), which uses runtime T(L)=2^O(d(L+d)). Thus, for constant-depth d=O(1) circuits with m=poly(n) gates and with the choice L=O(log(n)), the runtime of <ref> is poly(n).

Finally, we observe that <ref>, when applied to simulate a quantum circuit of depth d,  can be parallelized to depth O(d) if we allow classical gates that act on K=O(L^2) bits. To see this, let us describe how to parallelize all updates to the bit string x in step 2. of the algorithm for a single layer (depth-1 circuit) of quantum gates. To this end note that all updates of x corresponding to gates whose support is within a square region of side length L only depend on bits of x in a larger square region of side length ∼ 3L. These updates can therefore be performed by a single classical “gate" that acts on bits in the larger square region of side length 3L. By partitioning the grid into square regions with side length 3L we can update 1/9 of the
bits of x using a single layer of such gates in parallel. We can then simulate the entire depth-1 circuit using a classical circuit of depth 9.  






§ LONG-RANGE TRIPARTITE MIE IMPLIES UNCONDITIONAL QUANTUM ADVANTAGE IN ANY ARCHITECTURE


In this section we prove <ref>. We begin by describing what we mean by a circuit architecture and the associated family of random Clifford circuits. Then we consider the long-range tripartite MIE property (<ref>) and specialize it to Clifford circuits, using the fact that a three qubit stabilizer state that is entangled with respect to any bipartition of the qubits (Cf. <ref>), is locally Clifford equivalent to the GHZ state. Finally, we show random Clifford circuits which satisfy the long-range tripartite MIE property cannot be simulated by shallow probabilistic classical circuits. 

Our proof strategy for lower bounding classical circuits which is based on identifying GHZ-type measurement-induced entanglement is similar to the proof strategy from Ref. <cit.>. The difference is that here we aim to make this strategy work with Clifford circuits composed of uniformly random gates, and we find that long-range tripartite MIE is all that we require for quantum advantage. 




 §.§ Random Clifford circuits


An n-qubit circuit template 𝒯 is a tuple of subsets (m_1,m_2,…, m_R) such that m_i⊆ [n] for 1≤ i≤ R. A quantum circuit in this template is a product U_RU_R-1… U_2U_1 of gates such that U_i acts nontrivially only on qubits in m_i. The template is said to be k-local if |m_i|=k for all 1≤ i≤ R, and it is said to have depth d if any quantum circuit in this template can be implemented with circuit depth d.


A k-local, depth-d circuit architecture 𝒜={𝒯_n}_n∈Γ is a sequence of k-local, depth-d circuit templates with an increasing number of qubits Γ={n_1,n_2,n_3…}.


Any template 𝒯=(m_1,m_2,…, m_R) defines a random Clifford circuit, obtained by choosing the gates U_1,U_2,…, U_R to be uniformly random Clifford gates acting on their support. Similarly, an architecture 𝒜 is associated with a famiy of random Clifford circuits.


We will be particularly interested in random Clifford circuits which produce long-range tripartite entanglement in the sense of <ref>. We can specialize this property to Clifford circuits as follows.

We say that a three-qubit stabilizer state |Φ⟩ is GHZ-type entangled iff it is locally Clifford equivalent to the GHZ state. That is, there exist single-qubit Clifford unitaries C_1,C_2,C_3 such that 

    C_1⊗ C_2⊗ C_3|Φ⟩= 1/√(2)(|000⟩+|111⟩).

It is a well-known fact that any three qubit stabilizer state which is not a product state with respect to any bipartition of the qubits, is GHZ-type entangled. 

If ϕ is an n-qubit stabilizer state, we say that three qubits {h,i,j}⊆ [n] exhibit GHZ-type MIE iff they share GHZ-type entanglement after measuring all qubits in [n]∖{h,i,j}.
[Long-range tripartite MIE, Clifford version]
The architecture 𝒜={𝒯_n}_n∈Γ is said to satisfy the long-range tripartite MIE property if there is an absolute constant c>0 such that the following holds. Suppose n∈Γ and consider the circuit template 𝒯_n∈𝒜. Suppose U is a random Clifford circuit with template 𝒯_n, i.e. it is composed of gates U_1,U_2,…, U_R chosen uniformly at random from the set of k-qubit Clifford gates. Let |ψ⟩=U|0^n⟩.  Choose a uniformly random triple of qubits {h,i,j}⊆ [n]. Then, with probability at least c>0 (over the random choice of U and the triple h,i,j), the qubits h,i,j exhibit GHZ-type MIE in the state ψ.



It is not hard to see that <ref> is equivalent to <ref> for the random Clifford circuit family defined by an architecture 𝒜. The condition <ref> in the definition of long-range tripartite MIE property implies that the three-qubit state ρ_Ω is not a product state with respect to any bipartition of its qubits. Since in the case at hand it is also a stabilizer state, it is GHZ-type entangled. 



 §.§ Requirements for classical simulation of GHZ-type MIE




Let us begin by considering what kinds of classical circuits are capable of reproducing the measurement statistics of quantum states with GHZ-type MIE. 

Suppose U is an n-qubit Clifford circuit and |ψ⟩=U|0^n⟩.  Recall that we say that three qubits {h,i,j}⊆ [n] exhibit GHZ-type measurement-induced entanglement iff the state ρ_h,i,j after measuring all other qubits in the computational basis is locally Clifford equivalent to the GHZ state. We can rephrase this definition in terms of the stabilizer group of ψ, as follows: three qubits {h,i,j} of ψ   exhibit GHZ-type measurement-induced entanglement iff,  for some tensor product C_h⊗ C_i⊗ C_j of single-qubit Clifford unitaries on h,i,j, the stabilizer group of C_h⊗ C_i⊗ C_j|ψ⟩ contains elements

    g_1   =X_hX_iX_jZ(s)
    
    g_2   =Z_hZ_i I_jZ(t) 
    
    g_3   =I_h Z_iZ_jZ(u),

for some binary strings s,t,u∈{0,1}^n such that s_r=t_r=u_r=0 for r∈{h,i,j}. 


In the following Lemma we consider a classical function m:{0,1,2}^3→{0,1}^n which takes input b∈{0,1,2}^3 describing measurement bases X,Y,Z for qubits h,i,j respectively, and produces output m(b)∈{0,1}^n. We are interested in whether this function can simulate measurement of ψ in the given bases b on qubits h,i,j and the computational basis on all other qubits. Let Q(0),Q(1),Q(2) denote the single-qubit Clifford unitaries that change basis from the Z basis to the X,Y,Z bases respectively (i.e., Q(0)=(X+Z)/√(2), Q(1)=(Y+Z)/√(2), Q(2)=I). Let W(b)=Q(b_h)⊗ Q(b_i) ⊗ Q(b_j) be the tensor product of single-qubit unitaries that change basis to the one described by b. A necessary condition for such a simulation of ψ is that

    ⟨ m(b)|W(b)|ψ⟩≠ 0,    b∈{0,1,2}^3.

We show a condition under which <ref> is not possible.

Suppose h,i,j exhibit GHZ-type measurement-induced entanglement in the state |ψ⟩=U|0^n⟩. Let m:{0,1,2}^3→{0,1}^n and write m=m(b) with b=b_hb_ib_j∈{0,1,2}^3. Suppose that each output bit m_r depends on at most one of the input variables b_h,b_i,b_j, and that 


  * m_h is independent of b_i, b_j

  * m_i is independent of b_h, b_j

  * m_j is independent of b_h, b_i.

Then 

    Pr_b∼unif({0,1,2}^3)[ ⟨ m(b)|W(b)|ψ⟩= 0]≥ 1/27.



The proof of <ref> is given in the Appendix.
It says that classical circuits that are capable of simulating GHZ-type measurement induced entanglement must have certain kind of input-output correlations. In the next section we will see that this constraint is even more restrictive when ψ is an n-qubit state and many triples of qubits exhibit GHZ-type MIE.



 §.§ Lower bound for classical probabilistic simulation


In the following we shall view a random Clifford circuit as a fixed circuit with a random input that specifies the gates. That is, we construct a controlled Clifford circuit which takes the specification of all Clifford gates U_1,U_2,…, U_R as an input and then applies the corresponding Clifford circuit to an n-qubit register.

For any  k-local, n-qubit circuit template 𝒯_n=(m_1,m_2,…, m_R) we define a controlled-Clifford circuit as follows. The circuit has a data register of n qubits which are initialized in the |0^n⟩ state. In addition, there is one input register I_j of size O(k^2) for each gate j∈ R in the circuit, that contains enough qubits to store the description of an arbitrary k-qubit Clifford gate. The jth gate in the circuit applies Clifford unitary C on qubits m_j of the data register, controlled on the state |C⟩ of the input register I_j. An architecture 𝒜 defines a family of controlled Clifford circuits in this way, which we denote CC(𝒜).


Suppose 𝒜={𝒯_n}_n∈Γ is a depth-d, k-local architecture. As noted above we can construct an associated family CC(𝒜) of depth-d quantum circuits composed of O(k^2)-local controlled-Clifford gates.  Under what conditions can the input/output behaviour of this family of quantum circuits, with random inputs,  be simulated by a constant-depth classical circuit?  


Consider a family of probabilistic classical circuits which implement functions {F_n}_n∈Γ with the following input/output properties:


  * For each n∈Γ, F_n takes as input a list U_1,U_2,…, U_R of k-local Clifford gates that define a quantum circuit U=U_R… U_2U_1 of depth d in the template 𝒯_n.

  * F_n also takes as input a random string r∈{0,1}^ℓ(n) drawn from some probability distribution q_n. Here ℓ(n) can be any function of n and q_n can be any probability distribution over ℓ(n) bit strings.

  * The function outputs a binary string

    z=F_n(U_1,U_2,…, U_R,r)∈{0,1}^n.

For ease of notation, we sometimes write z=F_n(U,r).


We shall be interested in the case where each function F_n is implemented by a classical circuit with gates of fan-in K and circuit depth D. In this case each output bit z_j of F_n(U,r) depends on at most K^D input bits. In particular this implies that each output bit z_j depends on at most K^D of the gates in the circuit (each gate is specified by O(k^2) bits).

If we draw r∼ q_n and compute z=F_n(U,r) then z∈{0,1}^n samples from a distribution that we denote 

    P_U(z)      z∈{0,1}^n.

Let us also write Q_U(z)=|⟨ z|U|0^n⟩|^2 for the true output distribution. 



Suppose 𝒜 is a k-local, depth-d circuit architecture with the long-range tripartite MIE property (<ref>). Suppose n∈Γ satisfies n≥ (10d)^4 and let F_n be a function with input/outputs as described above. Suppose further that each output bit of z=F_n(U_1,U_2,…, U_R,r) depends on at most n^1/4 of the k-qubit Clifford gates U_1,U_2,…, U_R. Then for any r∈{0,1}^ℓ(n), we have

    Pr_U_1,U_2,…, U_R ∼𝒟_k[ ⟨ F_n(U,r)|U|0^n⟩=0]≥c/27-O(n^-1/4),

where c>0 is the absolute constant from <ref> and 𝒟_k is the uniform distribution on k-qubit Clifford unitaries.




In the following we fix r∈{0,1}^ℓ(n) and write z=F_n(U)=F_n(U,r).

Let U_1,U_2,…, U_R be the k-qubit gates that describe U. Here R≤ nd/k. These are arranged in d layers. For each t∈ [n] there is exactly one k-qubit gate that is the last one to act on qubit t. Call this gate V_t. Note that we may have V_t=V_ℓ for t≠ℓ.

For each gate w∈ [R], let ℒ^→(U_w)⊆ [n] consist of supp(U_w) as well as all output bits of F_n that depend on gate U_w. Here supp(U_w) denotes the support of U_w (the 
 k qubits on which it acts nontrivially). Let us consider a bipartite graph G with a vertex for each gate U_w with 1≤ w≤ R,  and a vertex for each output bit v∈ [n], and an edge {U_w,v} whenever v∈ℒ^→(U_w). Since each output bit depends on at most n^1/4 input gates, and each output bit can be in the support of at most d gates in the circuit, the maximum degree of any output bit is at most d+n^1/4≤ 1.1n^1/4 (since n≥ (10d)^4), and therefore the total number of edges in G is 

    ∑_w=1^R |ℒ^→(U_w)|≤ 1.1n^5/4.

Let 

    Bad={t∈ [n]: |ℒ^→(V_t)|≥ n^1/2}  and  Good=[n]∖Bad.

Then from <ref> we have

    |Bad|≤ 1.1n^3/4

and therefore 

    |Good|≥ n(1-1.1n^-1/4).

Now let us consider a graph G' with vertex set Good and an edge {s,t} iff

    ℒ^→(V_s)∩ℒ^→(V_t)≠∅.

The maximum degree Δ of G' satisfies 

    Δ≤max_t∈Good|ℒ^→(V_t)|· (d+n^1/4)≤  n^1/2· 1.1n^1/4=1.1n^3/4.

Now suppose we choose a triple of qubits {h,i,j}⊆ [n] uniformly at random. From <ref> we see that the probability that {h,i,j}⊆Good is 1-O(n^-1/4). Conditioned on this event, the probability that {h,i,j} form an independent set in G' can also be lower bounded as 1-O(n^-1/4). To see this note that since the maximum degree of G' is at most O(n^3/4), two randomly chosen vertices in Good have an edge between them with probability at most O(n^-1/4). Then apply this to the three pairs {h,i},{i,j} and {h,j} and use the union bound.

Let ℰ be the event that our three qubits h,i,j are an independent set in G'. Then we have shown Pr[ℰ]≥ 1-O(n^-1/4). Moreover, if ℰ occurs then 


    ℒ^→(V_h)∩ℒ^→(V_i)   =∅
    ℒ^→(V_i)∩ℒ^→(V_j)   =∅
    ℒ^→(V_h)∩ℒ^→(V_j)   =∅.



Now suppose that U_1,U_2,…, U_R ∼𝒟_k are uniformly random k-qubit Clifford gates. Let |ψ⟩=U|0^n⟩ be the output state of the circuit U. Let ℰ' denote the event that qubits h,i,j exhibit GHZ-type measurement-induced entanglement. That is, event ℰ' occurs iff there is GHZ-type entanglement between qubits h,i,j after measuring all other qubits of ψ in the computational basis. By the long-range GHZ property, event ℰ' occurs with probability at least c>0. 

By the union bound, the probability that both events ℰ and ℰ' occur is at least

    Pr[ℰ and ℰ']≥ c-O(n^-1/4).


Now let us suppose that events ℰ and ℰ' both occur. Starting with our circuit U_1,U_2,…, U_R we define a set of 27 quantum circuits as follows. For each of the qubits h,i,j there are three choices for a single-qubit Pauli basis X,Y,Z. Let us index these choices by a tuple b=b_hb_ib_j∈{0,1,2}^3, where 0,1,2 correspond to X,Y,Z respectively. Recall that we write Q(0),Q(1),Q(2) for the single-qubit Clifford unitaries that change basis from the Z basis to the X,Y,Z bases respectively (i.e., Q(0)=(X+Z)/√(2), Q(1)=(Y+Z)/√(2), Q(2)=I). For b_h,b_i,b_j∈{0,1,2}^3, let

    V_h(b_h)   =Q(b_h) V_h 
    
    V_i(b_i)   =Q(b_i) V_i 
    
    V_j(b_j)   =Q(b_j) V_j.

and let U(b) denote the circuit obtained from U_1,U_2,…, U_R by making the replacements V_h← V_h(b_h), V_i←  V_i(b_i), V_j← V_j(b_j).

Define a function m:{0,1,2}^3→{0,1}^n by m(b)=F_n(U(b)). From <ref> we see that each output bit m_j depends on at most one of the bits b_h,b_i,b_j, and that 


  * m_h is independent of b_i, b_j

  * m_i is independent of b_h, b_j

  * m_j is independent of b_h, b_i.

Therefore, from <ref> we conclude that 

    ⟨ F_n(U(b))|U(b)|0^n⟩=0     for some  b∈{0,1,2}^3.

Note that each of the circuits U(b) with b∈{0,1,2}^3 occurs in our random ensemble with the same probability as U. Therefore 

    Pr_U_1,U_2,…, U_R ∼𝒟_k[ ⟨ F_n(U)|U|0^n⟩=0]   ≥Pr[ℰ and ℰ']/27
       =c/27-O(n^-1/4).





Suppose 𝒜={𝒯_n}_n∈Γ is a k-local, depth-d circuit architecture with the long-range tripartite MIE property (<ref>). There exists a positive constant α >0 and positive integer n_0 such that the following holds. Suppose n∈Γ satisfies n≥max{n_0,(10d)^4}. Suppose there is a depth-D probabilistic classical circuit composed of gates of fan-in K with output distribution P_U on input circuit U satisfying

    𝔼_U_1,U_2,…, U_R ∼𝒟_k[P_U-Q_U_1]< α,

where 𝒟_k is the uniform distribution on k-qubit Clifford unitaries, and Q_U(z)=|⟨ z|U|0^n⟩|^2 is the true output distribution. Then D≥1/4log(n)/log(K).


Let F_n be the function which describes the classical circuit. Recall that F_n takes as input the gates U_1,U_2,…, U_R in the circuit as well as a random string r∈{0,1}^ℓ(n), and outputs a binary string z=F_n(U_1,U_2,…, U_R, r)=F_n(U,r). 

Suppose D<1/4log(n)/log(K). Then every output bit of the classical circuit depends on at most K^D< n^1/4 input bits. In particular, each output bit depends on at most n^1/4 of the input gates U_1,U_2,…, U_R. Then

    𝔼_U_1,U_2,…, U_R ∼𝒟_k(∑_x∈{0,1}^n |P_U(x)-Q_U(x)|)
       ≥𝔼_U_1,U_2,…, U_R ∼𝒟_k(∑_x:Q_U(x)=0 P_U(x))
       =𝔼_U_1,U_2,…, U_R ∼𝒟_kPr_r[⟨ F_n(U,r)|U|0^n⟩=0]
       =𝔼_rPr_U_1,U_2,…, U_R ∼𝒟_k[⟨ F_n(U,r)|U|0^n⟩=0]
       ≥𝔼_rc-O(n^-1/4)/27
       = c-O(n^-1/4)/27

where c>0 is the constant appearing in the definition of <ref>, and where we used <ref>. We choose n_0 large enough so that O(n^-1/4) is at most c/2 and then set α=c/54.




§ COARSE-GRAINED ARCHITECTURE



Here we discuss the “coarse-grained" two-dimensional circuit architecture where we are able to establish rigorous statements concerning measurement-induced entanglement and quantum advantage. In particular, we prove <ref>. The coarse-grained architecture shares salient features of shallow quantum circuits in the usual brickwork architecture; most importantly, there is a locality parameter τ that determines the linear size of the lightcone of a given qubit.   


Consider a family of random quantum circuits, shown in <ref>, acting on n qubits arranged on a √(n)×√(n) grid lattice. Each circuit U in this family consist of two layers of gates V and W applied consecutively such that 

    U |0^n⟩ = (W_m^2⊗⋯⊗ W_1)_W·(V_(m-1)^2⊗⋯⊗ V_1)_V|0^n⟩.

We refer to the gates V_1,…, V_(m-1)^2 in V as the first layer and W_1,…, W_m^2 in W as the second layer gates.

We shall consider two families of quantum circuits in this architecture. The first is obtained by choosing each gate to be a Haar-random unitary acting on a square of τ×τ qubits such that m^2 τ^2 = n. We will call a circuit U drawn from this family a random coarse-grained universal circuit. The second family we consider is obtained by choosing each gate to be a uniformly random τ^2-qubit Clifford unitary. We refer a circuit D drawn from this family as a random coarse-grained Clifford circuit. 






 §.§ Subsystem anticoncentration and long-range MIE


Below we show that random coarse-grained Clifford circuits exhibit subsystem anticoncentration in the sense described in the Introduction. In particular, we consider a tripartition [n]=ABC of the qubits such that |A|=1, and the corresponding set S defined in <ref>. We then give an upper bound on 𝔼_D[S] that approaches zero as |C|,n→∞. From our upper bound it follows directly that the family of random coarse-grained Clifford circuits satisfies the long-range MIE property. We use <ref> to infer that the family of random coarse-grained universal circuits also satisfies the long-range MIE property (<ref>).

To upper bound the expected size of S, we first bound the probability that a random Pauli operator becomes Z-type when it is evolved backwards in time through a random coarse-grained Clifford circuit. 

Given a Pauli operator P defined on the qubits [n], let (P) ⊆ [n] be set of qubits on which P takes a value other than the identity. For any Pauli operator P and random coarse-grained Clifford circuit D, it is relatively easy to see that the probability of D^† P D being Z-type is determined entirely by the overlap of (P) with the second layer of gates W_1,…, W_m^2. To keep track of this overlap, for any Pauli P we let the cluster 𝒢(P) denote the set of all second layer gates with non-trivial overlap with (P). Such a cluster can be specified by filling in squares of an m × m grid. This grid representation leads naturally to the notion of the size and perimeter of a cluster 𝒢(P), which we define next. 


    Given any cluster 𝒢(P), identify it with a subset of the m × m grid in the natural way. Then let the size of the cluster, denoted 𝒢(P) be the number of grid squares contained in the cluster. Also let the perimeter of the cluster, denoted Per(𝒢(P)) be the number of grid edges on the boundary of the cluster, excluding edges running along the outside of the grid (see <ref>). 




The following theorem bounds the probability that a Pauli is Z-type after conjugation by a random coarse-grained Clifford circuit.



Let P be a Pauli operator, and let D be a random coarse-grained Clifford circuit. Also let a = 𝒢(P) and l = Per(𝒢(P)) denote the size and perimeter of 𝒢(P), respectively.  Then 

    [D^†  P D  is Z-type] ≤(1/2) ^a τ^2 + 1/4l (τ^2 / 12 - 1).



Now let us consider the measurement-induced entanglement scenario in the case where A and C are single qubits. We will use first-moment methods to establish the existence of measurement-induced entanglement in the coarse-grained architecture for τ=Ω(√(log(m))).

We first bound the expected number of binary strings s ∈ S such that the perimeter of the cluster 𝒢(DZ(s)D^†) has a given length l.

 
    Let D be a random coarse-grained Clifford circuit  and let l be a positive integer. Let A ⊆[n] contain a single qubit and C ⊆ [n] contain at least one qubit, with A and C disjoint. Then 
    
    𝔼_D[{s ∈ S: Per(𝒢(DZ(s)D^†)) = l}] 
        
       ≤ 2^l(2log(m)-τ^2/48+5).




    Expanding out the definition of S gives 
    
    {s ∈ S: Per(𝒢(DZ(s)D^†)) = l}
       =|{s∈{0,1}^n: DZ(s)D^†=(-1)^α P_A⊗ Z(r)_B⊗  I_C 
        for some  r∈{0,1}^|B|, α∈{0,1}, P ∈{X,Y,Z}
        and Per(𝒢(DZ(s)D^†)) = l}|.


    But for any fixed r and P we have
    
    ∃ s ∈{0,1}^n : DZ(s)D^†=(-1)^α P_A⊗ Z(r)_B⊗  I_C 
       ⇔ D^†(P_A⊗ Z(r)_B⊗  I_C) D  is Z-type.


    Then we also have 
    
    {s ∈ S: Per(𝒢(DZ(s)D^†)) = l}
       =|{(r,P): Per(𝒢(P_A⊗ Z(r)_B⊗  I_C)) = l,  and 
        D^†(P_A⊗ Z(r)_B⊗  I_C) D  is Z-type} |.

    We can then bound the expected size of this set as follows. For a given cluster 𝒢', let Γ(𝒢') denote the set of all pairs r∈{0,1}^|B| and P∈{X,Y,Z} such that 𝒢(P_A⊗ Z(r)_B⊗  I_C) = 𝒢'. Then
    
    𝔼_D[{s ∈ S: Per(𝒢(DZ(s)D^†)) = l}] 
       = ∑_𝒢' : 
    Per(𝒢') = l ∑_(r,P)
    ∈Γ(𝒢')[D^†(P_A⊗ Z(r)_B⊗  I_C) D  is Z-type.] 
       ≤∑_𝒢'  : 
    Per(𝒢') = l ∑_(r,P)∈Γ(𝒢') 2^-𝒢'τ^2 - 1/4l(τ^2/12 - 1)
       ≤ 3 ∑_𝒢'  : 
    Per(𝒢') = l 2^-1/4l(τ^2/12 - 1)
       ≤ 6 2m(m-1)l 2^-1/4l(τ^2/12 - 1)
       ≤ 2^l( 2log(m) - τ^2/48 + 5).

    where we used <ref> to go from the first line to the second and that there are at most 2^𝒢'τ^2 strings r ∈{0,1}^n  such that 𝒢(P_A⊗ Z(r)_B⊗  I_C) = 𝒢' to go from the second line to the third. To go from the third line to the fourth we count the number of clusters 𝒢' with perimeter l. Any such cluster can be specified by first choosing the location of the l edges of the cluster, and then specifying the interior and exterior of the cluster. A crude upper bound gives that there are at most 2m(m-1)l ways to chose the l edges of the cluster, and then two possible options for how to specify the interior and exterior of the cluster once the edges are fixed. This argument gives {𝒢'  : Per(𝒢') = l }≤ 2 2m(m-1)l, which is the bound used above.


Finally, we upper bound the expected size of S.



    In the setting of <ref> and choosing τ≥√(1000log_2(m)), we have
    
    𝔼_D[S] ≤3/2^C + 1 + 2· 2^-τ^2/100.






We first observe 

    𝔼_D[S] ≤∑_l=0^∞𝔼_D[{s ∈ S: Per(𝒢(DZ(s)D^†)) = l}].

Then <ref> and our assumption on the size of d gives

    ∑_l=1^∞𝔼_D[{s ∈ S:    Per(𝒢(DZ(s)D^†)) = l}] 
       ≤∑_l=1^∞ 2^l(2log_2(m)-τ^2/48+5)
       =∑_l=1^∞ 2^l(2log_2(m)-τ^2/96+5)-lτ^2/96
       ≤∑_l=1^∞ 2^l(-τ^2/96)
       ≤ 2· 2^-τ^2/100

where in the second-to-last line we used the fact that 2log_2(m)-τ^2/96+5)≤ 2log_2(m)-10log_2(m)+5≤  0.
It remains to show a bound when l = 0. In this case, by the same logic what was used in the proof of <ref>, we have

    𝔼_D[{s ∈ S: Per(𝒢(DZ(s)D^†)) = 0}] 
       =𝔼_D[ |{ r∈{0,1}^B, P ∈{X,Y,Z} : 
       Per(𝒢(P_A⊗ Z(r)_B⊗  I_C)) = 0  and 
        D^†(P_A⊗ Z(r)_B⊗  I_C) D  is Z-type} | ].

But we have Per(𝒢(P_A⊗ Z(r)_B⊗  I_C)) = 0 only if  the cluster 𝒢(P_A⊗ Z(r)_B ⊗  I_C) is the entire m × m grid. This cluster has size m^2 and there are at most 

    3 (2^B) = 3 ( 2^n - A - C ) = 3 ( 2^n - 1 - C)

Pauli strings satisfying this condition. Then a straightforward application of <ref> gives 

    𝔼_D[{s ∈ S: Per(𝒢(DZ(s)D^†)) = 0}] 
       ≤ 3 ( 2^n - 1 - C) (2^- m^2 τ^2) = 3 (2^-C - 1)

and the result follows. 



The family of random coarse-grained universal circuits with τ≥√(1000log_2(m)) and n sufficiently large satisfies the long-range MIE property. The same statement holds for random coarse-grained Clifford circuits.


Let [n]=ABC be a tripartition of the qubits such that |A|=1 and B is a square shielding region with side length L, centred at A (see <ref>). Here L is chosen so that |C|≠ 0. Note that due to the 2D geometry, this implies |C|=Ω(√(n)).

Let U be a random coarse-grained universal circuit with τ≥√(1000log_2(m)). Plugging the upper bound from <ref> into <ref> we get 

    𝔼_U 𝔼_xTr(ρ(x)^2)≤1/2+O(1/poly(m))=1/2+O(1/poly(n)).

Taking n sufficiently large we can ensure the RHS is less than any constant c∈ (1/2,1).

If we replace U with a random coarse-grained Clifford circuit, then we can use <ref> along with <ref> and Markov's inequality to arrive at the same conclusion. 



 §.§ Long-range tripartite MIE


We have shown that random coarse-grained universal or Clifford circuits have the long-range MIE property (<ref>), as a consequence of the subsystem anticoncentration expressed in <ref>.  Below we specialize to random coarse-grained Clifford circuits, and we show that <ref> also implies long-range tripartite MIE. As discussed in the Introduction, this gives rise to an unconditional quantum advantage.


 We first review some facts about entanglement in stabilizer states.  For any graph G = (V,E), the V-vertex graph state |Φ_G⟩ is defined to be the stabilizer state with stabilizer group generated by 

    g_v = X_v (∏_u : (u,v) ∈ E Z_u).

It is known that any (finite dimensional) stabilizer state is locally Clifford equivalent to a graph state <cit.>. 

The following lemma shows that GHZ-type entanglement can always be produced by making computational basis measurements on a connected graph state with sufficiently many vertices.



Let G = (V,E) be a connected graph with at least 3 vertices and let |Φ_G⟩ be the associated graph state. Then there exist indices h,i,j ∈ V such that, for any x ∈{0,1}^V - 3, the state 

    |Φ_h,i,j⟩ 
        = ⟨x|_V \{h,i,j}|Φ_G⟩

is locally Clifford equivalent to a GHZ state (up to normalization).


The post-measurement state after measuring a qubit v of a graph state Φ_G in the computational basis and obtaining outcome y∈{0,1} is

    (∏_w:{v,w}∈ E(G) Z^y_w)|Φ_G(V∖{v})⟩

where G(V∖{v}) is the induced subgraph of G on all V∖{v},  obtained from G by removing vertex v and all edges incident to it. Starting from a connected graph G with at least three vertices, we choose 3 vertices Ω={h,i,j} such that the subgraph G(Ω) of G induced by Ω is a connected three-vertex graph. Using the above fact about computational basis measurement, we see that after measuring all vertices in V∖{h,i,j} the postmeasurement state is locally Clifford equivalent to the graph state |Φ_G(Ω)⟩ where G(Ω) is a connected three-vertex graph. Finally, we use the well-known fact that any 3 vertex connected graph state is locally Clifford equivalent to a GHZ state (this can be confirmed by a direct calculation).


Now consider a random coarse-grained Clifford circuit with τ≥√(1000log_2(m)) as in <ref>. In this setting, a straightforward consequence of <ref> is that measuring all but two output qubits of the randomly chosen circuit in the computational basis produces a two-qubit entangled stabilizer state on the unmeasured qubits with probability at least 1/4 - exp(-Ω(τ^2)). The next theorem shows that tripartite GHZ-type entanglement is also ubiquitous.



Random coarse-grained Clifford circuits with τ≥√(1000log_2(m)) satisfy the long-range tripartite MIE property (<ref>). In particular, let D be an n-qubit random coarse-grained Clifford circuit, with τ≥√(1000log_2(m)). Let {h,i,j} be a uniformly random triple of qubits. Then {h,i,j} exhibit GHZ-type MIE with probability at least 0.005 over the random choice of D and the random choice of {h,i,j}.


Let D be an n-qubit random coarse-grained Clifford circuit with τ≥√(1000log_2(m)).

Let us say that {h,i,j}⊆ [n] is a good triple, iff qubits  {h,i,j} share GHZ-type entanglement after all other qubits have been measured with probability at least 0.05 over the random choice of D.

Below we prove the following: for any set of qubits V⊆ [n] with V = 5, there exists a good triple {h,i,j}⊆ V.

The theorem then follows from this statement. To see this, consider the following procedure. First, choose a uniformly random subset of qubits V⊆ [n] of size |V|=5, and then choose a uniformly random triple {h,i,j}⊆ V. By the above, the probability that we are lucky and choose a good triple is at least 1/53=1/10.

Moreover, clearly this procedure samples a triple {h,i,j} from the uniform distribution over all n3 triples. 

It remains to show that for any set of qubits V⊆ [n] with V = 5, there exists a good triple {h,i,j}⊆ V.

We first consider the state of the qubits in V after the circuit D is applied and all other qubits are measured in the computational basis. For any qubit v ∈ V, <ref> give that there is no post-measurement entanglement between qubit v and qubits in the set V\{v} with probability at most 

    3/2^5 + 2· 2^-τ^2/100≤ 3/2^5 + 2· 2^-10< 0.1.

Then the union bound gives that there exists a qubit v' ∈ V with no post-measurement entanglement between v' and the qubits V\{v'} with probability at most 

    V· 0.1 < 0.5.

Equivalently, there is post measurement entanglement between each qubit v' ∈ V and qubits V\{v'} with probability at least 1/2. But then the post measurement state of the qubits V is locally Clifford equivalent to a graph state |Φ_G⟩ corresponding to a 5 vertex graph with no isolated vertices (this is because there is no entanglement between qubits corresponding to disconnected vertices of a graph state). It follows that the graph G must have either a 3 or 5 vertex connected component. If G has a 3 vertex connected component the qubits corresponding to this component are in a state locally Clifford equivalent to a GHZ state, and we are done. Otherwise, the graph state has a 5 vertex connected component. But then by <ref> there are vertices h,i,j ∈ V such that, for any x ∈{0,1}^2 we have that

    |Φ_hij⟩ = ⟨x|_V\{h,i,j}|Φ_G⟩

is locally Clifford equivalent to a GHZ state. Now let |Ψ_V⟩ denote the post measurement state of the qubits in the set V. Since this state is locally Clifford equivalent to |Φ_G⟩ we have 

    |Ψ_V⟩ = ⊗_v ∈ V C_v |Φ_G⟩

where each C_v is a 1-qubit Clifford unitary. Furthermore, since the distribution of coarse-grained Clifford circuits is unchanged under local Clifford operations, we can assume the Clifford unitaries C_v are chosen uniformly at random. 
Then the post measurement state of qubits h,i,j when the remaining two qubits of V are measured in the computational basis and outcome x∈{0,1}^2 is observed is given by

    |Ψ_h,i,j⟩   = ⟨x|_V\{h,i,j}⊗_v ∈ V C_v |Φ_G⟩.

Comparing with <ref> we see this is locally Clifford equivalent to a GHZ state provided the state 

    ⊗_v ∈ V\{h,i,j} C_v^†|x⟩_V\{h,i,j}

is a computational basis state. But since we can take the C_v to be random Cliffords this occurs with probability exactly 1/9. Putting this all together we see that {h,i,j} is a good triple: the state |Ψ_h,i,j⟩ is locally Clifford to a GHZ state with probability at least 
1/9·1/2≥ 0.05
over the random choice of D.





 §.§ Approximate implementation of coarse-grained circuits with two-qubit gates


Lastly, we demonstrate that coarse-grained circuits can be approximately implemented using two-qubit gates, instead of gates that act on blocks of τ×τ qubits.
We shall see that this yields a family of O(log(n))-depth quantum circuits composed of Haar-random two-qubit gates that satisfy the long-range MIE property.

Our construction replaces each τ×τ Haar random gate in a random coarse-grained universal circuit by an ε-approximate unitary 2-design for a sufficiently small approximation error ε=2^-Ω(τ^2).
Such unitary 2-designs can, for instance, be implemented using a 1D brickwork quantum circuit of depth O(τ^2) composed of Haar random two-qubit gates which is arranged in a snake-like pattern (or more formally, along a Hamiltonian path) on the square of τ×τ qubits <cit.>.
This results in a distribution of n-qubit random quantum circuits with an overall depth of O(τ^2). We denote these compiled random coarse-grained circuits by 𝒰̃.
We can analogously define 𝒟̃ to be the family of random Clifford circuits obtained by replacing the Haar random two-qubit gates in  𝒰̃ by random two-qubit Clifford gates.

The family of complied random coarse-grained circuits 𝒰̃ satisfies the long-range MIE property when τ≥√(1000log_2(m)) and n sufficiently large. The same statement holds for compiled random coarse-grained Clifford circuits 𝒟̃.


We begin by applying <Ref>, which allows us to reduce the analysis of the long-range MIE property in 𝒰̃ to that in 𝒟̃. 
To establish the long-range MIE property in 𝒟̃, we focus on bounding the expected size of the setS. 
In the Appendix, we demonstrate that the family of random Clifford circuits 𝒟̃ satisfies the bound in <ref> of <Ref>. 
We can then employ the same reasoning as in the proof of <Ref> to establish the long-range MIE property for circuits drawn from 𝒰̃, as stated in <Ref>.




Note added: As we finalized this paper, we became aware of concurrent works <cit.> that also study the 2D coarse-grained architecture introduced here. While we focus on establishing quantum advantage,  long-range MIE, and anti-concentration properties, these works demonstrate that with a suitable choice of log(n)-sized coarse-grained gates, a similar architecture can also exhibit the unitary k-design property. 



§ ACKNOWLEDGMENTS

DG acknowledges the support of
 the Natural Sciences and Engineering Research Council of Canada through grant number RGPIN-2019-04198. DG is a fellow of the Canadian Institute for
Advanced Research, in the quantum information science program. Research at Perimeter
Institute is supported in part by the Government of Canada through the Department of
Innovation, Science and Economic Development Canada and by the Province of Ontario
through the Ministry of Colleges and Universities.
MS is supported by AWS Quantum Postdoctoral Scholarship and funding from the National Science Foundation NSF CAREER award CCF-2048204.
Institute for Quantum Information and Matter is an NSF Physics Frontiers Center. 

unsrt








§ ANALYSIS OF <REF>

Here we prove <ref>. The proof is obtained by combining the fidelity bound from <ref> with the robustness property of the gate-by-gate algorithm established in Ref. <cit.>. Our setting differs slightly from the one considered there (this is due to the fact that ρ̃_t(ω_B) is not a pure state in general); below we adapt the proof to our setting.   

The gate-by-gate algorithm of Ref. <cit.> is obtained from <ref> by making a small but consequential replacement: replacing q̃_t(y|x_B(t)) with q_t(y|x_B(t)x_C(t)) in line 9. We shall analyze <ref> by comparing it to the gate-by-gate algorithm and using the performance guarantee of the latter algorithm. Specifically, in Ref. <cit.> it is shown that the binary string x at the end of the tth iteration of the for loop of the gate-by-gate algorithm is distributed according to 

    P_t(x)≡ |⟨ x|U_tU_t-1… U_1|0^n⟩|^2      x∈{0,1}^n.


Write W_t(x) for the probability of sampling x∈{0,1}^n at the end of the tth iteration of the for loop of <ref>.


First suppose that t∈Γ, so that U_t is a single-qubit gate. Let A=A(t), B=B(t), and C=C(t). Then

    W_t(x)   =∑_y:y_BC=x_BC W_t-1(y) q̃_t(x_A|x_B)
       =∑_y:y_BC=x_BC(W_t-1(y)-P_t-1(y)) q̃_t(x_A|x_B)+∑_y:y_BC=x_BCP_t-1(y) q̃_t(x_A|x_B)
       =∑_y:y_BC=x_BC(W_t-1(y)-P_t-1(y)) q̃_t(x_A|x_B)+∑_y:y_BC=x_BCP_t-1(y) (q̃_t(x_A|x_B)-q_t(x_A|x_Bx_C))+P_t(x),

where we used the fact that P_t(x)=∑_y:y_BC=x_BC P_t-1(y)q_t(x_A|x_Bx_C). Now using <ref>, summing over x∈{0,1}^n, and using the triangle inequality, we get

    W_t-P_t_1   ≤∑_x_BC∈{0,1}^BC∑_y:y_BC=x_BC |(W_t-1(y)-P_t-1(y))|∑_x_A∈{0,1}^|A|q̃_t(x_A|x_B)
       +∑_x_BC∈{0,1}^BC∑_y:y_BC=x_BCP_t-1(y) ∑_x_A∈{0,1}^|A||q̃_t(x_A|x_B)-q_t(x_A|x_Bx_C)|
       = W_t-1-P_t-1_1+𝔼_z∼ P_t-1,BC(q_t(·|z_Bz_C) -q̃_t(· |z_B)_1).


Using <ref> and the relationship between trace distance and quantum fidelity, we have

    q_t(·|z_Bz_C) -q̃_t(· |z_B)_1≤ρ_t(z_Bz_C)-ρ̃_t(z_B)_1≤  2√(1-F(ρ_t(z_Bz_C), ρ̃_t(z_B))).

Plugging <ref> into <ref> and using the fact that 𝔼[√(X)]≤ (𝔼[X])^1/2 for any non-negative random variable X gives

    W_t-P_t_1   ≤W_t-1-P_t-1_1+2(𝔼_z∼ P_t-1,BC[1-F(ρ_t(z_Bz_C), ρ̃_t(z_B))])^1/2
       ≤W_t-1-P_t-1_1+2(1-𝔼_z_B∼ P_t-1,B[Tr(ρ̃_t(z_B)^2)])^1/2     t∈ S.

where we used <ref> and the fact that P_t-1,BC=P_t,BC (since the unitary gate U_t acts only on qubit A and does not change the marginal distribution on BC). 

Next suppose t∉Γ, so U_t=CNOT_ij for some pair of qubits i≠ j. Let f_t(x) be the function applied to x in line 4 of <ref>; that is, it updates x by replacing x_j← x_i⊕ x_j. Then

    W_t(x)=W_t-1(f_t(x))    and    P_t(x)=P_t-1(f_t(x)),

so in this case 

    W_t-P_t_1=W_t-1-P_t-1_1      t∉Γ.


Applying <ref> for t=1,2,…, m and using the fact that P_0=W_0, we arrive at

    W_m-P_m_1≤ 2∑_t∈Γ(1-_z∼ P_t-1,B[Tr(ρ̃_t(z)^2)])^1/2=2∑_t∈Γ(1-_z∼ P_t,B[Tr(ρ̃_t(z)^2)])^1/2

where in the last equality we used the fact that U_t is a single-qubit gate acting on qubit A, so P_t-1,B=P_t,B. This completes the proof of <ref>.



§ PROOF OF <REF>

Before proving this theorem we require a preliminary result about the effect of conjugating Pauli operators through random Clifford circuits. 




Let P_1, P_2,…, P_r be uniformly random non-identity Pauli strings of various lengths, chosen so that P_1 ⊗ P_2 ⊗⋯⊗ P_r defines a Pauli string of length n. Denote this Pauli string P_[n] and for any Q ⊆ [n] let P_Q denote the corresponding truncation of this string. Let C be a uniformly random t-qubit Clifford operator. Then

    
  *  For any Q ⊆ [n] with Q = t: 
    
    [ C^† P_Q C  is Z-type] ≤(1/2)^t.

    
  * 
    For any Q ⊆ [n] with Q = k < t:
    
    [ C^†( P_Q ⊗ I ^⊗ t-k) C  is Z-type] ≤(1/4)^k + (1/2)^t.


Moreover, these bounds still hold even after arbitrary conditioning on the value of Paulis in the complimentary string P_[n] \ Q. 




We first argue that for any Q the probability of P_Q being identity is at most 4^-Q, even after conditioning on an arbitrary value for P_[n] \ Q. If the Pauli's P_1, P_2,…, P_r were uniformly random (including the possibility of being the identity) then each individual Pauli in P_[n] would be chosen uniformly at random and this statement would be immediate, with the bound 4^-Q holding exactly. That is, if we let P_1', P_2', …, P_r' be uniformly random Pauli strings which may be the identity we have 

    [(P_1' ⊗ P_2' ⊗⋯⊗ P_r')_Q = I   |  (P_1' ⊗ P_2' ⊗⋯⊗ P_r')_[n] \ Q = K' ] = 4^-Q

for any choice of Q and Pauli string K'. But now we observe that further enforcing that each P_i' is non-identity in the above equation can only increase the probability that (P_1' ⊗ P_2' ⊗⋯⊗ P_r')_Q is non-identity since, by Bayes' rule,

    [(P_1' ⊗ P_2' ⊗⋯⊗ P_r')_Q = I   |  (P_1' ⊗ P_2' ⊗⋯⊗ P_r')_[n] \ Q = K'  and  P_1', P_2', … , P_r' ≠ I ]/[(P_1' ⊗ P_2' ⊗⋯⊗ P_r')_Q = I   |  (P_1' ⊗ P_2' ⊗⋯⊗ P_r')_[n] \ Q = K' ]
       = [P_1', P_2', … , P_r' ≠ I    |  (P_1' ⊗ P_2' ⊗⋯⊗ P_r')_[n] \ Q = K'  and (P_1' ⊗ P_2' ⊗⋯⊗ P_r')_Q = I]/[P_1', P_2', … , P_r' ≠ I    |  (P_1' ⊗ P_2' ⊗⋯⊗ P_r')_[n] \ Q = K' ]

and the quantity on the on the right hand side of the above equation is clearly bounded above by one. The statement follows. 


A similar logic can be applied to prove statements (a) and (b) directly but, for concreteness, we will take a slightly more involved route. We start with (a). The random Clifford C will map a non-identity Pauli to a uniformly random non-identity Pauli on t qubits. There are 4^t - 1 such Paulis and 2^t - 1 of them are Z-type, so

    [ C^† P_Q C  is Z-type]    = [P_Q  is identity] + [P_Q  is non-identity] (2^t - 1/4^t - 1) 
       ≤ 4^-t + (1 - 4^-t)(2^t - 1/4^t - 1) 
       = 1/4^t + 2^t - 1/4^t = 1/2^t,

where we used our upper bound on the probability of P_Q begin the identity along with the fact that 2^t - 1/4^t - 1≤ 1 to go from the first line to the second. 

To prove (b) we begin with a similar expression and find

    [ C^†(P_Q⊗ I^⊗ t-k)C  is Z-type]    = [P_Q  is identity] + [P_Q  is non-identity] (2^t - 1/4^t - 1) 
       ≤ 4^-k + (1 - 4^-k)(2^t - 1/4^t - 1) 
       ≤ 4^-k + (2^t - 1/4^t - 1) ≤ 4^-k + 2^-t,

as desired. 


We now proceed to the proof of <ref>.


We write D = W V where V and W denote the first and second layer of random Clifford circuits, respectively. We first consider the Pauli string obtained by conjugating P by just the second layer of Cliffords, which we can write as P'=W^† P W. This string consists of a random non-identity Pauli string acting on every τ×τ patch of qubits in the support of 𝒢(P), and is identity elsewhere. There are a such patches by definition, and so a τ^2 qubits on which this random Pauli string can potentially be non-identity. Let Q' denote the set of all these qubits.

Now we consider the first layer of random Clifford unitaries. Recall from the definition of random coarse-grained circuits that we have V = V_1 ⊗ V_2 ⊗⋯⊗ V_(m-1)^2, where each V_i is a random τ×τ Clifford. For any 1≤ i≤ (m-1)^2 let Q'_i = Q' ∩(V_i) and let P'_i be the restriction of the Pauli string P' to the support of V_i. Finally let Q_0' = Q' \ (∪_i Q'_i) and P'_0 be the restriction of P' to Q'_0. Also, as a notational convenience, let V_0 denote the identity gate acting on the qubits in Q_0'. (This is to accommodate qubits in Q' which lie on the boundary of the circuit, and thus are not in the support of any V_i for i ≥ 1). We want to bound the quantity

    [D^† P D  is Z-type]    = [ V_i^† P'_i V_i  is Z-type for all 0 ≤ i ≤ (m-1)^2]
       = ∏_i = 0^(m-1)^2[ V_i^† P'_i V_i  is Z-type  |   V_j^† P'_j V_j  is Z-type for all 0 ≤ j < i].

It is immediate from definitions that 

    [V_0^† P_0' V_0  is Z-type] = [P_0'  is Z-type] ≤ 2^-Q_0'.

Then let 𝒱_int be the set of all V_i with 0 < i ≤ (m-1)^2 whose support is contained entirely in Q', and 𝒱_perim be the set of all V_i ith 0 < i ≤ (m-1)^2 whose support overlaps partially with Q'. 
As an immediate consequence of <ref> item:random Pauli and Clifford fully contained Pauli we see that for any V_i ∈𝒱_int

    [ V_i^† P'_i V_i  is Z-type  |   V_j^† P'_j V_j  is Z-type for all j < i] ≤ 2^-τ^2 = 2^- Q_i'.

Similarly, for any V_i ∈𝒱_perim, <ref> item:random Pauli and Clifford partially contained Pauli gives that 

    [ V_i^† P'_i V_i  is Z-type  |   V_j^† P'_j V_j  is Z-type for all j < i] ≤ 4^-Q_i' + 2^-τ^2.

For any V_i ∈𝒱_perim we have that at least 1/4 and at most 3/4's of the qubits in the support of V_i overlap with Q', and hence τ^2/4 ≤Q_i'≤ 3τ^2/4. Then we can also write 

    [ V_i^† P'_i V_i  is Z-type  |   V_j^† P'_j V_j  is Z-type for all j < i]    ≤ 4^-Q_i' + 2^-4 Q_i' / 3
       = 2^-Q_i'(2^-Q_i' + 2^-Q_i'/3)
       ≤ 2^-Q_i' - (τ^2/12) + 1.

Combining <ref> gives 

    [D^† PD is Z-type]   ≤ 2^-Q_0'(∏_V_i ∈𝒱_int 2^-Q_i')(∏_V_j ∈𝒱_perim 2^-Q_j - τ^2/12 + 1)
       = 2^- 𝒱_perim (τ^2/12 - 1) - ∑_i=0^(m-1)^2Q'_i
       = 2^- 𝒱_perim (τ^2/12 - 1) - Q'.

We observed previously that Q' = aτ^2. Every gate V_i∈𝒱_perim intersects with four edges, and in the worst case, all four of the edges could be perimeter edges. At the same time, every perimeter edge intersects with at least one gate in 𝒱_perim. Therefore, |𝒱_perim|≥1/4l. Inserting those two values into the equation above completes the proof. 





§ COMPILING THE COARSE-GRAINED ARCHITECTURE TO RANDOM QUANTUM CIRCUITS



Here we consider the setup of <Ref> which involves compiled random coarse-grained circuits 𝒰̃ and their Clifford versions 𝒟̃ composed of two-qubit gates. 
We show that an equivalent version of <ref> holds for 𝒟̃ with sufficiently small ε. 
The gates in the non-compiled coarse-grained circuits act on blocks of τ×τ qubits. 
Define t=τ^2. Let 𝒟_t denote the uniform distribution over t-qubit Clifford operators. 
Let 𝒟̃_t be a distribution of t-qubit random Clifford circuits forming an ε-approximate unitary 2-design w.r.t the operator norm. The approximate 2-design property of 𝒟̃_t in this notion implies that for every 2^t-by-2^t matrix A,B with ‖ A‖_F=‖ B‖_F=1, it holds that

    |_C̃∼𝒟̃_t[(BC̃^† AC̃)^2]-_C∼𝒟_t[(BC^† AC)^2]|≤ε.

While the compilation is agnostic to any specific construction of ε-approximate 2-design, for concreteness, we could implement the circuits in 𝒟̃_t using 1D brickwork random Clifford circuits with depth O(log(1/ε)) <cit.>. Alternatively, we can consider a local random quantum circuit architecture with O(tlog(1/ε)) gates where at each time step, a random 2-qubit Clifford gate is applied to a randomly chosen qubit and one of its neighbours on the 2D grid <cit.>.
We note that the constructions in <cit.> are known to from an ε-approximate 2-design if the circuits are composed of Haar random two-qubit gates or random gates drawn from a universal gate set.
However, we observe that the same constructions yield ε-approximate 2-design if the gates are chosen randomly from 2-qubit Clifford gates. 
This is the case because the 2-design bounds in <cit.> are established by analyzing mathematical objects defined solely using the second moment operator associated with the 2-qubit Haar measure 𝒰_2

    _U∼𝒰_2[U^⊗ 2⊗(U)^⊗ 2],

and the architecture template—for example by lower bounding the spectral gap of a local Hamiltonian defined in terms of <ref>. Since the uniform distribution over 2-qubit Clifford gates 𝒟_2 forms an exact unitary 2-design,

    _C∼𝒟_2[C^⊗ 2⊗(C)^⊗ 2]=_U∼𝒰_2[U^⊗ 2⊗(U)^⊗ 2].


Mirroring the proof strategy for <ref>, we begin by establishing a direct analog of <ref> in <Ref>.


Let K be an arbitrary non-identity t-qubit Pauli operator, draw a C̃∼𝒟̃_t, and define P=C̃^† KC̃. Let h,s≥ 1 satisfy h+s=t. Decompose P as P=± P_1⊗ P_2 such that P_1 and P_2 are h- and s-qubit Pauli operators respectively. Let ∅≠𝒫⊆{I,X,Y,Z}^⊗ s. Then for every ε≤1/3(4^t-1), it holds that

    (P_1=I^⊗ h|P_2∈𝒫)≤1/4^h(1+3(4^t-1)ε).




In the following, we make use of the fact that for every non-identity Pauli W∈{I,X,Y,Z}^⊗ t∖{I^⊗ t},

    (P=± W)=1/4^t_C̃∼𝒟̃_t[(W C̃^† K C̃)^2].

Thus, by <ref>,

    1/4^t-1-ε≤(P=± W)≤1/4^t-1+ε.

Since 𝒫≠∅, either I^⊗ s∉𝒫 or I^⊗ s∈𝒫. First consider the case where I^⊗ s∉𝒫. In this case, we have

    (P_1=I^⊗ h|P_2∈𝒫)   =(P_1=I^⊗ h and P_2∈𝒫)/(P_2∈𝒫)
       ≤|𝒫|(1/4^t-1+ε)/|𝒫|4^h(1/4^t-1-ε)
       ≤1/4^h(1+3(4^t-1)ε)

where in the last line, we use the fact that 1+x/1-x≤ 1+3x for every 0≤ x≤1/3. Secondly, for the case where I^⊗ s∈𝒫, we have

    (P_1=I^⊗ h|P_2∈𝒫)
    
    =   (P_1=I^⊗ h and P_2∈𝒫)/(P_2∈𝒫)
    =   (P_1=I^⊗ h and P_2=I^⊗ s)+(P_1=I^⊗ h and P_2∈𝒫∖{I^⊗ s})/(P_2=I^⊗ s)+(P_2∈𝒫∖{I^⊗ s})
    ≤   (|𝒫|-1)(1/4^t-1+ε)/(4^h|𝒫|-1)(1/4^t-1-ε)
    ≤   1/4^h(1+3(4^t-1)ε).





Let r∈{1,2,3,4}. Let K_1,…,K_r be arbitrary non-identity t-qubit Pauli operators. For every i∈{1,…,r}, draw an independent C̃_i∼𝒟̃_t, define P_i=C̃_i^† K_iC̃_i, and write P_i=± E_i⊗ F_i such that E_i and F_i are t/4- and 3t/4-qubit Pauli operators respectively. Define P_Q=⊗_i=1^r E_i, P_Q'=⊗_i=1^r F_i, and k=rt/4. Let ∅≠𝒫⊆{I,X,Y,Z}^3k. Then 


  * for r=4,

    _(C̃_1,…,C̃_r,C̃)∼𝒟̃_t[C̃^† P_QC̃ is Z-type|P_Q'∈𝒫]≤(1/2)^t-2^2t+2ε,



  * for r∈{1,2,3}, for every ε≤1/8^t,

    _(C̃_1,…,C̃_r,C̃)∼𝒟̃_t[C̃^†(P_Q⊗ I^⊗ t-k)C̃ is Z-type|P_Q'∈𝒫]≤1/4^k+2/2^t.





For every non-identity t-qubit Pauli operator W∈{I,X,Y,Z}^⊗ t∖{I^⊗ t}, it holds that

    _C̃∼𝒟̃_t(C̃^† WC̃ is Z-type)=2^t_C̃∼𝒟̃_t[⟨0^n|C̃^†W/√(2^t)C̃|0^n⟩^2]≤1/2^t+1+2^tε.

By the previous lemma and the independence among F_1,…,F_r, 

    (P_Q=I^⊗ k|P_Q'∈𝒫)   ≤1/4^k(1+3(4^t-1)ε)^r
       ≤1/4^k+15(4^t-1)/4^kε

where we use the fact that (1+x)^r≤ (1+x)^4≤ 1+5x for every 0≤ x≤ 0.15. Thus, for r=4, we get by convexity that

    (C̃^† P_Q C̃ is Z-type|P_Q'∈𝒫)
    ≤   (P_Q=I^⊗ k|P_Q'∈𝒫)+(1-(P_Q=I^⊗ k|P_Q'∈𝒫))(1/2^t+1+2^tε)
    ≤   1/4^t+15ε+(1-1/4^t-15ε)(1/2^t+1+2^tε)
    ≤   1/2^t+(2^t+15)ε
    
    =    2^-t+log(1+2^t(2^t+15)ε)
    ≤    2^-t+2^2t+2ε

where in the last line, we use the fact that log_2(1+x)≤ 2x for every x≥ 0. For the second bound, we have

    (C̃^†(P_Q⊗ I^⊗ t-k)C̃ is Z-type|P_Q'∈𝒫)
    ≤   1/4^k+15(4^t-1)/4^kε+(1-1/4^k-15(4^t-1)/4^kε)(1/2^t+1+2^tε)
    ≤   1/4^k+1/2^t+(15(4^t-1)/4^k+2^t)ε
    ≤   1/4^k+1/2^t+1/2^t

since ε≤1/8^t.




Let P be an n-qubit Pauli operator, and let D̃ be a compiled random Clifford circuit. Also let a = 𝒢(P) and l = Per(𝒢(P)) denote the size and perimeter of 𝒢(P), respectively. Assume τ≥Ω(√(log(m))) and choose ε=2^-ct=2^-cτ^2 for some large enough c so that (m-1)^22^2t+2ε≤ 1. Then 

    [D̃^†  P D̃ is Z-type] ≤(1/2) ^a τ^2 + 1/4l (τ^2 / 12 - 2)-1.



It suffices to establish bounds analogous to the ones appearing in the proof of <ref> for compiled random Clifford circuits. We inherit all the definitions from the proof of <ref>. 
For every V_i∈𝒱_int,

    [ V_i^† P'_i V_i  is Z-type  |   V_j^† P'_j V_j  is Z-type for all j < i] ≤ 2^-τ^2+2^2t+2ε = 2^- Q_i'+2^2t+2ε.

For every V_i∈𝒱_perim,

    [ V_i^† P'_i V_i  is Z-type  |   V_j^† P'_j V_j  is Z-type for all j < i]    ≤ 4^-Q_i' + 2^-τ^2+1
       ≤  4^-Q_i' + 2^-4 Q_i' / 3+1
       =2^-Q_i'(2^-Q_i' + 2^-Q_i'/3+1)
       ≤ 2^-|Q'|-τ^2/12+2.

Finally,

    [D^† PD is Z-type]   ≤ 2^-Q_0'+2^2t+2ε(∏_V_i ∈𝒱_int 2^-Q_i'+2^2t+2ε)(∏_V_j ∈𝒱_perim 2^-Q_j - τ^2/12 + 2)
       ≤ 2^- 𝒱_perim (τ^2/12 - 2) - ∑_i=0^(m-1)^2Q'_i+(m-1)^22^2t+2ε
       ≤ 2^- 𝒱_perim (τ^2/12 - 2) - Q'+1.

To finish the proof, we recall from the proof of <ref> that |Q_i|=aτ^2 and |𝒱_perim|≥1/4l.




§ PROOF OF <REF> 


Without loss of generality let us fix a local basis so that C_h, C_i, C_j are identity, and the stabilizer group of ψ contains stabilizers g_1,g_2,g_3 in <ref>. By taking products of these stabilizers g_1,g_2,g_3  we see that the stabilizer group of ψ contains the following elements:

    g_1   = X_hX_iX_j Z(s) 
    
    g_2   =Z_hZ_i I_j Z(t)
    
    g_3   =I_h Z_iZ_j Z(u)
    
    g_4   = -Y_hY_iX_j Z(s⊕ t)
    
    g_5   =-X_hY_iY_j Z(s⊕ u)
    
    g_6   =-Y_hX_iY_j Z(s⊕ t⊕ u)
    
    g_7   =Z_hI_iZ_j Z(u⊕ t).

Let us assume, to reach a contradiction, that m satisfies <ref>. Write

    m_s=∏_v∈ [n]m_v^s_v     m_t=∏_v∈ [n]m_v^t_v     m_u=∏_v∈ [n]m_v^u_v.


Note that our function m must only output binary strings that are consistent with stabilizers that are diagonal in the measurement basis b. In this way each of the stabilizers <ref> places a constraint on the possible outputs of m. 

For example, from the stabilizer g_2 we infer that 

    m_h· m_i· m_t=1     whenever   b_h=b_i=2.

Since each output bit depends on only one input bit, and  m_h,m_i are independent of b_j,  we can infer from the above that m_t is independent of b_j. We now explain this in more detail. Let L_h,L_i,L_j⊆ [n] be the set of output bits m_v that depend on inputs b_h,b_i,b_j respectively, so that

    m_t=(-1)^a∏_v∈ L_h m_v(b_h)∏_v∈ L_i m_v(b_i)∏_v∈ L_j m_v(b_j)

for some a∈{0,1}. Then <ref> implies

    ∏_v∈ L_j m_v(b_j)=(-1)^a m_h(2)· m_i(2)·∏_v∈ L_h m_v(2)∏_v∈ L_i m_v(2)     b_j∈{0,1,2}.

where b∈{0,1} is fixed. Plugging <ref> into <ref> gives 

    m_t=∏_v∈ L_h m_v(b_h)∏_v∈ L_i m_v(b_i)[m_h(2)· m_i(2)·∏_v∈ L_h m_v(2)∏_v∈ L_i m_v(2)]

which shows that m_t is independent of b_j. 

In exactly the same way, from the stabilizers g_3, g_7 we infer that m_u is independent of b_h and that m_u· m_t is independent of b_i.

Next consider the subset of inputs b_h,b_i,b_j∈{0,1} which correspond to measuring these qubits in the X or Y bases. Since m_t is independent of b_j and m_U is independent of b_h, there exist affine functions e,f,g,h:{0,1}→{0,1} such that 

    m_t=(-1)^e(b_h)+f(b_i)   m_u=(-1)^g(b_i)+h(b_j)     b_h,b_i,b_j∈{0,1}.

Since m_t· m_u is independent of b_i we may set 

    g(b_i)=f(b_i).


From the stabilizers g_1,g_4,g_5,g_6 we infer

    i^b_h+b_i+b_jm_hm_im_j m_sm_t^b_hm_u^b_j=1      b_h⊕ b_i⊕ b_j=0      b_h,b_i,b_j∈{0,1}.


Plugging <ref> into <ref> gives, for b_h,b_i,b_j∈{0,1}, 

    i^b_h+b_i+b_jm_hm_im_j m_s(-1)^b_h(e(b_h)+f(b_i))(-1)^b_j(f(b_i)+h(b_j))=1      b_h⊕ b_i⊕ b_j=0.


Now since each output bit m_v depends on at most one of the variables b_h,b_i,b_j, from the above we infer that for b_h,b_i,b_j∈{0,1} we have

    i^b_h+b_i+b_j(-1)^α_0+α_1b_h+α_2b_i+α_3b_j(-1)^β b_hb_i(-1)^β b_ib_j=1      b_h⊕ b_i⊕ b_j=0

for some coefficients

    α_0,α_1,α_2,α_3,β∈{0,1}

The claim below shows that we have reached a contradiction and we conclude that <ref> is not satisfied by any function m of the form described in the theorem statement. <ref> then follows directly.

<ref> is not satisfied for any choice of coefficients <ref>.


Assume, to reach a contradiction, that <ref> holds. We'll  plug in values of b_h,b_i,b_j and infer a set of linear equations that must be satisfied by the coefficients α_0,α_1,α_2,α_3,β. By plugging in b_h=b_i=b_j=0 we infer α_0=0. Next setting (b_h,b_i,b_j)={(1,1,0),(1,0,1),(0,1,1)} we get

    α_1+α_2+β   =1  2
    α_1+α_3    =1  2
    α_2+α_3+β   =1  2

Summing these three equations we are led to a contradiction (0=1  2) and therefore there is no solution.





 Paper Title: Non-equilibrium dynamics of symmetry-resolved entanglement and entanglement asymmetry: Exact asymptotics in Rule 54.



Dynamics of symmetry-resolved entanglement in Rule 54]
Non-equilibrium dynamics of symmetry-resolved entanglement and entanglement
asymmetry: Exact asymptotics in Rule 54



School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK



k.klobas@bham.ac.uk




  July 2024


§ ABSTRACT 
Symmetry resolved entanglement and entanglement asymmetry are two measures of quantum correlations sensitive to symmetries of the system. Here we discuss their non-equilibrium dynamics in the Rule 54 cellular automaton, a simple, yet interacting, integrable model. Both quantities can be expressed in terms of the more analytically tractable “charged moments”, i.e. traces of powers of a suitably deformed density matrix, via a replica trick. We express them in terms of a tensor network, which we contract in space using a system of local algebraic relations. This gives the asymptotic form for the charged moments, valid in the regime of large but finite time that is shorter than all the relevant subsystem sizes. In this regime the charge moments decay exponentially with the rate given by the leading solution to a cubic equation.


This paper is dedicated to the memory of Marko Medenjak (1990–2022). In the two years that we overlapped as PhD students and shared the office, he made a deep impact on me in my formative years as a researcher. I have very fond memories of long coffee breaks, during which we finished all the leftover seminar biscuits, and heated physics discussions that led to many scientific bets resulting in free pizza for the whole group. Marko has taught me many useful skills, perhaps the most notable being how to most efficiently roll on the chair from the desk to the whiteboard. He was one of the brightest people I have met, and had a long and successful career ahead of him when he tragically passed away. He will be sorely missed.




§ INTRODUCTION

The dynamics of entanglement in non-equilibrium quantum many-body systems show
remarkable universality. For instance, after a quantum
quench <cit.> the entanglement entropy
between a subsystem and the rest typically shows linear growth followed by
saturation at a value extensive in the subsystem size. This happens under very
mild conditions, and can be observed in a variety of systems, ranging from
integrable <cit.>
to
chaotic <cit.>.
The reason for this ubiquity of entanglement growth is intimately connected to
thermalisation: a quantum system equilibrates after the entanglement has spread
through the system, and the entanglement entropy has saturated at a value that
coincides with the thermal entropy of the reduced density matrix. The latter is
typically extensive regardless of the number of (local) conservation laws, and
therefore the same general dependence can emerge for both integrable and
chaotic dynamics.

The standard measure of (bipartite) entanglement is the entanglement entropy,
but there are many other measures of quantum
correlations <cit.>.  Here we
consider two examples that probe the interplay between entanglement and
symmetries (local conserved charges) of the system. The first one is the
symmetry resolved entanglement
entropy <cit.>,
defined as the entanglement entropy of a symmetric state reduced to a given
symmetry sector. The second one is the entanglement
asymmetry <cit.>, given by the relative entropy between a
non-symmetric state and its symmetrised counterpart. Recently, these quantities
have been extensively studied, both theoretically <cit.>, and
experimentally <cit.>. One of their appealing features for theoretical
investigation is the fact that they can be — through a replica trick —
recast in terms of so-called charged moments (see
Sec. <ref> for precise definition), which are particularly
convenient for analytical manipulations. Despite that, the vast majority of
exact results come from free theories, as the exact non-equilibrium
dynamics of interacting systems is beyond the reach of existing methods.
Refs. <cit.> proposed an
effective description for the dynamics of charged moments in interacting
integrable systems. It is based on the idea of exchanging the roles of space
and time <cit.>, which maps the non-equilibrium dynamics of
charged moments to a stationary quantity accessible through standard
tools <cit.>. Even though this
approach has quantitative predictive power, it is hard to rigorously prove it
in full generality, and one has to test its predictions against exact results.
Here we provide an example where such exact results can be found in closed
analytical form.

Specifically, we consider the reversible cellular automaton Rule
54 <cit.>, which is a model on a discrete lattice of qubits
with time-evolution defined in discrete time. In 
recent years this model has been recognised as an example of 
interacting integrable model <cit.>, whose
simplicity allows for exact characterization of various non-equilibrium
properties <cit.>
(see also a review <cit.>). The main technical tool we use here is
a generalization of the techniques introduced in
Refs. <cit.>,
which are based on space-time-duality ideas <cit.>.
In essence, one first recasts the charged moments as a tensor network, and then
contracts it in the space direction, which can be done exactly due to a
set of local algebraic relations fulfilled by the model. This procedure gives access to the charged moments for short times, that is, times smaller than the sizes of all the subsystems, reproducing the results of Refs. <cit.>.

The rest of the paper is organized as follows. In Sec. <ref> we present the precise definitions of the quantities of interest. In Sec. <ref>, we specialise them to the system and quench protocol considered, and express them in terms of a transfer matrix implementing the evolution in space. In Sec. <ref> we characterise the fixed points of this transfer matrix, which encode all the information on the short-time regime. The fixed points are then used in Sec. <ref> to work out the asymptotic form of charged moments. Finally, Sec. <ref> contains our closing remarks. Some of the technical steps are relegated to the two appendices.



§ THE QUANTITIES OF INTEREST

We assume quantum quench protocol: initially a closed quantum system is
prepared in a state |Ψ_0⟩, and then it is left to evolve unitarily so
that at time t the system is described by a pure state |Ψ_t⟩. We
wish to characterize correlations and fluctuations of conserved charges in the
system, which we will capture by the charged moments
Z_α,β⃗(t), defined with respect to an extensive conserved
charge Q,

    Q=∑_j q_j,      q_j=^⊗ j-1⊗ q ⊗^⊗ L-j-r+1

where q is a local operator that acts nontrivially on r consecutive
sites[In the general discussion we will always assume r=1, as we
implicitly rely on Q=Q_A⊕ Q_A̅. If r>1 this does not hold
anymore, but the corrections are subleading in the sizes of subsystems and time
t, therefore most of the discussion carries over to finite r.].
Using Q, we define the charged moments as

    Z_α,β⃗(t)=[
        e^i β_1 Q_Aρ_A(t)
        e^i β_2 Q_Aρ_A(t)⋯
        e^i β_α Q_Aρ_A(t)
      ],    β⃗=
      [
      β_1    β_2    ⋯   β_α],

where α∈ℕ is analogous to the Rényi index,
β⃗∈ℂ^α, and ρ_A(t), Q_A are the state and
charge reduced to a subsystem A,

    ρ_A(t)=_A̅|Ψ_t⟩⟨Ψ_t|,    
      Q_A=-14mu
      ∑_ j [j,j+r-1]⊆ A-14mu
      q_j.


Charged moments contain information about how correlations are shared between
the subsystem A and the complement A̅. For instance, in the limit
β⃗→0⃗, one recovers Rényi entanglement entropies,

    S_A^(α)(t)=1/1-α[ρ_A^α(t)]=
      1/1-αZ_α,0⃗(t).

Another special point is α→ 1, β⃗→β, in which case
the charged moments reduce to the full counting statistics,

    Z_1,β(t)=[^β Q_Aρ_A(t)].


For α>1, and generic β⃗, the charged moments themselves do not
immediately reduce to known quantities, but they can be related to Rényi
entanglement entropies of the state reduced to different symmetry sectors. To
see this, let us first consider a case when the initial state |Ψ_0⟩ 
is an eigenstate of the charge, which means that at time t the state
|Ψ_t⟩⟨Ψ_t| commutes with Q. Then we define ρ_A,q(t) as
the reduced state projected to the subspace with the value of charge equal to
q,

    ρ_A,q(t)=Π_qρ_A(t) Π_q=Π_qρ_A(t),
          Π_q=1/2π∫_-π^πβ̣^ıβ(Q_A-q),

where in the Fourier decomposition of the projector Π_q, we implicitly
assumed Q to have an integer spectrum, and q∈ℤ. We denote by
S_A,q^(α)(t) the n-Rényi entropy of ρ_A,q,

    S_A,q^(α)(t)=1/1-α[ρ_A,q^α(t)],

which is referred to as symmetry resolved Rényi entanglement
entropy <cit.>.
Using the above expression for the projector Π_q, we can express it in
terms of charged moments as [The initial state commutes with Q,
therefore in the leading order also ρ_A(t) commutes with Q_A, and
Z_α,β⃗(t) is the same for all β⃗ with the same sum
∑_j=1^αβ_j.]

    [ρ_A,q^α(t)]=
      1/2π∫_-π^πβ̣Z_α,[β00…0](t).


On the other hand, if the initial state is not an eigenstate of the
charge, an interesting question is how quickly does the reduced density matrix
become block-diagonal in the charge-eigenstate basis. This is measured by the
entanglement asymmetry <cit.>, Δ S_A(t),
which is defined as the relative entropy between the reduced density matrix at time t, ρ_A(t), and its symmetrized counterpart, ρ̅_A(t),

    Δ S_A(t)=-[ρ_A(t)(logρ̅_A(t)-logρ_A(t))],

with 

    ρ̅_A(t)=∑_ q∈ℤΠ_q ρ_A(t) Π_q.

The asymmetry can be conveniently expressed as a α→1 limit of the
Rényi entanglement asymmetry

    Δ S_A(t)= lim_α→ 1Δ S^(α)(t),    Δ S^(α)(t)=1/1-αlog[ρ̅^α_A(t)]]/[ρ^α_A(t)],

which can in turn be expressed in terms of the charged moments by using  
the Fourier representation of the projector Π_q (cf. eq:defProj),

    [ρ̅^α_A(t)]=∫_-π^πdβ_1 dβ_2⋯ dβ_α/(2π)^α
      Z_α,β⃗(t) δ(β_1+β_2+…+β_α).

Note that for a charge symmetric initial state, we have
Z_α,β⃗(t)=Z_α,0⃗(t), and therefore Δ S^(α)(t)=0.



§ THE SETUP



 §.§ The definition of the dynamics


The dynamics is defined on a system of 2L qubits, with the time-evolution
defined in two discrete time-steps, so that at time t the state of the system
is given as 

    |Ψ_t⟩ = (𝕌_o𝕌_e)^t |Ψ_0⟩,
          𝕌_e=∏_j U_2j,    𝕌_o=∏_j U_2j-1.

Here, U_j is an operator that acts as U to the three sites centred around j,

    U_j=^⊗ j-2⊗ U ⊗^⊗ 2 L-j-1,

and U is a 3-site local unitary operator given by the following
computational-basis matrix elements,

    s_1^' s_2^' s_3^'Us_1^' s_2^' s_3^'=
      δ_s_1^',s_1^'δ_s_2^',χ(s_1,s_2,s_3)δ_s_3^',s_3^',
    χ(s_1,s_2,s_3)≡ s_1+s_2+s_3+s_1 s_32.

The operator U deterministically changes the middle site depending on the
full 3-site configuration, while the left and right sites are left unchanged.
This implies that all the terms in 𝕌_ e/o mutually commute and can
be applied at the same time. The time-evolution can therefore be naturally
given in terms of the staggered tensor network pictured in Figure <ref>,
where the two circles represent two distinct tensors defined as

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
      -0.7500.7500-0.7500.7500colU at (-1,0) 0.8s_1;
       at (0,-1) 0.8s_2;
       at (1,0) 0.8s_3;
       at (0,1) 0.8s_4;
    =δ_s_1,s_2δ_s_2,s_3δ_s_3,s_4,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
      -0.7500.7500-0.7500.7500colU at (-1,0) 0.8s_1;
       at (0,-1) 0.8s_2;
       at (1,0) 0.8s_3;
       at (0,1) 0.8s_4;
    =δ_s_4,χ(s_1,s_2,s_3).




 §.§ The simplest conserved charges

The model is integrable <cit.> and
exhibits an infinite number of conservation laws. Here we will
consider the two conserved quantities with the shortest support,
Q^(+)=∑_j q^+_j, and Q^(-)=∑_j q^-_j,

    q_j^(+)=1-σ_j+1^z/4(2+σ_j+2^z+σ_j^zσ_j+2^z)
      ,    
      q_j^(-)=(-1)^j-8mu/4(1+σ_j^zσ_j+1^z-σ_j^z-σ_j+1^z),

which can be interpreted as the total number of particles, and their current respectively.
The support of the two operators is larger than 1, which implies that the object
e^iβ Q^±_A (where A is a finite subsystem) is not in a product form
(contrary to U(1) charges considered in <cit.>).
However, their finite support implies an efficient MPO representation with a finite bond
dimension. In particular, in the case of Q^(-), the auxiliary space is two-dimensional,
and the MPO has the following staggered structure,

    e^iβ Q^-_A=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        60.50+0.500.50colObsLines+0.50colObsLines in 1,3,...,-0.50.5+1-0.5+10.50colObscolObsLines+10colObscolObsLines,

where the vertical legs represent 1-qubit physical degrees of freedom, while the two
sets of matrices are

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        -0.500.500-0.500.500colObscolObsLines at (0,0.75) 0.8s;
         at (0,-0.75) 0.8b;
      =
      δ_s,0δ_b,0[ 1     0  1     0 ]
      +δ_s,1δ_b,1[0     1  0     e^i β]
      ,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        -0.500.500-0.500.500colObscolObsLines=
      .
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        -0.500.500-0.500.500colObscolObsLines|_β↔-β,

and the boundary vectors in the auxiliary space are chosen as

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        -0.5000-0.50colObsLines=
        .[1 0]-4mu.^T,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        0.50000.50colObsLines=[1 1].


Similarly, e^iβ Q^(+) can be also represented as an MPO,
but the larger support implies a larger bond dimension (in particular, it is 3),
and there is no even-odd staggering effect,

    e^iβ Q^+_A=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        60.50+0.500.50colObsLines+0.50colObsLines in 1,3,...,-0.50.5+1-0.5+10.50colObscolObsLines+10colObscolObsLines,

with 

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        -0.500.500-0.500.500colObscolObsLines at (0,0.75) 0.8s;
         at (0,-0.75) 0.8b;
      =
      δ_s,0δ_b,0[1   0   0 e^2 i β   0   0 1   0   0]
      +δ_s,1δ_b,1[0   1   0 0   0   e^i β 0   0   e^iβ]
      -6mu, [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        -0.5000-0.50colObsLines=
      .[0 e^-iβ/1+e^iβ1/1+e^iβ]-4mu.^T-14mu, [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.6]
        0.50000.50colObsLines=
      [
        1 
      1 
      e^-iβ]-6mu.

Note that for simplicity we use the same notation for the auxiliary degrees of freedom
and boundary vectors in the two cases, but it is always clear from the context which one
is used.



 §.§ Initial states

The reduced density matrix ρ_A(t) is the state ρ(t)=|Ψ_t⟩⟨Ψ_t|
reduced to the subsystem A, where the initial state |Ψ_0⟩ is chosen to be
from the family of solvable initial states introduced
in <cit.>,

    |Ψ_0⟩=(|ψ_1⟩⊗|ψ_2⟩)^⊗ L-8mu,  |ψ_1⟩=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        1010.7510colIstcolLines=
      [1 0 ]-4mu,  |ψ_2⟩=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        1010.7510colIstcolLines
      =[√(1-ϑ)√(ϑ)]-4mu,  
      0<ϑ<1.

After being initialized in one of these states, the system can be
shown <cit.> to relax to the family of Gibbs states
determined by the first one of the conserved quantities introduced above,
ρ∼ e^μ Q^(+), with the chemical potential μ=log(ϑ^-1-1).



 §.§ Space evolution

The definitions above completely specify our setup, and by denoting the Hermitian
conjugates of time-evolution tensors eq:defTETens with red,[We note that in our case the red and green tensors are the same, as the model is left-right symmetric, deterministic, and reversible. However, the formulation with two different colours naturally generalises to a more general setting, therefore we decided to keep the distinction between the two sets of tensors.]

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
      -0.7500.7500-0.7500.7500colUc=
    .
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        -0.7500.7500-0.7500.7500colU.^†,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
      -0.7500.7500-0.7500.7500colUc=
    .
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        -0.7500.7500-0.7500.7500colU.^†,

we are able to represent e.g. Z_1,β^(+)(l,t) as shown in Fig. <ref>. 
The graphical representation of the tensor network suggests that Z_1,β^(+) can
be equivalently contracted in space rather than in time. In particular, we introduce
Hilbert space of the temporal chain  of 2t qubits,

    𝒱_t=(ℂ^2)^⊗ 2t,

and define transfer matrices 𝕎_t, 𝕎_β,t^(±) that act on
2× 2t vertical lattice sites, and (possibly) the auxiliary degree of freedom, 

    𝕎_t∈(𝒱_t⊗𝒱_t),  𝕎_β,t^(+)∈(𝒱_t⊗ℂ^3⊗𝒱_t),  𝕎_β,t^(-)∈(𝒱_t⊗ℂ^2⊗𝒱_t).

In particular, space transfer matrices are built by two consecutive rows of 
the diagram in Fig. <ref> (and an analogous one corresponding to
Z_1,β^(-)(l,t)),

    𝕎_t=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        41012*+22022*+2 in 1,2,...,0.252.750.25+1+2.75+1+ in 1,3,...,1colU2colU1+1colU2+1colU1++1colUc2++1colUc1++2colUc2++2colUc10colIstcolLines20colIstcolLines12*+2colIstCcolLines22*+2colIstCcolLines,    𝕎_β,t^(+)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        41012*+22022*+2 in 1,2,...,0.252.750.25+1+2.75+1+ in 1,3,...,1colU2colU1+1colU2+1colU1++1colUc2++1colUc1++2colUc2++2colUc10colIstcolLines20colIstcolLines12*+2colIstCcolLines22*+2colIstCcolLines0.25+12.75+11+1colObscolObsLines2+1colObscolObsLines,    𝕎_β,t^(-)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        41012*+22022*+2 in 1,2,...,0.252.750.25+1+2.75+1+ in 1,3,...,1colU2colU1+1colU2+1colU1++1colUc2++1colUc1++2colUc2++2colUc10colIstcolLines20colIstcolLines12*+2colIstCcolLines22*+2colIstCcolLines0.25+12.75+11+1colObscolObsLines2+1colObscolObsLines.

In terms of these transfer matrices the moments 
Z_1,β^(±)(l,t) take the following form,

    Z_1,β^(±)(l,t)=
      [ 
      𝕎^L-l_t
      𝔹_ L,β,t^(±).
      𝕎_β,t^(±).^l𝔹_ R,β,t^(±)],

where 𝔹_ L/R,β,t^(±) act as identity on both 𝕍_t,
and as boundary vectors
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
    0.50000.50colObsLines,
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
    -0.5000-0.50colObsLines
on the auxiliary degree of freedom (see Fig. <ref>).



More complicated charged moments Z_α,β⃗^(±)(l,t) are obtained
by considering α copies of transfer matrices that are connected in a staggered way,

    Z_α,β⃗^(±)(l,t)=
      [ 
      ℙ_α,t^†(𝕎_t^⊗α)^L-lℙ_α,t𝔹_ L,α,β⃗,t^(±)(
      𝕎_β_1,t^(±)⊗𝕎_β_2,t^(±)⊗⋯⊗𝕎_β_α,t^(±))^l𝔹_ R,α,β⃗,t^(±)],

where 𝔹_ L/R,α,β⃗,t^(±) is the
multi-copy generalization of 𝔹_ L/R,β,t^(±),

    𝔹_ L/R,α,β⃗,t^(±)=
      𝔹_ L/R,β_1,t^(±)⊗𝔹_ L/R,β_2,t^(±)⊗⋯⊗𝔹_ L/R,β_α,t^(±),

and ℙ_α,t∈(𝒱_t^⊗ 2α) 
is a permutation operator that periodically shifts even vectors in
a tensor product of 2α states from 𝒱_t,

    ℙ_α,t|x_1⟩⊗|x_2⟩⊗⋯⊗|x_2 α⟩ =
      |x_1⟩⊗|x_2α⟩⊗|x_3⟩⊗|x_2⟩⊗⋯⊗|x_2α-2⟩.




§ FIXED-POINTS OF THE SPACE TRANSFER MATRIX

The representation of charged moments in terms of space transfer matrices (cf.eq:CMspace) in general offers no benefit as it is completely equivalent
to the definition eq:genChargedMomentDef.  However, the locality and
unitarity of time-evolution imply that 𝕎_t has a unique maximal
eigenvalue 1, all the other eigenvalues are equal to 0, and the largest
Jordan block is of size at most 2t+1. Therefore, a high-enough power of
𝕎_t can be substituted by a projector to its fixed points
(i.e. leading left and right eigenvectors),

    𝕎_t |r_t⟩=|r_t⟩,    ⟨l_t|𝕎_t=⟨l_t|,    .𝕎_t^x|_x>2t+1=|r_t⟩⟨l_t|.

The additional auxiliary degree of freedom due to e^iβ Q^(±)_A
changes the spectral properties of the transfer matrix
𝕎_β,t^(±), so that the leading eigenvalue is no longer
necessarily 1. Nonetheless, as is shown in <ref>, the
leading eigenvalue Λ_β^(±) is t-independent and unique, and
a repeated application of the transfer matrix again projects to the leading
eigenvectors, 

    𝕎_β,t^(±)r_β,t^(±)=Λ_β^(±)r_β,t^(±),    l_β,t^(±)𝕎_β,t^(±)=Λ_β^(±)l_β,t^(±),
    lim_x→∞(1/Λ_β^(±)𝕎_β,t^(±))^x
        =r_β,t^(±)l_β,t^(±)/r_β,t^(±)l_β,t^(±).

Therefore in the regime of large system and subsystem sizes (compared to t)
the behaviour of charged moments Z_α,β⃗^(±)(l,t) is
completely determined by the properties of leading eigenvalues Λ_β^(±),
and the corresponding eigenvectors
r_β,t^(±), l_β,t^(±), r_t, 
l_t.



 §.§ Fixed points of 𝕎_t

We start by summarizing the construction of the fixed points of 𝕎_t,
which was introduced in Refs. <cit.>.
The fixed points can be conveniently expressed in the following MPO form,

    ⟨l_t|=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        [thick,colLines,rounded corners] (0.75,4) – (-0.75,4) – (-0.75,6) – (0.75,6);
        [thick,colLines,rounded corners] (0.75,3) – (-0.875,3) – (-0.875,7) – (0.75,7);
        [thick,colLines,rounded corners] (0.75,2) – (-1,2) – (-1,8) – (0.75,8);
        [thick,colLines,rounded corners] (0.75,1) – (-1.125,1) – (-1.125,9) – (0.75,9);
        000500colMPS01colMPS02colMPS03colMPS04colMPS05,    |r_t⟩=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        [thick,colLines,rounded corners] (-0.75,4) – (0.75,4) – (0.75,6) – (-0.75,6);
        [thick,colLines,rounded corners] (-0.75,3) – (0.875,3) – (0.875,7) – (-0.75,7);
        [thick,colLines,rounded corners] (-0.75,2) – (1,2) – (1,8) – (-0.75,8);
        [thick,colLines,rounded corners] (-0.75,1) – (1.125,1) – (1.125,9) – (-0.75,9);
        000500colMPS01colMPS02colMPS03colMPS04colMPS05,

where the ticker vertical line represents the auxiliary vector space. It is
straightforward to see that the ansatz for ⟨l_t| gives the left fixed
point of the transfer matrix, if the following set of algebraic relations is
fulfilled,

    [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504.75[thick,colLines,rounded corners](1.75,4) – (-0.5,4) – (-0.5,5.5) – (1.75,5.5);
        [thick,colLines,rounded corners](1.75,3) – (-0.625,3) – (-0.625,6.5) – (1.75,6.5);
        [thick,colLines,rounded corners](1.75,2) – (-0.75,2) – (-0.75,7.5) – (1.75,7.5);
        [thick,colLines,rounded corners](1.75,1) – (-0.875,1) – (-0.875,8.5) – (1.75,8.5);
        012colMPS03colMPS04colMPS121415.517.512colU13colU14colU15.5colUc16.5colUc17.5colUc=
      [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504.75[thick,colLines,rounded corners]
        (0.75,4) – (-0.5,4) – (-0.5,5.5) – (0.75,5.5);
        [thick,colLines,rounded corners]
        (0.75,3) – (-0.625,3) – (-0.625,6.5) – (0.75,6.5);
        [thick,colLines,rounded corners]
        (0.75,2) – (-0.75,2) – (-0.75,7.5) – (0.75,7.5);
        [thick,colLines,rounded corners]
        (0.75,1) – (-0.875,1) – (-0.875,8.5) – (0.75,8.5);
        034colMPS02colMPS01colMPS,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0002.75[thick,colLines,rounded corners] (1.75,2) – (-0.5,2) – (-0.5,3.5) – (1.75,3.5);
        [thick,colLines,rounded corners] (1.75,1) – (-0.625,1) – (-0.625,4.5) – (1.75,4.5);
        00colMPS01colMPS02colMPS101213.515.512colU11colU13.5colUc14.5colUc10colIstcolLines15.5colIstCcolLines=
      [baseline=([yshift=1.5ex]current bounding box.center),scale=0.5]
        0002.75[thick,colLines,rounded corners] (0.75,2) – (-0.5,2) – (-0.5,3.5) – (0.75,3.5);
        [thick,colLines,rounded corners] (0.75,1) – (-0.625,1) – (-0.625,4.5) – (0.75,4.5);
        00colMPS012colMPS,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0003.75[thick,colLines,rounded corners] (1.75,3) – (-0.5,3) – (-0.5,4.5) – (1.75,4.5);
        [thick,colLines,rounded corners] (1.75,2) – (-0.625,2) – (-0.625,5.5) – (1.75,5.5);
        [thick,colLines,rounded corners] (1.75,1) – (-0.75,1) – (-0.75,6.5) – (1.75,6.5);
        00colMPS01colMPS02colMPS03colMPS101314.517.513colU12colU11colU14.5colUc15.5colUc16.5colUc10colIstcolLines17.5colIstCcolLines=
      [baseline=([yshift=1.5ex]current bounding box.center),scale=0.5]
        0003.75[thick,colLines,rounded corners] (0.75,3) – (-0.5,3) – (-0.5,4.5) – (0.75,4.5);
        [thick,colLines,rounded corners] (0.75,2) – (-0.625,2) – (-0.625,5.5) – (0.75,5.5);
        [thick,colLines,rounded corners] (0.75,1) – (-0.75,1) – (-0.75,6.5) – (0.75,6.5);
        00colMPS01colMPS023colMPS,
    [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (1.75,3) – (-0.5,3) – (-0.5,5) – (1.75,5);
        [thick,colLines,rounded corners] (1.75,2) – (-0.625,2) – (-0.625,6) – (1.75,6);
        [thick,colLines,rounded corners] (1.75,1) – (-0.75,1) – (-0.75,7) – (1.75,7);
        04012colMPS03colMPS121612colU13colU15colUc16colUc=
      [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (0.75,3) – (-0.5,3) – (-0.5,5) – (0.75,5);
        [thick,colLines,rounded corners] (0.75,2) – (-0.625,2) – (-0.625,6) – (0.75,6);
        [thick,colLines,rounded corners] (0.75,1) – (-0.75,1) – (-0.75,7) – (0.75,7);
        0401colMPS02colMPS03colMPS,    [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (1.75,2) – (-0.5,2) – (-0.5,4) – (1.75,4);
        [thick,colLines,rounded corners] (1.75,1) – (-0.625,1) – (-0.625,5) – (1.75,5);
        03012colMPS121412colU14colUc=
      [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (0.75,2) – (-0.5,2) – (-0.5,4) – (0.75,4);
        [thick,colLines,rounded corners] (0.75,1) – (-0.625,1) – (-0.625,5) – (0.75,5);
        0302colMPS01colMPS,

and an analogous set of left-right flipped relations ensures that |r_t⟩ is
the right fixed point. A solution to these relations exists for the 3-dimensional
auxiliary vector space <cit.>,

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62500.625-0.5500.550 at (-0.75,0) 0.80;
           at (0.75,0) 0.80;
          00colMPS=
        [
          1-ϑ   1-ϑ   -1+ϑϑ   ϑ   1-ϑϑ   -ϑ^2/1-ϑ   -ϑ],
       [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62500.625-0.5500.550 at (-0.75,0) 0.80;
           at (0.75,0) 0.81;
          00colMPS=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62500.625-0.5500.550 at (-0.75,0) 0.81;
           at (0.75,0) 0.80;
          00colMPS=
        [
          0   1-ϑ   -1+ϑϑ   0   0ϑ   0   0
        ],       [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
            0-0.62500.625-0.5500.550 at (-0.75,0) 0.81;
             at (0.75,0) 0.81;
            00colMPS=
          [
            0   1   0
            1   0   0
            0   0   0
          ],
       [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
            0-0.62500.625-0.5500.550 at (-0.75,0) 0.8s;
             at (0.75,0) 0.8b;
            00colMPS=
          δ_b,s[δ_s,0   0   0
            0    δ_s,1    0 
            0     0    δ_s,1],

while the two-step bulk tensor
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
  00.750-0.75-0.5-0.250.5-0.25-0.50.250.50.250-0.250.25colMPS,
and the boundary vectors 
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
  000-0.500,
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
  0000.500colMPS, and
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
  0000.500colMPS, are reported in <ref>.
Here the parameter ϑ is set by the initial state eq:defSolvableIS.



 §.§ Leading eigenvectors of generalized transfer matrices

We use the ideas summarized above to find leading eigenvectors of
𝕎_β,t^(±). We again assume an MPO representation similar
to eq:defAnstazFPmpo,

    l_β,t^(+)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        [thick,colLines,rounded corners] (0.75,4) – (-0.75,4) – (-0.75,6) – (0.75,6);
        [thick,colLines,rounded corners] (0.75,3) – (-0.875,3) – (-0.875,7) – (0.75,7);
        [thick,colLines,rounded corners] (0.75,2) – (-1,2) – (-1,8) – (0.75,8);
        [thick,colLines,rounded corners] (0.75,1) – (-1.125,1) – (-1.125,9) – (0.75,9);
        0005050.75500colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta04colMPSBeta05colMPSBeta in 0,...,50,    r_β,t^(+)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        [thick,colLines,rounded corners] (-0.75,4) – (0.75,4) – (0.75,6) – (-0.75,6);
        [thick,colLines,rounded corners] (-0.75,3) – (0.875,3) – (0.875,7) – (-0.75,7);
        [thick,colLines,rounded corners] (-0.75,2) – (1,2) – (1,8) – (-0.75,8);
        [thick,colLines,rounded corners] (-0.75,1) – (1.125,1) – (1.125,9) – (-0.75,9);
        000505-0.75500colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta04colMPSBeta05colMPSBeta in 0,...,50,    l_β,t^(-)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        [thick,colLines,rounded corners] (0.75,4) – (-0.75,4) – (-0.75,6) – (0.75,6);
        [thick,colLines,rounded corners] (0.75,3) – (-0.875,3) – (-0.875,7) – (0.75,7);
        [thick,colLines,rounded corners] (0.75,2) – (-1,2) – (-1,8) – (0.75,8);
        [thick,colLines,rounded corners] (0.75,1) – (-1.125,1) – (-1.125,9) – (0.75,9);
        0005050.75500colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta04colMPSBeta05colMPSBeta in 0,...,50,    r_β,t^(-)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        [thick,colLines,rounded corners] (-0.75,4) – (0.75,4) – (0.75,6) – (-0.75,6);
        [thick,colLines,rounded corners] (-0.75,3) – (0.875,3) – (0.875,7) – (-0.75,7);
        [thick,colLines,rounded corners] (-0.75,2) – (1,2) – (1,8) – (-0.75,8);
        [thick,colLines,rounded corners] (-0.75,1) – (1.125,1) – (1.125,9) – (-0.75,9);
        000505-0.75500colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta04colMPSBeta05colMPSBeta in 0,...,50.

Note that in the case of Q^(+) we assume the left and right eigenvectors to
be made up of the same bulk tensors, while for Q^(-) they change. This is
due to the symmetry of charges: Q^(+) is even under left-right reflection,
while Q^(-) is odd.[On the other hand, the top boundary vectors do not share this property: they are generically assumed to be different since the matrices defining the MPO representation of e^iβ Q^(±) (cf. eq:MPOqmDef and eq:MPOqpDef) are not symmetric.]

To find tensors that constitute fixed points we again assume a set of appropriate local algebraic relations. In particular, the relations fulfilled by the bottom and bulk tensors only (i.e. those given by Eq. eq:localRelationsLeftBulkBottom) stay unchanged, while the ones involving top boundary vectors (cf. Eq. eq:localRelationsLeftTop) have to be amended to allow for the additional MPO auxiliary space,

    [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (1.75,3) – (-0.5,3) – (-0.5,5) – (1.75,5);
        [thick,colLines,rounded corners] (1.75,2) – (-0.625,2) – (-0.625,6) – (1.75,6);
        [thick,colLines,rounded corners] (1.75,1) – (-0.75,1) – (-0.75,7) – (1.75,7);
        041.75404colMPSBeta012colMPSBeta03colMPSBeta121612colU13colU14colObscolLines15colUc16colUc in 1,...,40=
      [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (0.75,3) – (-0.5,3) – (-0.5,5) – (0.75,5);
        [thick,colLines,rounded corners] (0.75,2) – (-0.625,2) – (-0.625,6) – (0.75,6);
        [thick,colLines,rounded corners] (0.75,1) – (-0.75,1) – (-0.75,7) – (0.75,7);
        040.75404colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta in 1,...,40,  [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (1.75,2) – (-0.5,2) – (-0.5,4) – (1.75,4);
        [thick,colLines,rounded corners] (1.75,1) – (-0.625,1) – (-0.625,5) – (1.75,5);
        031.75303colMPSBeta012colMPSBeta121412colU13colObscolLines14colUc in 1,...,30=
      Λ_β^(+)[baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (0.75,2) – (-0.5,2) – (-0.5,4) – (0.75,4);
        [thick,colLines,rounded corners] (0.75,1) – (-0.625,1) – (-0.625,5) – (0.75,5);
        030.75303colMPSBeta02colMPSBeta01colMPSBeta in 1,...,30,    [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (1.75,3) – (-0.5,3) – (-0.5,5) – (1.75,5);
        [thick,colLines,rounded corners] (1.75,2) – (-0.625,2) – (-0.625,6) – (1.75,6);
        [thick,colLines,rounded corners] (1.75,1) – (-0.75,1) – (-0.75,7) – (1.75,7);
        041.75404colMPSBeta012colMPSBeta03colMPSBeta121612colU13colU14colObscolLines15colUc16colUc in 1,...,40=
      [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (0.75,3) – (-0.5,3) – (-0.5,5) – (0.75,5);
        [thick,colLines,rounded corners] (0.75,2) – (-0.625,2) – (-0.625,6) – (0.75,6);
        [thick,colLines,rounded corners] (0.75,1) – (-0.75,1) – (-0.75,7) – (0.75,7);
        040.75404colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta in 1,...,40,  [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (1.75,2) – (-0.5,2) – (-0.5,4) – (1.75,4);
        [thick,colLines,rounded corners] (1.75,1) – (-0.625,1) – (-0.625,5) – (1.75,5);
        031.75303colMPSBeta012colMPSBeta121412colU13colObscolLines14colUc in 1,...,30=
      Λ_β^(-)[baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (0.75,2) – (-0.5,2) – (-0.5,4) – (0.75,4);
        [thick,colLines,rounded corners] (0.75,1) – (-0.625,1) – (-0.625,5) – (0.75,5);
        030.75303colMPSBeta02colMPSBeta01colMPSBeta in 1,...,30,

while the corresponding set of relations for the top boundary vectors of right fixed points read as

    [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (-1.75,3) – (0.5,3) – (0.5,5) – (-1.75,5);
        [thick,colLines,rounded corners] (-1.75,2) – (0.625,2) – (0.625,6) – (-1.75,6);
        [thick,colLines,rounded corners] (-1.75,1) – (0.75,1) – (0.75,7) – (-1.75,7);
        04-1.75404colMPSBeta012colMPSBeta03colMPSBeta-12-16-12colU-13colU-14colObscolLines-15colUc-16colUc in 1,...,40=
      [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (-0.75,3) – (0.5,3) – (0.5,5) – (-0.75,5);
        [thick,colLines,rounded corners] (-0.75,2) – (0.625,2) – (0.625,6) – (-0.75,6);
        [thick,colLines,rounded corners] (-0.75,1) – (0.75,1) – (0.75,7) – (-0.75,7);
        04-0.75404colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta in 1,...,40,  [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (-1.75,2) – (0.5,2) – (0.5,4) – (-1.75,4);
        [thick,colLines,rounded corners] (-1.75,1) – (0.625,1) – (0.625,5) – (-1.75,5);
        03-1.75303colMPSBeta012colMPSBeta-12-14-12colU-13colObscolLines-14colUc in 1,...,30=
      Λ_β^(+)[baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (-0.75,2) – (0.5,2) – (0.5,4) – (-0.75,4);
        [thick,colLines,rounded corners] (-0.75,1) – (0.625,1) – (0.625,5) – (-0.75,5);
        03-0.75303colMPSBeta02colMPSBeta01colMPSBeta in 1,...,30,    [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (-1.75,3) – (0.5,3) – (0.5,5) – (-1.75,5);
        [thick,colLines,rounded corners] (-1.75,2) – (0.625,2) – (0.625,6) – (-1.75,6);
        [thick,colLines,rounded corners] (-1.75,1) – (0.75,1) – (0.75,7) – (-1.75,7);
        04-1.75404colMPSBeta012colMPSBeta03colMPSBeta-12-16-12colU-13colU-14colObscolLines-15colUc-16colUc in 1,...,40=
      [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2504[thick,colLines,rounded corners] (-0.75,3) – (0.5,3) – (0.5,5) – (-0.75,5);
        [thick,colLines,rounded corners] (-0.75,2) – (0.625,2) – (0.625,6) – (-0.75,6);
        [thick,colLines,rounded corners] (-0.75,1) – (0.75,1) – (0.75,7) – (-0.75,7);
        04-0.75404colMPSBeta01colMPSBeta02colMPSBeta03colMPSBeta in 1,...,40,  [baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (-1.75,2) – (0.5,2) – (0.5,4) – (-1.75,4);
        [thick,colLines,rounded corners] (-1.75,1) – (0.625,1) – (0.625,5) – (-1.75,5);
        03-1.75303colMPSBeta012colMPSBeta-12-14-12colU-13colObscolLines-14colUc in 1,...,30=
      Λ_β^(-)[baseline=([yshift=0.7ex]current bounding box.center),scale=0.5]
        00.2503[thick,colLines,rounded corners] (-0.75,2) – (0.5,2) – (0.5,4) – (-0.75,4);
        [thick,colLines,rounded corners] (-0.75,1) – (0.625,1) – (0.625,5) – (-0.75,5);
        03-0.75303colMPSBeta02colMPSBeta01colMPSBeta in 1,...,30.

Combining the relevant equations we obtain a set of algebraic relations
involving a finite number of variables. Upon solving these we again find
a solution with a 3-dimensional representation, where the bulk tensors
are simply related to the original ones eq:originalFixedPointTensors,

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.62500.625-0.5500.55000colMPSBeta00=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.62500.625-0.5500.55000colMPSBeta00=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.62500.625-0.5500.55000colMPSBeta00=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.62500.625-0.5500.55000colMPS,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.62500.625-0.5500.55000colMPSBeta00=
      [baseline=([yshift=-1.2ex]current bounding box.center),scale=0.5]
        0-0.62501-0.5500.55000colMPS00.625colMPSBeta00.625,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.62500.625-0.5500.55000colMPSBeta00=
      [baseline=([yshift=-1.2ex]current bounding box.center),scale=0.5]
        0-0.62501-0.5500.55000colMPS00.625colMPSBeta00.625,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.62500.625-0.5500.55000colMPSBeta00=
      [baseline=([yshift=-1.2ex]current bounding box.center),scale=0.5]
        0-0.62501-0.5500.55000colMPS00.625colMPSBeta00.625,

where
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
  00.100.900.5colMPSBeta00.5,
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
  00.100.900.5colMPSBeta00.5, and
[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
  00.100.900.5colMPSBeta00.5, are diagonal matrices in the auxiliary space,

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        0-0.500.500colMPSBeta=
      [
        1     0     0 
        0    γ    0 
        0     0    γ],    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.500.500colMPSBeta00=
        .
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.500.500colMPSBeta|_γ→ e^-iβ,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.500.500colMPSBeta00=
        .
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.500.500colMPSBeta|_γ→ e^iβ,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.500.500colMPSBeta00=
        .
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.500.500colMPSBeta|_γ→ e^iβΛ_β^(+).

Note that the bulk tensors in l_β,t^(-) and 
r_β,t^(-) are related to each other through a flip of sign of β.
The remaining tensors and boundary vectors are reported in <ref>.
The corresponding eigenvalues are determined as (see <ref>),

    logΛ_β^(±)=
      lim_L→∞1/LlogΨ_0e^iβ Q^(±)Ψ_0,

and are

    Λ_β^(+)=1-ϑ+ϑ e^2iβ,    Λ_β^(-)=1.


Finally, we note that the fixed points are for convenience not normalized, but rather
we keep the overlap between the right and left eigenvectors to be 

    l_β,t^(+)r_β,t^(+)=.Λ_β^(+).^2t,
          l_β,t^(-)r_β,t^(-)=1=.Λ_β^(-).^2t.

See <ref> for details.



§ ASYMPTOTICS OF CHARGED MOMENTS

When the subsystem size l and the system size L are sufficiently large
compared to the time t (in particular, when l,L>2t+1) the charged moments
factorize into the two contributions corresponding to the edges of the
subsystem,

    lim_l,L→∞Z_α,β⃗^(±)(l,t)/∏_j=1^α.Λ_β_j^(±).^l
      =
      z_α,β⃗^ L (±)(t) 
      z_α,β⃗^ R (±)(t)/∏_j=1^α.Λ_β_j^(±).^2t,

where z_α,β⃗^ L/R  (±) are the contributions
given by the (un-normalized) leading eigenvectors,

    z_α,β⃗^ L  (±)=
        l_t^⊗α ℙ_α,t(
        𝔹_ L,β_1,t^(±)r_β_1,t^(±)⊗𝔹_ L,β_2,t^(±)r_β_2,t^(±)⊗⋯⊗𝔹_ L,β_α,t^(±)r_β_α,t^(±)),
    
        z_α,β⃗^ R  (±)=
        (
        l_β_1,t^(±)𝔹_ R,β_1,t^(±)⊗l_β_2,t^(±)𝔹_ R,β_2,t^(±)⊗⋯⊗l_β_α,t^(±)𝔹_ R,β_α,t^(±)) ℙ_α,t^†r_t^⊗α.

In terms of the fixed-point tensors, these expressions take form of
2t× 2α tensor networks,

    z_α,β⃗^ L  (-)(t)
      =
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
         in 1,...,40.46.60.4-0.46.6-0.40.46.6 in 1,...,605 in 2,4,6 in 1,3
            +0.45 in 1,3,50colMPS5+10colMPS in 1,3colMPS+1colMPS+1colMPS+1+1colMPS in 2,4,65-0.55-0.55.25-0.54.755colMPSBeta5 in 1,3+0.45colMPSBeta+0.45 at (2,-0.75) 0.8β_1;
         at (4,-0.75) 0.8⋯;
         at (6,-0.75) 0.8β_α;
      ,    
      z_α,β⃗^ R  (-)(t)
      =
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
         in 1,...,40.46.60.4-0.46.6-0.40.46.6 in 1,...,605 in 1,3,5 in 2+0.45 in 1,3,50colMPS+15+10colMPS in 1,3colMPS+1colMPS+1colMPS+1+1colMPS in 1,3,55+0.55+0.55.25+0.54.755colMPSBeta5 in 2+0.45colMPSBeta+0.45 in 4+0.5colMPSBeta+0.5 at (1,-0.75) 0.8β_1;
         at (3,-0.75) 0.8⋯;
         at (5,-0.75) 0.8β_α;
      ,

where the label under a column denotes the value of β in which the
tensors in that column are evaluated. The diagrams representing
z_α,β⃗^ L/R (+)(t) are analogous, only the tensors
with  and  are replaced with the corresponding  tensors.

These tensor networks are hard to contract for finite t, but we are able to
characterize their large-t asymptotics. We start by introducing the transfer
matrix 𝒯_α,γ⃗, defined as

    𝒯_α,γ⃗:=
      [baseline=([yshift=0ex]current bounding box.center),scale=0.5]
         in 1,20.46.60.4-0.46.6-0.40.46.6 in 1,...,60.252.75 in 2,4,61.45 in 1,3,5 in 1colMPS+1colMPS+1colMPS+1+1colMPS in 2,4,61.45colMPSBeta at (2,0) 0.8γ_1;
         at (4,0) 0.8⋯;
         at (6,0) 0.8γ_α;
      ,

with generic values of γ_j.  All four relevant tensor networks can be
reproduced by repeatedly applying 𝒯_α,γ⃗ and
appropriately setting bottom and top vectors. Moreover, assuming that
𝒯_α,γ⃗ has a unique leading eigenvalue
λ_α,γ⃗, the boundary vectors play no role in the
asymptotic behaviour of left and right contributions. In this case the
asymptotic slopes of the logarithms of z^ L/R
(±)_α,β⃗ are completely determined by the leading
eigenvalue of the transfer matrix,

    s_α,β⃗^(+):=
      lim_t→∞log z^ R (+)_α,β⃗(t)/t=
      lim_t→∞log z^ L (+)_α,β⃗(t)/t=
      .
      logλ_α,γ⃗|_γ_j→ e^iβ_jΛ_β_j^(+),

and similarly

    s_α,β⃗^(-):=
      lim_t→∞log z^ R (-)_α,β⃗(t)/t=
      lim_t→∞log z^ L (-)_α,-β⃗(t)/t=
      .
      logλ_α,γ⃗|_γ_j→ e^-iβ_j.

Note that the symmetry of Q^(-) again implies the flip of the sign of
β⃗ between the contributions of the left and right boundary.



 §.§ Spectrum of the transfer matrix

The spectrum of 𝒯_α,γ⃗ can be found by following
and appropriately modifying the approach of Ref. <cit.>. We first 
clump together two vertical legs and two horizontal legs into single degrees of freedom,

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
         in 1,20.42.6 in 1,20.252.75 in 1 in 1colMPS+1colMPS+1colMPS+1+1colMPS21.45colMPSBeta at (0,2) 0.8s_1;
         at (0,1) 0.8s_2;
         at (3,2) 0.8b_1;
         at (3,1) 0.8b_2;
         at (1,0) 0.8v_1;
         at (2,0) 0.8v_2;
         at (1,3) 0.8w_1;
         at (2,3) 0.8w_2;
      =:
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        -0.800.800-0.800.800colM at (0,1) 0.83 w_1+w_2;
         at (0,-1) 0.83 v_1+v_2;
         at (-1,0) 0.82 s_1+s_248mu;
         at (1,0) 0.848mu 2 b_1+b_2;
      ,

so that the transfer matrix is expressed as [Note that we are
suppressing the subscripts γ_j to lighten the notation. Everywhere in
this section each column is assumed to correspond to a different value of γ_j
unless explicitly stated.]

    𝒯_α,γ⃗=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        -0.5-0.52.5-0.5-0.502.50-0.502.50 in 0,...,2[white] (-0.2,-1) rectangle (+0.2,1);
          -110colM.

We then perform a convenient local similarity transformation,

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        -0.800.800-0.800.800colMt
      =
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        -0.800.800-1.7501.75[colObsLines,fill=white] (0,1) circle (0.35);
        [colObsLines,fill=white] (0,-1) circle (0.35);
         at (0,1) -;
        00colM,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        00.750-0.75[colObsLines,fill=white] (0,0) circle (0.35);
      =P,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        00.750-0.75[colObsLines,fill=white] (0,0) circle (0.35);
         at (0,0) -;
      =P^-1,
 
with the local matrix P given as,

    P=
      [
        1     0     0     0     0     0     0     0     0
        0     1     1     0     0     0     0     0     0
        0     1    -ϑ/1-ϑ    0     0     0     0     0     0
        0     0     0     1     0     0     1     0     0
        0     0     0     0     1     1     0     1     1
        0     0     0     0     1    -ϑ/1-ϑ    0    
        1    -ϑ/1-ϑ
        0     0     0     1     0     0    -ϑ/1-ϑ    0     0
        0     0     0     0    -ϑ/1-ϑ   -ϑ/1-ϑ    0     1     1 
        0   0   0   0   -ϑ/1-ϑ   ϑ^2/(1-ϑ)^2   0   
        1   -ϑ/1-ϑ].

This gives us a transformed transfer matrix
𝒯̃_α,γ⃗ that has the same spectrum as
𝒯_α,γ⃗,

    𝒯̃_α,γ⃗=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
        -0.5-0.52.5-0.5-0.502.50-0.502.50 in 0,...,2[white] (-0.2,-1) rectangle (+0.2,1);
          -110colMt.

The transformed tensors eq:transformedTensorsENT satisfy Lemma 1 from
Ref. <cit.>. Namely, for any integer power m≥ 1
the following holds,

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
         in 0,2,6,8,100.252.751+0.751-0.752+0.752-0.751colMt2colMt at (1,11) 0.8v_1;
         at (2,11) 0.8w_1;
         at (1,9) 0.8v_2;
         at (2,9) 0.8w_2;
         at (1,7) 0.8v_3;
         at (2,7) 0.8w_3;
         at (1,5) 0.8v_4;
         at (2,5) 0.8w_4;
         at (1.5,4.25) ⋮;
         at (0.875,3) 0.78v_m-1;
         at (2.125,3) 0.78w_m-1;
         at (1,1) 0.8v_m;
         at (2,1) 0.8w_m;
         at (1,-1) 0.8v_1;
         at (2,-1) 0.8w_1;
      
      =∏_j=1^mδ_w_j,v_j[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
         in 0,2,6,8,100.252.751+0.751-0.752+0.752-0.751colMt2colMt at (1,11) 0.8v_1;
         at (2,11) 0.8v_1;
         at (1,9) 0.8v_2;
         at (2,9) 0.8v_2;
         at (1,7) 0.8v_3;
         at (2,7) 0.8v_3;
         at (1,5) 0.8v_4;
         at (2,5) 0.8v_4;
         at (1.5,4.25) ⋮;
         at (0.875,3) 0.78v_m-1;
         at (2.125,3) 0.78v_m-1;
         at (1,1) 0.8v_m;
         at (2,1) 0.8v_m;
         at (1,-1) 0.8v_1;
         at (2,-1) 0.8v_1;
      .

The proof follows from a straightforward modification of the proof given in
Ref. <cit.>, and will be not repeated here.


The property eq:Lemma1 directly implies that traces of powers of
𝒯̃_α,γ⃗ (and therefore also
𝒯_α,γ⃗) are given as 

    [𝒯_α,γ⃗^m]=
      [𝒯̃_α,γ⃗^m]=
      [τ̃_α,γ⃗^m],

where  τ̃_α,γ⃗ is a 9× 9 matrix given by following
matrix elements,

    [τ̃_α,γ⃗]_w,v=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
         in 00.54.750.5-0.54.75-0.56.257.56.25-0.57.5-0.50.57.5 in 1,2,3,4,7[white] (+0.2,+0.75) rectangle (-0.2,-0.75);
            +0.75-0.75colMt at (5.5,-0.25) ⋯;
         in 1,2,3,4,7 at (,1) 0.8w;
           at (,-1) 0.8v;
        .

Explicitly, the matrix τ̃_α,γ⃗ reads as

    τ̃_α,γ⃗=
      [
        (1-ϑ)^2α   0   0   0   0   Γ   (1-ϑ)^α   0   Γϑ^α(1-ϑ)^α   0   0   0   0   0   ϑ^α   0   0
        0   0   0   0   Γ   0   0   Γ   0
        0   ϑ^αΓ   ϑ^αΓ   0   0   0   0   0   0
        0   0   0   ϑ^α   0   0   0   0   0
        0   0   0   (1-ϑ)^α   0   0   0   0   0
        0   -32mu(1-ϑ)^αΓ   (1-ϑ)^αΓ   0   0   0   0   0   0ϑ^2α   0   0   0   0   0   0   0   0ϑ^α(1-ϑ)^α   0   0   0   0   0   0   0   0
      ]-4mu,

where Γ is a shorthand notation for the product of γ_j,

    Γ=∏_j=1^αγ_j.


This matrix has three non-zero eigenvalues with algebraic and
geometric multiplicities equal to 1,

    Spect(τ̃_α,γ⃗)=
      {λ_α,γ⃗,λ^(1)_α,γ⃗,λ^(2)_α,γ⃗,0},

which are obtained as solutions to the following cubic equation,

    x^3=(x(1-ϑ)^α+ϑ^αΓ)^2.

Here we use λ_α,γ⃗ to denote the leading eigenvalue,
and we assume it is isolated

    λ_α,γ⃗>
      λ^(1)_α,γ⃗,
      λ^(2)_α,γ⃗,

which is not always the case. However, as long as Γ does not take a negative real value, there is a gap between the leading and the largest subleading eigenvalue for any ϑ≠ 0,1.



§ CONCLUSIONS

In this paper we extended the approach of
Refs. <cit.> to study
time-evolution of charged moments in Rule 54. We formulated the charged moments
of the two simplest charges (the total particle number, and the imbalance
between right and left movers) in terms of a space transfer matrix, which can
be understood as a deformation of the transfer-matrix capturing the dynamics of
local observables. We then found the fixed points of this deformed transfer
matrix, which are made of tensors that fulfil a set of local algebraic
relations analogous to the non-deformed case. This enabled us to fully characterize
the charged moments in the early-time regime. We showed that they decay exponentially
with a rate given in terms of a solution of a cubic equation. As was argued in
Refs. <cit.>, this is a special
case of a more general non-linear equation applying for interacting integrable
systems, which can be obtained from an equation for equilibrium charged moments
upon flipping the roles of space and time.

This work leaves several open questions. Perhaps the most urgent one is to understand what are the limits of the algebraic approach to characterize space evolution. As is showed here, the fixed points can be explicitly constructed also when the space transfer matrix is deformed by applying e^iβ Q^(±)_A at time t, where Q^(±) are the two simplest charges. The natural next step is to understand for which Q is the deformed transfer matrix of Rule 54 still solvable. Does Q need to be conserved, or can it be a more general operator? More generally, a key question is to find a broader class of systems that admit analogous solutions.

I thank Bruno Bertini, Pasquale Calabrese, Mario Collura, and Colin Rylands for collaborations on related topics, and Bruno Bertini for the careful reading of the manuscript. This work was supported by the Leverhulme Trust through the Early Career Fellowship No. ECF-2022-324. I warmly acknowledge the hospitality of the University of Ljubljana where this manuscript was finished.


iopart-num-mod






§ SPECTRAL PROPERTIES OF SPACE TRANSFER MATRICES

To find the spectrum of the space-transfer matrices we first note that by
definition we have for any L≥ 1

    [𝕎_β,t^(±)  L] = 
      Ψ_te^i β Q^(±)Ψ_t =
      Ψ_0e^i β Q^(±)Ψ_0 =
      [𝕎_β,0^(±)  L],

where the r.h.s. corresponds to the system defined on 2L sites with periodic
boundary conditions, in the second equality we used that Q^(±) is
conserved under time-evolution, and the last equality again follows from the definition
of the transfer matrix. Transfer matrices 𝕎_β,0^(±) act
only on the auxiliary space of the e^iβ Q MPO, and take the following form,

    𝕎_β,0^(+)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        01012*+22022*+210colIstcolLines20colIstcolLines12*+2colIstCcolLines22*+2colIstCcolLines0.25+12.75+11+1colObscolObsLines2+1colObscolObsLines=
      [
        1-ϑ   ϑ    0 
        e^2iβ(1-ϑ)     e^2iβϑ    0 
        (1-ϑ)    ϑ    0 
      ],  𝕎_β,0^(-)=
      [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.55]
        01012*+22022*+210colIstcolLines20colIstcolLines12*+2colIstCcolLines22*+2colIstCcolLines0.25+12.75+11+1colObscolObsLines2+1colObscolObsLines=
      [
        1-ϑ   ϑ
        1-ϑ   ϑ].

Both these matrices have exactly one non-zero eigenvalue, which we denote by 
Λ_β^(±), and are

    Λ_β^(+)=1-ϑ+e^2iβϑ,    Λ_β^(-)=1.


Note that the matrices W_β,0^(±) have trivial Jordan structure, and therefore

    1/Λ_β^(±)W_β,0^(±)=
      r_β,0^(±)l_β,0^(±)/r_β,0^(±)l_β,0^(±).

This ceases to be the case for t>0, when the Jordan structure of the transfer
matrix becomes richer, but using the conservation of Q^(±) together with 
locality of time-evolution and eq:TM0eigen one can show that we have

    .(1/Λ_β^(±)W_β,0^(±))^x
      |_x > 2t+1=
      r_β,0^(±)l_β,0^(±)/r_β,0^(±)l_β,0^(±).




§ FIXED-POINT TENSORS



 §.§ Auxiliary two-step tensors

Here we report the remaining tensor needed for
Eqs. eq:localRelationsLeftBulkBottom, eq:localRelationsLeftBeta,
and eq:localRelationsRightBeta to hold. The blue tensors take the
following form,

    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.80;
           at (0.875,1) 0.80;
           at (-0.875,0) 0.80;
           at (0.875,0) 0.80;
        =
        [
          (1-ϑ)^2     (1-ϑ)^2     -(1-ϑ)^2
     
          (1-ϑ)ϑ    (1-ϑ)ϑ    -(1-ϑ)ϑ
     
          (1-ϑ)ϑ    -ϑ^2    ϑ^2
        ],       [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.81;
           at (0.875,1) 0.81;
           at (-0.875,0) 0.80;
           at (0.875,0) 0.80;
        =
        [ϑ    0     0
     
          0    ϑ    (1-ϑ)
    
          0     0     0
        ],-32mu 
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.80;
           at (0.875,1) 0.80;
           at (-0.875,0) 0.80;
           at (0.875,0) 0.81;
        =
        [
          0     (1-ϑ)^2     -(1-ϑ)^2
     
          0     (1-ϑ)ϑ    -(1-ϑ)ϑ
     
          0     -ϑ^2    ϑ^2
        ],
           [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.81;
           at (0.875,1) 0.81;
           at (-0.875,0) 0.80;
           at (0.875,0) 0.81;
        =
        [ϑ    0     0
     
          0     0     0
    
          0     0     0
        ],
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.80;
           at (0.875,1) 0.80;
           at (-0.875,0) 0.81;
           at (0.875,0) 0.81;
        =
        [
          0     1-ϑ    0
     
          0    ϑ    0
     
          0     0    ϑ],
           [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.81;
           at (0.875,1) 0.81;
           at (-0.875,0) 0.81;
           at (0.875,0) 0.81;
        =
        [ϑ    0     0
     
          1-ϑ    0     0
    
          0     0     0
        ],
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.80;
           at (0.875,1) 0.81;
           at (-0.875,0) 0.80;
           at (0.875,0) 0.81;
        =
        [
          0     1-ϑ    0
     
          0    ϑ    0
     
          0     0    ϑ],
           [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.8s;
           at (0.875,1) 0.818mu 1-s;
           at (-0.875,0) 0.8b;
           at (0.875,0) 0.8b;
        =0,
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.80;
           at (0.875,1) 0.81;
           at (-0.875,0) 0.81;
           at (0.875,0) 0.80;
        =
        [
          0     0     0
     
          0    ϑ^2     (1-ϑ)ϑ
     
          0    ϑ^2     (1-ϑ)ϑ
    ],
           [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.8s_1;
           at (0.875,1) 0.8b_1;
           at (-0.875,0) 0.8s_2;
           at (0.875,0) 0.8b_2;
        =
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.8b_1;
           at (0.875,1) 0.8s_1;
           at (-0.875,0) 0.8b_2;
           at (0.875,0) 0.8s_2;
        .

  The orange tensors are very similar, with the majority of the tensors being the same as blue,
  
    .[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPSBeta at (-0.875,1) 0.8s_1;
           at (0.875,1) 0.8b_1;
           at (-0.875,0) 0.8s_2;
           at (0.875,0) 0.8b_2;
        |_s_1+s_2<2=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.8s_1;
           at (0.875,1) 0.8b_1;
           at (-0.875,0) 0.8s_2;
           at (0.875,0) 0.8b_2;
        ,

  while for s_1=s_2=1 one needs to multiply the relevant tensors with a transformation analogous to eq:defPentagon
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.62501.625-0.5500.550-0.5510.551001colMPSBeta at (-0.875,1) 0.81;
           at (0.875,1) 0.81;
           at (-0.875,0) 0.8s;
           at (0.875,0) 0.8b;
        =
        [baseline=([yshift=-1.2ex]current bounding box.center),scale=0.5]
          0-0.62502-0.5500.550-0.5510.551001colMPS at (-0.875,1) 0.81;
           at (0.875,1) 0.81;
           at (-0.875,0) 0.8s;
           at (0.875,0) 0.8b;
          01.625colMPSBeta,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0-0.500.500colMPSBeta=
        [γ    0     0 
          0     1     0 
          0     0     1
          ],    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
            0-0.501.2500colMPSBeta00.75colMPSBeta=
          [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
            0-0.501.2500colMPSBeta00.75colMPSBeta=
          γ[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
            0-0.501.25.

  Note that this holds for any label , , and  upon replacing γ with e^-iβ, e^iβ, and e^iβΛ_β^(+) respectively.

  

 §.§ Boundary tensors

  The bottom boundary tensors are all the same and do not depend on the value of β,
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPS=
        [
          1-ϑϑ-ϑ^2/1-ϑ]-6mu,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          0000.62500colMPS=
        [
          1  0  0
        ]-6mu.

  On the other hand, the top boundary vectors act on both the auxiliary space of
  the MPO describing e^iβ Q^(±), and the auxiliary space of the fixed-point MPO,
  and therefore must obtain the dependence on the charge in question and the value of β.
  In particular, the top boundary vector in the absence of the charge is
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
           at (-0.75,0) 0.8s;
          000-0.62500=[1   1   0],

  while the top boundary vector included in l_β,t^(-) is
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.625000.625000colMPSBeta00 at (0.875,0) 0.8x;
        -8mu=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.625000.625000colMPSBeta00 at (0.875,0) 0.8x;
        -8mu=
        [δ_x,1   δ_x,2    0
        ].

  Unlike the bulk tensors (cf. eq:newFixedPointTensors) the top boundary
  vectors of r_β,t^(-) are not obtained just by the flip of the
  sign β↦ -β, but take a completely different form,
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500-0.625000colMPSBeta00 at (-0.875,0) 0.8x;
        =
        [
          1    δ_x,1 + e^iβδ_x,2    0
        ],    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500-0.625000colMPSBeta00 at (-0.875,0) 0.8x;
        =
        [
          1    δ_x,1 + e^-iβδ_x,2    0
        ].

  We note that the overlap between the top boundary vectors
  of fixed-points and the boundary vectors from eq:boundaryMPO reduce to the
  top boundary vector of the fixed-point l_t,
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500-0.6250-0.625-0.25-0.6250.2500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500-0.6250-0.625-0.25-0.6250.2500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.625000.62500.625-0.250.6250.2500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.625000.62500.625-0.250.6250.2500colMPSBeta00=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500.


  In the case of Q^(+), the top boundary vectors take the following form,
  
    -11mu
          [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
            000-0.62500-0.625000colMPSBeta00 at (-0.875,0) 0.8x;
          =
          [δ_x,1+e^2iβδ_x,2+δ_x,3δ_x,1+e^2iβ/γ_+(δ_x,2+δ_x,3)(e^3iβ/γ_+-1)δ_x,1]^T-10mu,
       [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
            000-0.625000.625000colMPSBeta00 at (0.875,0) 0.8x;
          -8mu=
          [e^iβ/γ_+δ_x,1e^2iβ(1-ϑ)/γ_+^2(ϑ/1-ϑδ_x,2+δ_x,3)e^2iβ(1-ϑ)/γ_+^2(δ_x,2-δ_x,3)
          ]^T-10mu,       -4mu
            [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
              000-0.62500-0.625000colMPSBeta00 at (-0.875,0) 0.8x;
            =
            [δ_x,1+e^2iβδ_x,2+δ_x,3
              e^2iβδ_x,1+γ_+(δ_x,2+δ_x,3)(e^3iβ-γ_+)(δ_x,2+δ_x,3)
            ]^T-10mu,
       [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
              000-0.625000.625000colMPSBeta00 at (0.875,0) 0.8x;
            -8mu=
            [e^iβ/γ_+δ_x,1e^iβ(1-ϑ)/γ_+(δ_x,2+ϑ e^2iβ/1-ϑδ_x,3)e^iβ(1-ϑ)/γ_+(-δ_x,2+e^2iβδ_x,3)
            ]^T-10mu,

  where 
  
    γ_+=e^iβΛ_β^(+),

  and Λ_β^(+) is given in Eq. eq:defLambda. The two overlaps
  between these vectors and the boundary vectors of the Q^(+) MPO that are needed
  to evaluate Eq. eq:defTNOneEdgeContrib are given as
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.625000.62500.625-0.250.6250.2500colMPSBeta00=
        [e^iβ/γ_+e^iβ(1-ϑ+ϑ e^iβ)/γ_+^2
          -e^iβ(1-e^iβ)(1-ϑ)/γ_+^2]^T,    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500-0.6250-0.625-0.25-0.6250.2500colMPSBeta00=
        [
          1γ_+ e^-iβ
          e^2 iβ-γ_+ e^-iβ]^T.

  Note that in the limit of β→ 0 all of the overlaps reduce to
  
    .
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.625000.62500.625-0.250.6250.2500colMPSBeta00|_β→ 0
        =
        .
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.625000.62500.625-0.250.6250.2500colMPSBeta00|_β→ 0
        =
        .
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500-0.6250-0.625-0.25-0.6250.2500colMPSBeta00|_β→0
        =
        .
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500-0.6250-0.625-0.25-0.6250.2500colMPSBeta000-0.625colMPSBeta0-0.625|_β→0=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          000-0.62500.

  

 §.§ Normalization

  The normalization of boundary vectors above was chosen so that the following holds,
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02031213031303colMPSBeta13colMPSBeta02colMPS12colMPS in 301=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02031213031303colMPSBeta13colMPSBeta02colMPS12colMPS in 301=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02031213031303colMPSBeta13colMPSBeta02colMPS12colMPS in 301=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02031213031303colMPSBeta13colMPSBeta02colMPS12colMPS in 301=1,

  which together with 
  
    [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          [thick,colLines,rounded corners] (0,2) – (-0.75,2) – (-0.75,3.5) – (1.75,3.5) – (1.75,2) – cycle;
          01.50311.513031303colMPSBeta02colMPSBeta12colMPSBeta13colMPSBeta in 2,301=Λ_β^(+)[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02.250312.2513031303colMPSBeta13colMPSBeta in 301,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          [thick,colLines,rounded corners] (0,2) – (-0.75,2) – (-0.75,3.5) – (1.75,3.5) – (1.75,2) – cycle;
          01.50311.513031303colMPSBeta02colMPSBeta12colMPSBeta13colMPSBeta in 2,301=Λ_β^(+)[baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02.250312.2513031303colMPSBeta13colMPSBeta in 301,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          [thick,colLines,rounded corners] (0,2) – (-0.75,2) – (-0.75,3.5) – (1.75,3.5) – (1.75,2) – cycle;
          01.50311.513031303colMPSBeta02colMPSBeta12colMPSBeta13colMPSBeta in 2,301=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02.250312.2513031303colMPSBeta13colMPSBeta in 301,  [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          [thick,colLines,rounded corners] (0,2) – (-0.75,2) – (-0.75,3.5) – (1.75,3.5) – (1.75,2) – cycle;
          01.50311.513031303colMPSBeta02colMPSBeta12colMPSBeta13colMPSBeta in 2,301=
        [baseline=([yshift=-0.6ex]current bounding box.center),scale=0.5]
          02.250312.2513031303colMPSBeta13colMPSBeta in 301,

  implies
  
    l_β,t^(+)r_β,t^(+)=.Λ_β^(+).^2t,
            l_β,t^(-)r_β,t^(-)=1.




 Paper Title: Homogeneous Magnetic Flux in Rydberg Lattices.







jeix@iastate.edu
Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA

rukmani.bai@itp3.uni-stuttgart.de
Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Université Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau Cedex, France

Université Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau Cedex, France

Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

sebastian.weber@itp3.uni-stuttgart.de
Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

+These authors contributed equally to this work.

§ ABSTRACT 
We present a method for generating homogeneous and tunable magnetic flux for bosonic particles in a lattice using Rydberg atoms. Our setup relies on Rydberg excitations hopping through the lattice by dipolar exchange interactions. The magnetic flux arises from complex hopping via ancilla atoms. Remarkably, the total flux within a magnetic unit cell directly depends on the ratio of the number of lattice sites to ancilla atoms, making it topologically protected to small changes in the positions of the atoms. This allows us to optimize the positions of the ancilla atoms to make the flux through the magnetic unit cell homogeneous. With this homogeneous flux, we get a topological flat band in the single-particle regime. In the many-body regime, we obtain indications of a bosonic fractional Chern insulator state at ν = 1/2 filling.


Homogeneous Magnetic Flux in Rydberg Lattices
    S. Weber
    
=============================================








§ INTRODUCTION

Charged particles in the presence of magnetic fields exhibit many fascinating phenomena <cit.> including integer <cit.> and fractional <cit.> quantum Hall states, as well as topological fractional Chern insulating states <cit.>. The observation of these many-body states requires strong interactions between the charged particles and high magnetic fields. In this context, quantum simulators comprised of ultracold atoms trapped in optical lattices have emerged as a pathway to achieve these conditions, owing to their many tunable experimental parameters <cit.>.  
However, ultracold atoms are charge neutral and do not experience the Lorentz force when exposed to external magnetic fields.
Despite this obstacle, magnetic fields for neutral atoms can be implemented artificially <cit.>. There are several ways to generate effective magnetic fields, including rotating the optical lattice <cit.>, using laser assisted tunneling <cit.>, implementing spatially inhomogeneous light fields <cit.>, periodically perturbing the Hamiltonian <cit.>, and lattice shaking <cit.>. In this work, we propose an alternative method to generate homogeneous magnetic flux in rectangular lattices of charge neutral particles using only lattice geometry and Rydberg atoms.

Rydberg atoms trapped in optical tweezer arrays provide a convenient platform to study many body physics for charge neutral particles in effective magnetic fields due to the experimental control of their long-range dipole-dipole interactions <cit.>, the tunable geometry of the arranged atoms, and dipolar exchange interactions <cit.>. Effective magnetic fields arise from complex hopping, known as the Peierls phase <cit.>, which can be realized solely from dipolar exchange interactions in Rydberg atoms without requiring inhomogeneous light fields or lattice shaking <cit.>. Rydberg atom arrays have been identified as a tool to explore different quantum states of matter such as symmetry-protected topological phases <cit.>, fractional Chern insulating states <cit.>, and quantum spin liquid states <cit.>. However, it remains an open challenge to find setups where the effective magnetic field corresponds to a homogeneous magnetic flux.

Here, we provide a way to generate robust, homogeneous, and tunable magnetic flux in rectangular lattices of Rydberg atoms. Our method makes use of ancilla atoms in a background lattice which induce an effective magnetic flux arising from dipolar exchange interactions. Strikingly, the total flux per magnetic unit cell is determined by the ratio of ancilla atoms to Rydberg atoms. This ratio is a bulk characteristic of the system and robust to perturbation; hence, the total flux through the magnetic unit cell is a topological property, but the flux can be tuned locally by varying the lattice geometry. Using this method, flat topological bands can be generated with arbitrary flux values. After establishing the robustness of the magnetic flux, we find indications for a bosonic fractional Chern insulator state at ν = 1/2 filling in the many body regime, in agreement with previous studies that theoretically predicted such a state in a bosonic Harper Hofstadter model <cit.>.

This article is organized as follows. In section II, we discuss our setup along with the Rydberg level scheme. In section III, we examine obtainable magnetic flux patterns and show that the total flux through the magnetic unit cell is determined by the ratio of sites between the background and rectangular lattices. Furthermore, we demonstrate how the magnetic flux can be made homogeneous by optimizing the lattice. In section IV, we discuss the topological flat bands in the single-particle picture. Finally, in section V, we investigate the many-body regime and obtain a topological state at ν = 1/2 filling using the robust flux generated from our method.






§ SETUP


We begin with the microscopic description of our setup. We focus on Rydberg atoms 
trapped individually by optical tweezers, where each atom can occupy one of two Rydberg states which are energetically isolated from all other Rydberg states. The atoms are arranged to form a rectangular lattice A (blue sites) and background lattice B of ancilla atoms (red sites), as depicted in Fig.<ref>.

We start by describing lattice A in detail. Lattice A has horizontal spacing b and vertical spacing a. Rydberg atoms in lattice A have quantum number n and an internal level structure consisting of two different Rydberg states, namely a Rydberg S-state |0⟩ and a Rydberg P-state |-⟩, as shown in Fig.<ref> <cit.>. The states can be isolated energetically from other states using, for example, a combination of electric and magnetic fields <cit.>.
The state |0⟩ can be interpreted as the vacuum state, and the state |-⟩ can be interpreted as a bosonic excitation. Therefore, the model is described by hard-core bosons with bosonic operators satisfying b^†_-|0⟩ = |-⟩ <cit.>. Within the lattice, dipolar exchange interactions give rise to hopping of |-⟩ excitations, which can in general be described by the Hamiltonian

    ℋ =
    ∑_ijt^-_ijb^†_-,jb_-,i ,

where t^-_ij is the hopping element
between site i and j of lattice A.



We use the ancilla lattice B to generate effective magnetic fields. We do so by allowing off-resonant hopping processes that occupy the ancilla atoms virtually, making the hopping elements t^-_ij complex. To achieve these hopping processes, lattice B hosts Rydberg atoms with quantum number n' and an internal level structure consisting of a Rydberg S-state |0⟩' and a Rydberg P-state |+⟩, as shown in Fig. <ref>. Having two different quantum numbers  n and n' for the atoms forming the rectangular lattice A and the background lattice B, allows us to choose level schemes where virtual hopping processes among atoms within the same lattice are negligible. Note that we specifically chose the |-⟩ and |+⟩ state to differ in the magnetic quantum number by two. All lattice B sites are initialized in the ground state |0⟩'. To ensure that the lattice B sites are only occupied virtually, we apply an energy shift μ = [E(|+⟩)-E(|0⟩') - E(|-⟩)+E(|0⟩)]. This shift allows us to treat the hopping from a site i to a site j through an ancilla atom in second-order perturbation theory. We have illustrated such a second-order process in Fig.<ref>, where a |-⟩ excitation at site i hops to site j by first virtually occupying the ancilla atom s with a |+⟩ excitation. The spin-flip hopping processes couple the |+⟩ to the |-⟩ state, changing the internal angular momentum by two which yields a complex phase due to the conservation of angular momentum <cit.>. 

Taking into account these processes, the hopping element t^-_ij becomes

    t^-_ij = |t^-_ij| e^iϕ_ij 
    	= -t^-/|𝐑_ij|^3 
    	- ∑_s w^2/μ|𝐑_is|^3
    	|𝐑_sj|^3e^2iϕ_isj ,

where the first term is the direct hopping from i to j with hopping strength t^-, and the second term arises from second-order hopping processes. The phase factor of a second-order process is determined by the in-plane azimuthal angle ϕ_isj shown in Fig. <ref>. The sign of ϕ_isj is determined by 𝐑_ij×𝐑_is where 𝐑_ij is the vector from i to j. The amplitude of a second-order process is determined by the strength w of the hopping between lattice A and B and the energy difference μ <cit.>. Because the value of t^-/w is highly implementation dependent and the results of our paper stay qualitatively the same for different values of t^-, we consider the limit w≫ t^- and set t^-=0 for the rest of this work.

The ancilla lattice B is chosen such that it has the same periodicity in the y direction as lattice A. The periodicity of lattice B in the x direction is commensurable with the periodicity of lattice A. This allows us to specify a unit cell of the combined lattice with p lattice B sites and q lattice A sites, being invariant under translations by qbx̂ and aŷ. For an example of a unit cell with q=4, p=7, see the gray shaded region in Fig. <ref>. This unit cell coincides with the magnetic unit cell of the lattice A. We align lattice B so that, in the y direction, sites of lattice A are centered between sites of lattice B. This guarantees real hopping elements along x since pairs of lattice B sites contribute to the second-order hopping with conjugate phases 
<cit.>.The integer-valued number of flux quanta through the magnetic unit cell is fully determined by the complex hoppings along y. Remarkably, as we will show in the next section, the number of flux quanta is directly related to the numbers q and p of lattice sites in the magnetic unit cell.








§ GENERATING HOMOGENEOUS MAGNETIC FIELDS



In this section, we show that the effective magnetic flux can be tuned to be homogeneous, and arbitrary rational numbers of flux quanta can be realized in the Rydberg lattice A. As explained in the previous section <ref>, only the hopping along y direction is complex-valued. To characterize this hopping, we consider two sites of lattice A within an infinite lattice B, as shown in Fig. <ref> (a). For our argument and without loss of generality, lattice B is taken to be rectangular with horizontal lattice spacing c and vertical spacing a. The two lattice A sites are a pair of vertical nearest-neighbors separated by a. 

While our discussion holds for arbitrary ratios of c/a, we consider c = 4/7a for illustration, being consistent with the setup depicted in Fig. <ref> that realizes this value for q=4, p=7, and a=b. In the following, we move the pair of lattice A sites along x while keeping track of the complex hopping between them. The hopping is thus a function of horizontal distance x and is denoted as t(x). The phase of t(x) is denoted as ϕ(x).

The phase ϕ(x) is plotted in Fig. <ref> (b) as the pair of lattice A sites are translated along x from 0 to c. This translation reveals a few general and key properties of ϕ(x). First, due to translational symmetry, t(x) is invariant when x changes by an integer multiple of c. Accordingly, ϕ(x) is invariant with ϕ(x+c)=ϕ(x) 2π. Second, we observe that ϕ(mc+x) = -ϕ(mc-x) ∀  m ∈1/2ℤ because the corresponding geometrical arrangements are related by a reflection. The result here can be understood from the fact that t_ij = t_ji^* from Eq. (<ref>), and that a reflection effectively reverses the hopping direction. Third, the phase ϕ(x) is a continuous function in the interval [0,c] as long as t(x) ≠ 0.
Importantly, these properties imply that ϕ(x) can only change by 2π n when x increases from 0 to c. We call the integer value n the winding number.

It turns out that the winding number is dependent on the ratio c/a. For example, for c/a = 4/7, we observe that ϕ(x) varies from π to -π as x varies from 0 to c, giving a winding number of 1. In contrast, for c/a ≲ 0.41, the winding number is found to be 0. For c/a ≳ 2.00, the winding number is 2. In between, 0.41 ≲ c/a ≲ 2.00, the winding number is 1, see Fig. <ref> (b). The values c/a ≈ 0.41 and c/a ≈ 2.00 are critical values which separate different regimes. These critical values correspond to special cases in which the hopping t(x) vanishes at the reflection symmetric point x=c/2; thus, the phase ϕ(c/2) and the winding number are no longer defined. 



This behavior of the phase ϕ(x) has implications for the flux through a magnetic unit cell. Let us consider that the lattice A has a rational horizontal spacing b = p/qc. Then, the magnetic unit cell will contain p lattice B sites and q lattice A sites. Our analysis of the winding number showed that the phase ϕ(x) of the hopping along y increments by 2π n with  n ∈{0,1,2} between each lattice B site. Accordingly, the total flux through the magnetic unit cell must be 2π pn because we have p lattice B sites. The average flux through each lattice A plaquette is thus 2π pn/q.



Notably, our argument did not rely on specific positions of the lattice sites. Thus, rearranging the positions of the sites in the magnetic unit cell does not change the total flux through it as long as t(x) does not vanish. This allows us to optimize the homogeneity of the flux and hopping strengths by adapting the positions of the lattice sites. 

While tuning the positions of each site is a high-dimensional optimization problem that is challenging to solve exactly, we use a heuristic that provides reasonably homogeneous magnetic fields and hoppings. Our heuristic takes place in several stages. At the beginning, lattice A is initialized with a=b, and the background lattice B is initialized with vertical spacing a and horizontal lattice spacing c = (q/p)a. This fixes the average flux to 2π p/q, provided that 0.41 ≲ q/p ≲ 2.00. As a first optimization step, the flux is optimized for homogeneity by rearranging lattice B sites within the magnetic unit cell along x. During this process, we keep the magnetic unit cell symmetric under reflection. As a second optimization step, we optimize for homogeneous hopping strengths. To do so, we tune the ratio between a and b and, in addition, allow lattice B sites to move out of the plane in z direction. Again, we only allow for rearrangements that keep the reflection symmetry. Optimizing for homogeneous flux or hopping strength are not independent processes; however, iterating the processes converges in that flux variance and hopping strength variance may both be minimized, as shown in Fig. <ref>. As we will see in the next section, having a homogeneous flux and nearly homogeneous hoppings allow for obtaining topological flat bands in our model.






§ TOPOLOGICAL BAND STRUCTURE

In this section, we present the topological band structure of lattice A for the optimized and not optimized positions of lattice B sites.
In Fig. <ref>, we show the resulting topological band structure and density of states (D.O.S.) in the single particle picture. The considered unit cell with p=7, q = 4 lattice sites provides four bands as lattice A has q = 4 sites in its magnetic unit cell. 
We observe that the optimized lattice exhibits topological flat bands as shown in Fig. <ref> (b) for nearest-neighbor hoppings. Fig. <ref> (a) depicts the results if we do not optimize the positions of lattice B sites.
Notably, for the optimized lattice, the density of states is gapped and remains gapped if we take into account the long-range character of the dipolar interaction as shown in Fig. <ref> (c).

In summary, Fig. <ref> shows that we have obtained a flat band for a system with p/q = 7/4. However, we want to point out that flat topological bands can be achieved for arbitrary flux values by adjusting the ratio p/q and optimizing the magnetic unit cell through our proposed method.






§ MANY-BODY PHYSICS

In the single-particle picture, we have obtained nearly homogeneous flux and flat bands in the Rydberg lattices. Now, we extend our analysis to the many-body regime. To be specific, we aim to study a fractional Chern insulator (FCI) state, using the flux generated in the lattice through our method. The properties of our model in the single-particle regime combined with the strong interaction between particles provided by the hard-core constraint, make our approach a promising tool for realizing FCI states using Rydberg atoms.

We use exact diagonalization (ED) method to obtain the many-body ground state of our Hamiltonian provided by Eq. (<ref>). In our implementation of ED, the Hilbert space dimension is reduced due to particle number conservation. To obtain the FCI state, we consider finite lattices with p/q = 7/4 and a filling fraction ν = 1/2. This corresponds to a lattice with 1/4 flux per plaquette and one particle for every two flux. 
We consider lattice sizes from 4 × 4 to 8 × 6 with periodic boundary conditions and obtain the many-body ground state. Due to periodic boundary conditions, the topology of the system is a torus. The observed ground state is degenerate which is the signature of a FCI ν = 1/2 state. To further support this interpretation of the ground state, we calculate the many-body Chern number for the ground state. For this, we use twisted periodic boundary conditions along x and y directions with twist angles θ_x and θ_y

    T_x(L_x)ψ(x,y) = e^iθ_xψ(x,y) , 
    
        T_y(L_y)ψ(x,y) = e^iθ_yψ(x,y) ,

where T_i(L_i) are operators which translate single-particle wavefunctions ψ around the ith direction of the torus. The twist angles θ_x and θ_y correspond to additional fluxes which are threaded through the hole of the torus or through its interior. Thus, the many-body Hamiltonian and its eigenstates Ψ(θ_x,θ_y) become functions of the twist angles. Within this parameter space, the non-abelian Berry connection and the many-body Chern number are defined as:

    𝒜_j^α,β(θ_x,θ_y) = i⟨Ψ^α|∂_j|Ψ^β⟩ , 
    𝒞 = 1/2π∫_0^2π∫_0^2π∂_x𝒜_y^α,β - ∂_y𝒜_x^α,β dθ_x dθ_y ,

where ∂_j is the partial derivative with respect to θ_j, and α, β correspond to degenerate eigenstates. We calculate the many-body Chern number in addition to the ground state degeneracy to verify topological equivalence to Laughlin ν = 1/2 states.

The lowest six eigenenergies for nearest-neighbor hopping as a function of twist angles is shown in Fig. <ref> (a) and (b) for lattices of size 4 × 4 and 8 × 6, respectively. Here, the results are computed across an 11×11 discretization of the twist angle space. We observe that the ground state is degenerate and it is gapped from the first excited state for all twist angles. The doubly-degenerate ground state manifold has a Chern number 𝒞=1. This corresponds with the FCI ν=1/2 state. The nearest-neighbor model also maps to the known models of FQH states considered in Ref. <cit.>. Therefore, we have established that our approach exhibits FCI state if we consider hopping up to nearest-neighbor.
To make a conclusive statement about the effect of long-range hoppings, larger systems would be required, which are not accessible via exact diagonalization. However, already the reduced flatness in the single-particle band structure, see Fig <ref> (c), indicates that the topological state might be destabilized. On the other hand, it might be that more sophisticated and numerically challenging optimization strategies than those applied by us can find geometries that result in flatter bands. Thus, it is an open question whether, in the thermodynamic limit, the topological state survives in the presence of long-range hoppings.








§ CONCLUSION

We have proposed a method to generate arbitrary homogeneous flux and hopping in rectangular lattices of Rydberg atoms using dipolar exchange interactions and a background lattice of ancilla atoms.
Note that while we focused on Rydberg atoms, our proposal can be easily adapted to polar molecules, where similar level structures and interactions can be obtained.
We showed that within a magnetic unit cell, the flux is proportional to the ratio between the number of sites of the background lattice, p, and the number of sites of the rectangular lattice, q. We demonstrated that the flux is topologically robust and unaffected by changes in the atomic positions within the magnetic unit cell as long as hopping amplitudes do not vanish. To make the flux homogeneous, we made use of the freedom to change the atomic positions. With this homogeneous flux, we obtained topological flat bands in the single-particle regime. Further, in the many-body regime, we obtained a bosonic fractional Chern insulator state at ν = 1/2 filling with nearest neighbor hopping. This many-body state has the same topological properties as of the Laughlin state ν = 1/2 with many body Chern number 𝒞 = 1.
To fully understand the effect of long-range hoppings, larger system sizes must be explored. These were inaccessible by the exact diagonalization method applied by us and require numerical methods such as DMRG. Alternatively, to suppress long-range hoppings altogether, one can implement our model using Rydberg S and D states that are only dipole-coupled in second order, giving rise to exchange interactions that scale with 1/R^6 where R is the interatomic distance. Future work can be done to explore the entire parameter space of our model.





We gratefully acknowledge financial support by the Baden-Württemberg Stiftung via Grant BWST ISF2019-017
under the program Internationale Spitzenforschung. This project has also received funding  from the French-German collaboration for
joint projects in NLE Sciences funded by the Deutsche Forschungsgemeinschaft (DFG) and the Agence National de la Recherche (ANR, project RYBOTIN). A.B. and T.L. acknowledge funding by the Agence Nationale de la Recherche (ANR-22-PETQ-0004 France 2030, project QuBitAF), Horizon Europe programme HORIZON-CL4-2022-QUANTUM-02-SGA via the project 101113690 (PASQuanS2.1), and the European Research Council (Advanced grant No. 101018511-ATARAXIA).





 Paper Title: Adaptive Quantum Generative Training using an Unbounded Loss Function.











Adaptive Quantum Generative Training using an Unbounded Loss Function



This work was funded by grants from the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-Design Center for Quantum Advantage under contract number DE-SC0012704.

    Kyle Sherbert
Department of Physics 

Virginia Tech

Blacksburg, VA 

kyle.sherbert@vt.edu
Jim Furches^*
Department of Physics 

Virginia Tech

Blacksburg, VA 

jfurches@vt.edu
Karunya Shirali^*
Department of Physics 

Virginia Tech

Blacksburg, VA 

karunyashirali@vt.edu
Sophia E. Economou
Department of Physics 

Virginia Tech

Blacksburg, VA 

economou@vt.edu
Carlos Ortiz Marrero
AI & Data Analytics Division 

Pacific Northwest National Laboratory

Seattle, WA 

carlos.ortizmarrero@pnnl.gov

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



*These authors contributed equally to this work

§ ABSTRACT 
We propose a generative quantum learning algorithm, Rényi-ADAPT, using the Adaptive Derivative-Assembled Problem Tailored ansatz (ADAPT) framework in which the loss function to be minimized is the maximal quantum Rényi divergence of order two, an unbounded function that mitigates barren plateaus which inhibit training variational circuits. We benchmark this method against other state-of-the-art adaptive algorithms by learning random two-local thermal states. We perform numerical experiments on systems of up to 12 qubits, comparing our method to learning algorithms that use linear objective functions, and show that Rényi-ADAPT is capable of constructing shallow quantum circuits competitive with existing methods, while the gradients remain favorable resulting from the maximal Rényi divergence loss function.



Quantum Algorithm, Qubit, Circuit Synthesis, Logic Gates




§ INTRODUCTION

Many promising quantum machine learning algorithms almost exclusively use a linear bounded operator as their objective functions during training <cit.>. These loss functions are typically estimated by measuring the expectation values of Hermitian operators directly during training, computing their gradients, and updating the parameters. Despite optimism, training using linear objective functions has proved computationally difficult due to a concentration of measure phenomenon known as a barren plateau that leads to exponentially small gradients as we scale the system size of our problem <cit.>. These results paint a bleak picture for the future of quantum machine learning, and finding a scalable approach to train generic parametrized models has become a central problem in the field.  Existing approaches to overcome barren plateaus are based on empirical evidence <cit.> or are restricted to a specific architecture <cit.> but none provide a generic way to train quantum models that is guaranteed to avoid these no-go results. Recent work by Kieferová et al. <cit.> suggests the use of a Quantum Divergence, an unbounded, nonlinear loss function with simple gradients, in place of a linear objective function, when performing generative training to address the barren plateau problem. The standard arguments for barren plateau theorems do not apply to divergences because these experience a logarithmic divergence when the two states are nearly orthogonal. This causes the gradients of the divergence between nearly orthogonal quantum states to be large, and thereby provides a workaround for all known barren plateau results. 

In our work, we extend Kieferová et al. <cit.> method and propose an adaptive training algorithm to optimize the maximal quantum Rényi divergence between two quantum states, that is, the output of a quantum circuit and some quantum data state. We provide evidence showing how our adaptive algorithm performs compared to other generative training algorithms. In particular, we explore the gradient evolution of each method and argue that our Rényi-ADAPT has favorable gradients at scale, when compared to methods such as Overlap-ADAPT <cit.> and ADAPT-VQE-Gibbs<cit.>. 



§ ADAPT-VQE ALGORITHM

Variational quantum eigensolvers (VQEs) <cit.> employ parametrized quantum circuits to divide the computational task of optimizing an objective function f(θ_1, θ_2, …θ_n) (most often the ground state energy of a physical system, estimated as the expectation value of the system Hamiltonian ⟨ H ⟩) between quantum and classical processors, in which the quantum processor performs measurements of the objective function and the classical processor updates the variational parameters {θ_i} at each iteration. The choice of parametrized circuit (the “ansatz”) is critical to the performance of the algorithm, and the limited coherence times available on current quantum hardware prompt the use of shallow circuits. In general, a good ansatz needs to be sufficiently expressive to represent the solution accurately, but also have a low circuit depth. 

The Adaptive Derivative-Assembled Problem Tailored ansatz (ADAPT)-VQE algorithm <cit.> employs an adaptive strategy to construct the variational ansatz dynamically using operators in a predefined operator pool, starting from an initial state |ψ_ ref⟩. Defining the variational parameters θ^(k) = (θ_1,…,θ_k) and the operator pool 𝒜 = { A^(1), A^(2), … A^(N)}, the ansatz in iteration k+1 of the algorithm may be written as

    |ψ_k+1(θ^(k+1))⟩ = e^-iθ_k+1 A_k+1|ψ_k(θ^(k))⟩.

In (<ref>), the ansatz at iteration k is grown by appending operator A_k+1 with coefficient θ_k+1; the operator is chosen by measuring the energy gradients | . ∂⟨ H ⟩ / ∂θ_k+1|_θ_k+1=0| for each operator in the pool and selecting the one with the largest gradient. For this step, it can be shown that 

    | . ∂⟨ H ⟩ / ∂θ_k+1|_θ_k+1=0| =  | ⟨ψ_k(θ^(k)) | [ A_k+1, H ] | ψ_k(θ^(k)) ⟩|,

where the right hand side of (<ref>) can be efficiently measured on a quantum processor. The pool operator gradient-measurement step is followed by a convergence check: if the pool operator gradient norm is smaller than a threshold ε, the calculation is terminated; if not, the ansatz is grown as in (<ref>). The ansatz-growing step is followed by a VQE optimization of all variational parameters, where 

the parameters (θ_1,…,θ_k) take their previously optimized values, and the newly added parameter is initialized as 0.

In this work, we are interested in training mixed states, so we extend ADAPT following the strategy in <cit.>,
    in which we use (<ref>) to prepare a pure state on a set of visible and hidden (ancilla) qubits.
The trial mixed state is taken as the state on the visible qubits when the hidden qubits are traced out:

    σ_k(θ^(k)) = _H|ψ_k(θ^(k))⟩⟨|

We hereafter write σ_k(θ^(k)) as σ(θ) for notational simplicity, or even simply as σ when context allows.

The ADAPT-VQE algorithm was shown to reach arbitrarily accurate energies and yield shallow ansätze when applied to small molecules, and inspired adaptive techniques for studying other problems, including time evolution and dynamics <cit.>, excited-state preparation <cit.>, and optimization <cit.>. In particular, Overlap-ADAPT-VQE <cit.>, a method that we use in this work, dynamically constructs an ansatz by maximizing the overlap between a trial state and the target state. It was found to produce ultracompact ansätze, thereby providing accurate reference states for subsequent ADAPT runs, and resulting in shallower circuit depths of the final ansatz compared to regular ADAPT. 

The loss function in Overlap-ADAPT-VQE <cit.>, formulated for pure states, is 

    ℒ_O(θ) = ⟨Ψ_ target | ψ(θ) ⟩.

For mixed states ρ and σ, the fidelity has the following generalization:

    F(ρ, σ) = Tr√(√(ρ) σ(θ)  √(ρ))

In this work, we extend Overlap-ADAPT-VQE to prepare mixed states, using the loss-function 1 - F(ρ, σ(θ))^2,
    where ρ is the target state and σ(θ) is the trial state.

    ℒ_O(ρ, σ(θ)) = 1 - (Tr√(√(ρ) σ(θ)  √(ρ)))^2.







§ THERMAL STATE PREPARATION USING THE ADAPT-VQE-GIBBS ALGORITHM

An adaptive protocol for the preparation of Gibbs thermal states ρ_G = e^-β H/Z was proposed in <cit.> in which an easily measurable customized objective function C(σ) was introduced; it replaced the Gibbs free energy F(σ) = E(σ) - k_B TS(σ) as the objective function to minimize. Using this objective function, in combination with the ADAPT-VQE strategy of dynamically constructing ansätze from a predefined operator pool, it was shown that accurate representations of the Gibbs state could be obtained across a range of temperatures using low-depth circuits. The protocol makes use of a purifying ancilla system A in addition to the data system D to generate a pure state |ψ⟩∈ℋ_D ⊗ℋ_A, where the Gibbs state ρ_G of system D is obtained by tracing out the ancillary system (the visible and hidden qubits in this work correspond to the data and ancilla qubits, respectively, in ADAPT-VQE-Gibbs). The modified objective function

    C(σ(θ)) = - Tr(ρ_G σ) + 1/2Tr(σ^2),

where σ = Tr_A |ψ⟩⟨ψ| is the variational trial state, and ρ_G is an approximation of the target Gibbs state. Using C(σ) as the objective function is convenient because it circumvents the need to measure the von Neumann entropy, which is hard to measure on current quantum hardware. The approximation of ρ_G in (<ref>) is obtained via a truncated Taylor expansion of e^-β H/Z. 


In our calculations, we truncate the Taylor expansion of e^-β H at order m=5, i.e., set e^-β H≈∑_n=0^51/n! (-β)^n H^n, which was found to lead to Gibbs states of fidelities comparable to those obtained when using the exact operator produced by summing the infinite series <cit.>.




§ RÉNYI-ADAPT


The loss function which we are interested in comparing is a generalization of the quantum relative entropy known as the quantum Rényi divergence or “sandwiched” Rényi relative entropy  <cit.>. The Quantum Rényi divergence inherits many of the mathematical properties of the Rényi divergence, and it reduces to the quantum relative entropy (quantum analog of the KL-divergence) and the limit where the dominant parameter of the loss function goes to 1.
We are particularly interested in using an upper bound to the quantum Rényi divergence called the maximal Rényi divergence of order 2, 

    D_2(σ(θ)|ρ) = log( Tr(σ^2 ρ^-1) ).

Here, ρ is the density of the training data state and σ(θ) corresponds to the density of the output state of a parameterized quantum circuit, where θ are the circuit parameters. These loss functions have not been widely studied in a quantum computing context and were introduced in <cit.> as a way to do quantum generative training. 
The main reason for using D_2(σ|ρ) as a loss function is that it upper-bounds the quantum relative entropy and its gradients are considerably simpler to measure on quantum hardware than those of the ordinary Rényi divergence and quantum relative entropy <cit.>. Please consult Appendix <ref> for more information on the gradients.




Note that, in general, D_2(ρ|σ) ≠D_2(σ|ρ). However, if both ρ and σ are full rank, D_2(ρ|σ) = D_2(σ|ρ) = 0 if and only if ρ=σ <cit.>. This makes D_2(σ(θ)|ρ) is a reasonable loss function to consider for generative training.









As described in Section <ref>, the ADAPT algorithm selects an operator from a pool with the largest energy gradient at each iteration. We can apply a similar logic, but instead use the gradient of the maximal Rényi divergence to select operators from a pool {H_1, …, H_k} to form a trial state σ(θ) following the logic outlined in equations (<ref>) and (<ref>), which we can then optimize at each iteration. This technique is what we call Rényi-ADAPT.



§ EXPERIMENTS


Following the notation of <cit.>, we denote the n-qubit target thermal state as ρ, the controllable parameters as θ, and the trial state as σ(θ). In the context of the algorithm, the trial state starts as a pure state acting on a register of n_V visible qubits and n_H hidden (ancilla) qubits, within a full Hilbert space of ℋ_σ = ℋ_V ⊗ℋ_H.
The final (mixed) state output of σ(θ) is obtained by tracing out the hidden register.
Unless otherwise noted, all experiments were carried out with n_H = n_V, and we use the letter n to denote both.


All multivariate optimization steps were performed using the BFGS algorithm, as implemented in the  Julia package <cit.>.



 §.§ Adaptive optimization


In order to demonstrate that the adaptive optimization process works, we ran ADAPT on a single random two-local thermal state ρ of n = 3 qubits.
We trained ansätze with three different loss functions (<ref>,<ref>,<ref>) to learn ρ.
For the operator pool, we include all one- and two-local Paulis acting on any of the six qubits (three visible and three hidden).
There are a total of 6  1·3^1=18 one-local operators and 62·3^2=135 two-local operators, for a total of 153 operators.
In addition to the three ADAPT variants investigated, we ran VQE with a fixed-structure ansatz (“VQE”) consisting of all 153 Pauli rotations (with arbitrary ordering) for all three loss functions, to compare against previous work <cit.>.






Like the other adaptive methods, Rényi-ADAPT successfully learned the target state (Fig. <ref>). Because the VQE ansätze start out sufficiently expressive to reach the target state with just one optimization, they required drastically fewer iterations to converge compared to the adaptive methods, implying the fixed-structure approach would require drastically fewer circuit evaluations in a quantum experiment. In contrast, adaptive methods achieved excellent loss values with at least a 60% reduction in parameter count (Table <ref>) compared to the VQE ansätze. This translates to a commensurate reduction of circuit depth in a quantum experiment, making the adaptive approach the preferred method for devices with limited coherence times.







 §.§ Increasing system size


Next, we investigate the robustness of our method for larger system sizes. Similarly to the previous experiment, each algorithm was tested in 20 random thermal states, where the number of visible qubits now varied in the range n ∈ [1, 4]. For each method, we recorded the worst-case infidelity over all instances after each ADAPT iteration, enabling a direct comparison of compactness between the three loss functions.



For all system sizes, overlap-ADAPT achieved a given infidelity threshold with the fewest parameters, but performance was competitive among the loss functions (Fig. <ref>). At the largest system size of n = 4, ADAPT-VQE-Gibbs and Rényi-ADAPT stopped near infidelities of 10^-3 while overlap-ADAPT reached 10^-4. However, we were only able to run 1 trial at this size due to computational constraints, and so this behavior may not be representative. Interestingly, all methods required exponentially deeper circuits as n increased due to the larger Hilbert space dimension and the locality of the operator pool.
These simulations suggest that the Rényi divergence does not provide an advantage in terms of circuit depth.

It is worth mentioning here that the loss functions for Rényi divergence D_2(ρσ(θ)) (<ref>) and overlap ℒ_O(ρ, σ(θ)) (<ref>) were evaluated using the exact thermal state ρ, while the loss of ADAPT-VQE-Gibbs C(σ(θ)) (<ref>) was calculated using a truncated Taylor expansion of the thermal state, thus limiting the fidelity it could achieve (although it still achieves > 99.8% fidelity for the n=4 trial shown). This approximation was intentional; in most quantum machine learning applications, the target state ρ is known, whereas in ADAPT-VQE-Gibbs, which was designed for quantum chemical and condensed matter systems, ρ is the desired output for which a priori estimates are not guaranteed, necessitating the need for an approximation.




    
    
    
    
    







 §.§ Gradient results


The unbounded nature of the maximal Rényi divergence lends it resilience to barren plateaus, as discussed by Kieferova et al. <cit.>, providing better trainability for variational circuits. We investigated this claim by analyzing the gradients of the loss functions.

In a similar setup as before, using 20 random combinations of the target state ρ and the reference state σ_0, we calculated the initial pool gradients used to select the first operator. We denote the magnitude of the largest gradient as ||g||_∞, which we refer to as the L_∞ norm of the pool gradient. Because this did not require the full ADAPT procedure, we were able to run experiments on a system size of n=6 (12 total qubits, with n_H = n_V). We expected that barren plateaus would manifest as an exponential decay of ||g||_∞ as n increased, and therefore hypothesized that the Rényi divergence would not decay.







The results are shown in Fig. <ref> with the lines indicating the median  of the initial pool gradient ∞-norm over the trials and the shaded regions depicting the full range of values. The Rényi divergence exhibited larger, nearly constant gradients as the system size n increased, while the overlap and Gibbs loss functions decayed exponentially at rates similar to each other. 



Table <ref> contains exponential functions fitted to the medians of each loss function with respect to n. While Rényi-ADAPT still suffers exponential decay in the large n limit, the decay is significantly slower. To illustrate by how much slower, let us imagine that the gradient measurement has a resolution of 10^-5, such that when all operators have gradients below this threshold (i.e. ||g||_∞ < 10^-5), the ADAPT protocol cannot meaningfully begin. We denote this in Table <ref> as the “predicted failure". Using the loss functions for either overlap-ADAPT or ADAPT-VQE-Gibbs, the algorithm would fail to start at n ≈ 15 qubits (16 and 14, respectively), while with the Rényi divergence, this threshold is not reached until n = 67, a 4x improvement.
For our lower-precision criteria of ||g||_∞ = 10^-3, we expect that overlap-ADAPT and ADAPT-VQE-Gibbs will become untrainable at system sizes of n ≈ 10. Note that our simulations (n ≤ 6) are well below this regime, such that this advantage of Rényi-ADAPT is not reflected in the experiments whose results are shown in Figs. <ref> and <ref>.









 §.§ Initial state




The choice of reference state is important for the performance of ADAPT, with good reference states being close to the target <cit.>. For poor reference states, ADAPT may require many more operators, removing its compact-circuit advantage, or even fail to converge altogether. Some examples of good reference states include the Hartree-Fock state for the ground state of molecular electronic structure problems <cit.> and the Néel antiferromagnetic state for the ground state of lattice spin systems <cit.>. For overlap-ADAPT and Rényi-ADAPT, the ideal reference state is, of course, σ_0 = ρ.

An interesting possibility of the Rényi divergence is that it could avoid concentrating (or flattening out) far away from ρ. If true, this would improve the trainability of circuits and reduce the need to choose a good reference state, which would be helpful when one is either hard to produce or completely unknown for the given problem.



After processing the data in Fig. <ref>, we calculated the fidelities F(ρ, σ_0) between the random target and reference states. The initial pool gradient ||g||_∞ is shown in Fig. <ref> as a function of F. We observed that further away from ρ, the gradient landscapes for ADAPT-VQE-Gibbs and overlap-ADAPT decayed exponentially, suggesting that trainability could become an issue. In contrast, Rényi-ADAPT again demonstrated favorable near-constant gradients (indicative of the loss landscape not flattening), supporting the idea that it functions effectively with a wider range of reference states.






 §.§ Completion





ADAPT-VQE experiments require a specific choice of the value ε to take as the gradient norm threshold.
In this work, we adopted the somewhat arbitrary choice of ε=1e-3.
To assess the degree to which this choice was appropriate, we examine the L_∞ norm of the pool gradient ||g||_∞ for each ADAPT iteration, for all 20 trials where n=3 qubits.
As a general trend, pool gradients tend to fall exponentially as each parameter increases the expressivity of the ansatz.
However, when no further improvements can be made to the trial state, all pool gradients will rapidly drop to zero.
To highlight this behavior, we plot in Fig. <ref> the L_∞ norm of the pool gradient as a function of the percent completion of the ADAPT run, where a “completed” run is one where the final ||g||_∞<ε.

Our results show that, on average, ||g||_∞ values for both ADAPT-VQE-Gibbs and overlap-ADAPT decay only exponentially all the way through the ||g||_∞<ε stopping condition.
Meanwhile, the Rényi-ADAPT curves exhibit super-exponential decay roughly 80% of the way to the same stopping condition, followed by several runs dropping from ||g||_∞>1e-3 to ||g||_∞<1e-4 in a single ADAPT iteration.
We reiterate that Rényi-ADAPT tended to require deeper circuits to attain this convergence (as observed in Figs. <ref> and <ref>), so this observation does not necessarily indicate an advantage for Rényi-ADAPT, especially in terms of near-term viability.
However, this does suggest that Rényi-ADAPT requires a lower minimum resolution when measuring pool gradients to attain a region of strong convergence, implying that each pool operator could be selected in Rényi-ADAPT using a lower shot count than in the other methods, helping to offset the major cost of measurement overhead incurred by the ADAPT protocol.







§ DISCUSSION


In this work, we characterize Rényi-ADAPT as a method resistant to barren plateaus for preparing thermal states. It combines the trainability advantages of the maximal Rényi divergence <cit.> with the adaptive optimization procedure of ADAPT-VQE <cit.>. We did not observe the gradients obtained in Rényi-ADAPT to decay in our experiments. We conclude with a discussion of our results and how they relate to the existing literature on barren plateaus.

A recent review of barren plateaus by Larocca et al. <cit.> summarizes the types and origins of barren plateaus. They can arise from highly expressive circuits, poor choice of initial state, global measurements, and noise. The relationship between these is studied for 2-designs in <cit.>.

Expressiveness By iteratively constructing circuits, ADAPT balances the trade-off between constructing a sufficiently expressive circuit to reach the objective minimum and overparameterizing the circuit, which would induce barren plateaus.

Our results in Figs. <ref> and <ref> show overlap-ADAPT producing the most compact circuits in terms of parameter count. On devices where producing the most compact circuits is a priority (like near-term devices with limited coherence times), we surmise that the overlap would be the best choice of the algorithms we tested, provided a suitable reference state.

Note that the expressiveness of the circuit is characterized by the size of its dynamical Lie algebra (DLA) <cit.>. Since ADAPT can, in principle, append a previously-added operator (which would not increase the DLA), the DLA may be smaller than expected by assuming unique operators for each parameter. However, similar to the existing ADAPT literature <cit.>, we use parameter counts as a proxy for circuit expressiveness and leave the quantification of DLA to future work.



Measurement The compactness of ADAPT comes at the considerable expense of measuring the operator pool at each iteration. The 1- and 2-local pool (which has 3(n_V + n_H) + 9·n_V+n_H2 operators) we used is complete
. However, by using these small, local operators, constructing a large unitary requires many more iterations. We could alternatively choose a minimal operator pool, which has been shown to have a number of operators linear in the number of qubits. <cit.> This minimizes the measurement overhead in each ADAPT iteration, at the cost of requiring an even longer circuit to reach arbitrary states.

On the other hand, by expanding the pool to include all k-local Paulis, we can reduce the number of parameters required in the circuit, at the cost of significantly more measurements required to measure the full pool gradient. According to current barren plateau research, the increased globality of the measurements could be problematic, suggesting there is a trade-off between including larger pool operators and trainability. We did not investigate this in our work.





Initial State Choosing a random reference state increases the chances of a barren plateau, following the curse of dimensionality <cit.>. As detailed in Appendix <ref>, we had to choose a random reference state to allow ADAPT to construct mixed states. 

Our experiments show that Rényi-ADAPT has larger gradients for our reference states both with respect to increasing system size (Fig. <ref>) and larger distances from the objective (Fig. <ref>). Therefore, at least in the absence of a good reference state, overlap-ADAPT and ADAPT-VQE-Gibbs should have to make exponentially more measurements of the gradient, or they might possibly not start at all, a reason to favor Rényi-ADAPT.



As we used a state vector simulator with infinite shots, this was not shown in our experiments. Additionally, our simulations did not reach the scale where we anticipate overlap-ADAPT and ADAPT-VQE-Gibbs to start below the convergence criteria.

Noise Noise-induced barren plateaus are perhaps the most significant obstacle to leveraging near-term quantum devices that lack error correction. Longer circuits are generally noisier, which is the original motivation behind ADAPT-VQE <cit.>.

We did not test with a noisy quantum simulator, which we leave to future work. Our noiseless results show overlap-ADAPT producing more compact circuits, which one would expect to have less noise. However, as noise concentrates the loss function, this restricts the space of good reference states, which we found to be a potential obstacle to the scalability of overlap-ADAPT. Investigating this interplay would be interesting. Additionally, since Rényi-ADAPT was more trainable (initially) with poor reference states, one could imagine that a hybrid algorithm where Rényi-ADAPT prepares a reference state for overlap-ADAPT might perform well, combining the advantages of each algorithm we observed.



§ DATA AVAILABILITY


The code and data used for the simulations and figures can be found at <https://github.com/kmsherbertvt/RenyiADAPT>.



§ ACKNOWLEDGMENT

The authors would like to thank Ada Warren for her helpful comments and discussion on the ADAPT-VQE-Gibbs code.
This research was supported by the DOE Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704.

unsrt  






 §.§ Hyperparameters



We give the hyperparameters of our experiments in Table <ref>. 𝒫_k^(n) denotes the set of all k-local Pauli operators that act on n qubits. The gradient convergence expressions refer to the gradient of the operator pool.





The Hamiltonians used in the experiments were generated by assigning random coefficients drawn from a normal distribution with zero mean and unit variance to one- and two-local Pauli operators on the n_V visible qubits. To ensure that the energy scale remained the same with different system sizes, the Hamiltonians H = ∑_i c_i P_i (where each P_i is one- or two-local) were subsequently normalized such that c_i →c_i/||𝐜||. Twenty such Hamiltonians were generated to sample various two-local models. The operator pool consisted of all one- and two-local Pauli operators on the total number of qubits n_V + n_H.

The reference states used were produced by generating states that are partially-entangled between the visible and hidden qubits. As discussed in previous work <cit.>, reference states without any entanglement between the visible and hidden qubits maximize the purity Tr(ρ^2) of ρ = Tr_A |ψ⟩⟨ψ|. Non-zero gradients of the ADAPT-VQE-Gibbs objective function are then obtained only from the first term in (<ref>); the structure of the pool operators then implies that the operator with the largest gradient will always be local to the visible system qubits, without the ability to generate entanglement between it and the hidden system. This would result in the true Gibbs state (a mixed state) being unattainable. A similar argument can be made against the use of maximally mixed reference states. Correspondingly, for Rényi-ADAPT, it was found that for product reference states, individual Pauli operators were incapable of generating entanglement between the visible and hidden qubits, and that operators consisting of linear combinations of diagonal and off-diagonal Pauli operators (`diagonal' here refers to operators whose eigenvectors the product reference is composed of) would be necessary in order for such entanglement to be created.

In order to overcome the constraint on the pool operators arising from using product states, partially-entangled reference states were generated by first applying random single-qubit R_y rotations to each qubit in the state |0⟩^⊗ n_V+n_H, with the angle being drawn from [ - π, π), followed by a layer of CNOT gates between each visible-hidden qubit pair, with the control being on the hidden qubits. Twenty such reference states were generated to sample various initializations. We leave the investigation of operator pools consisting of linear combinations of Pauli operators to future work.

The convergence criterion for ADAPT was set such that the calculation was ended when the gradient norm of the pool operators was below 1e-3.



 §.§ Gradients

In order to select operators and conduct gradient descent, it is essential to have a formula for gradients that can be measured in a quantum computer. Below we provide a discussion of the closed form expression of the gradients of each loss function that we considered in our work:





  §.§.§ Overlap-ADAPT


The expression for the pool operator gradients at step k+1 may be written as <cit.>

    ∂/∂θ_k+1. ⟨Ψ_ target | e^-iθ_k+1 A_k+1ψ_k(θ^(k))⟩|_θ_k+1=0
     = ⟨Ψ_ target | i A_k+1ψ_k(θ^(k))⟩.




  §.§.§ ADAPT-VQE-Gibbs

In the case where the ansatz is composed of operators e^-iθ G whose Hermitian generators G have exactly two distinct eigenvalues ϵ_0 and ϵ_1, the exact gradients of the objective function can be efficiently obtained using the parameter-shift rule. Defining an auxiliary function 

    C̃(θ,ϕ) = - Tr(ρ_G σ(θ)) + 1/2Tr(σ(θ) σ(ϕ)),

the gradient of C(σ(θ)) may be written as

    ∂/∂θ C(σ(θ)) = r [ C̃(θ + π/4r,θ) - C̃(θ - π/4r,θ) ],

where r = 1/2(ϵ_1 - ϵ_0). 



  §.§.§ Rényi-ADAPT

This divergence has closed-form gradients that can be efficiently computed on a Quantum Computer. Define a parameterized quantum circuit by σ_v := Tr_h[ Π_j=1^N e^-iH_jθ_j|ψ_ ref⟩⟨ψ_ ref|Π_j=N^1 e^iH_jθ_j], where we measure the output density by taking a partial trace, denoted by Tr_h, over a hidden subspace of some set of ancilla qubits. In this case, the gradients of equation <ref> take the form,

    ∂_θ_kD_2(σ_vρ) = -i Tr({ Tr_h([H_k, σ]),σ_v}ρ^-1)/ Tr(σ_v^2 ρ^-1).
 
where H_k = ∏_j=1^k-1 e^-iH_j θ_j H_k ∏_j=k-1^1 e^iH_j θ_j. 
See Ref. <cit.> for more details on the algorithms and access model needed for this calculation.


 Paper Title: Pulse family optimization for parametrized quantum gates using spectral clustering.






Department of Applied Physics, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
Eindhoven Hendrik Casimir Institute, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
[Corresponding author: ]r.j.p.t.d.keijzer@tue.nl 


Eindhoven Hendrik Casimir Institute, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
Q-CTRL, Berlin, Germany

Q-CTRL, Berlin, Germany


Department of Applied Physics, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
Eindhoven Hendrik Casimir Institute, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands




§ ABSTRACT 
Parametrized gate circuits are used in plentiful applications in the current NISQ era of quantum computing. These parametrized gates are chiefly implemented using analytically found pulse protocols, often yielding suboptimal gate times, and consequently, fidelities. Alternatively, gate optimization algorithms are designed to construct high fidelity pulses for individual, fixed points in continuous parameter space. Gates for intermediate parameters can subsequently be found by some form of interpolation between previously constructed pulses. Nevertheless, it is not guaranteed (as with analytic protocols) that the pulses found by the optimization algorithms belong to the same family of solutions and thus show resemblance. Interpolation between two pulses of differing solution families often leads to high infidelities, as the pulse strays away from the minimum in the parameter/fidelity landscape. In this work, we introduce a spectral clustering method to sort high-fidelity, optimized pulses in families, and interpolating solely between pulses of the same family. Accordingly, interpolations will always approach maximal fidelity. Furthermore, as more than one pulse family is constructed, the parameter space can be partitioned according to which family prevails fidelity-wise. This work provides a meticulous demonstration of our constitutive continuous gate family construction by applying it to a universal gate set for Rydberg and Cat qubits under noise.


Pulse family optimization for parametrized quantum gates using spectral clustering
    S.J.J.M.F. Kokkelmans
    August 5, 2024
==================================================================================





§ INTRODUCTION



In order for a quantum computer to be able to perform all possible computations, it has to possess the ability to execute a universal gate set, consisting of all possible single qubit rotations and at least one entangling two-qubit operation <cit.>. The canonical single qubit gate set are the rotations gates over angle θ: R_X(θ), R_Y(θ), and R_Z(θ), which form the basis for many variational quantum algorithms (VQAs) <cit.>. These parametrized gates are executed in physical systems using some control function on the qubits, i.e. laser pulses or electrical currents, specific to each individual parameter. Current era quantum computing systems are in the NISQ-era, where qubits are highly susceptible to noise <cit.>. This means that it is important to choose the optimal control function to mitigate noise, which is often not a standard analytical protocol, but some faster control pulse as constructed by a pulse optimization algorithm <cit.>. These variational methods often lead to faster, and consequently higher fidelity, pulses.

The problem with these optimization algorithms is that, unlike analytical protocols, they only prescribe pulses for one fixed point in the parameter space at a time, and not for the entire continuous set of parameters, which is especially relevant for VQAs <cit.>, where copious different parameter realizations are necessary. Constructing control pulses every time a new unique parameter is required is far too computationally expensive, making it infeasible in the majority of VQAs.



The task at hand is to extend a collection of solutions for discrete points in parameter space to the entire continuous set by some form of interpolation. However, this approach has an underlying complication. Assume a set of M subsequent, close points in the gate parameter space {θ_i}_i^M for which one wishes to optimize the pulses. Initially, the pulse optimization algorithm will find a local minimizing pulse for θ_0, and because the optimization problem is regular, there will exist a corresponding local minimizing pulse for θ_1 as long as the two values for θ are adequately close. In general, there will exist corresponding pulse solutions for each θ_i, which we then call a pulse family, as introduced in Ref. <cit.>. Interpolating within a pulse family will yield pulses for intermediate values of θ, with comparable high fidelities. The problem, however, is that there is no guarantee that for every value of θ_i, matching local minima will be found, and thus there is no guarantee that all the optimized pulses found will belong to a single family. Interpolating between two pulse families, will generally yield lower fidelities because the resulting pulses will not be in local minima of the pulse/fidelity landscape, see Fig. <ref>.

Previous work has been done on the construction of high fidelity continuous gate sets. References <cit.> construct pulses to mitigate hardware errors based on a parameterized Hamiltonian. These results are extended to the continuous case in <cit.>. Reference <cit.> is the first to mention the problem of similarity between pulses, but only related to single parameter optimization. References <cit.> all devise methods to optimize continuous gate sets while maintaining pulse similarity across different parameters for interpolation purposes. In <cit.>, a neural network approach is used, where the networks are trained by randomly sampling the parameter space. The disadvantage of such a black box method is that verification of the resulting pulses is non-trivial. In <cit.>, Trotterization is used to implement continuous parameter exponential matrices, but does not account for time optimization. The work in <cit.> utilizes Tikhonov regularization and feedforward to enforce pulse similarity, which is partly employed in our method. However, our work is the first to utilize clustering methods from graph theory <cit.> to find multiple families of gates, which has three main advantages over all previous methods:

    
  – Foremost, partitioning the parameter space according to where certain pulse families outperform the others has the potential to yield exceedingly higher fidelities than a singular family approach;
    
  – Finding multiple families allows for the selection of pulses which are best suited to experimental procedures after verification on a setup;
    
  – The clustering method provides a more well-suited manner of interpolating pulses using Wasserstein-2 distances from optimal transport (see Sec. <ref>).


The layout of this paper is as follows. Section <ref> describes optimal control methods used to construct optimal pulses for discrete parameter values θ_i. Section <ref> prescribes our pulse family clustering algorithm, its relation to other methods, as well as its advantages. Section <ref> details the interpolation methods employed in this work. In Sec. <ref>, we show initial results for our approach, starting with results on the Wasserstein-2 distance to characterize pulse similarity in Sec. <ref>. Section <ref> illustrates the increases in fidelities our method has for interpolation. Finally, Sec. <ref> summarizes and presents a future outlook.



§ PULSE OPTIMIZATION


The application analyzed in this work is creating a universal parametrized gate set of R_X(θ), R_Y(θ), R_Z(θ) and R_ZZ(θ), where θ∈[0,π]. This will be performed on Rydberg qubits <cit.>, an architecture that recently has become exceedingly mature, and also on Cat qubits <cit.>, which recently are gaining traction within the quantum computing community. This illustrates the versatility of our methods. In this section, we briefly describe the optimization procedure for fixed parameters θ_i, which has been well-described in literature before <cit.>. The evolution of the qubit system density matrix ρ_t follows a Lindblad equation <cit.> of the form

    ∂_tρ_t   =-i[H_sys[z(t)],ρ_t]+ℒ(ρ_t),   ρ(0)=ρ_0,
    ℒ(ρ)   =∑_k γ_k V_k ρ V_k-1/2γ_k{V_k^† V_k,ρ},

where H_sys[z(t)] is the system Hamiltonian controlled by user-defined, and optimizable pulses z(t), and ℒ is the Lindblad operator responsible for the different sources of decoherence in the system. V_k and γ_k respectively are the jump operators with corresponding strengths, defining the Lindblad operator. The pulses are functions on [0,T], where T is the gate end time, bounded by experimental limitations as z_min≤ z(t) ≤ z_max. The physical interpretation of the pulse type is system specific. In Rydberg systems, we can control the transitions between the |0⟩, |1⟩ and |r⟩ states on qubit j using coupling strengths Ω_ab,j and detunings Δ_b,j on transition |a⟩↔|b⟩ to get z(t)∈{Ω_01,j(t),Δ_1,j(t),Ω_1r,j(t),Δ_r,j} <cit.>. Meanwhile, Cat qubits offer control on the single photon drive E_j(t), the detuning Δ_j(t), interaction strength g(t), and the two photon drive G_j(t) <cit.>. This gives  z(t)∈{E_j(t),Δ_j(t),G_j(t),g(t)}. For the Rydberg and the Cat qubit systems, we introduce characteristic timescales τ_Ryd=1/Ω_max and τ_Cat=1/K respectively, where Ω_max is a maximal coupling strength in the Rydberg system and K is the Kerr non-linearity in the Cat qubit system. Since our methods are agnostic to the qubit architecture and the physical implementation is not integral to understanding our results, we do not further discuss details on the specific systems here. For detailed Hamiltonians, Lindblad operators and a discussion on the pulse end times T, see App. <ref>.

A solution to the Lindblad equation (<ref>) for a specific set of pulses z(t) is described by a trace preserving, completely positive operator called a quantum channel ℰ_z, s.t. ρ_T=ℰ_z(ρ_0) <cit.>. For each specific gate U∈{R_X(θ_i), R_Y(θ_i), R_Z(θ_i), R_ZZ(θ_i)} to be created, the goal is to optimize the fidelity given by

    min_z F(ℰ_z, Û):=min_z∑_j Tr(U P_j^† U^†ℰ_z(P_j))+d^2/4^Nd+d^2,

as in <cit.>. Here N is the number of qubits and d is the dimension of the qubit system. For Rydberg systems d=3^N and for Cat systems d=2^N. P_j are the Pauli matrices (for Rydberg extended to a 3-dimensional system) of which there are 4^N. Using the software package Boulder Opal, we optimize the pulses z(t) for the cost function in (<ref>), using the package's standard convergence criteria <cit.>. For all problems in this work, we optimize M=20 equidistant predefined angles {θ_i}_i^M in the parameter space [0,π].





§ PULSE CLUSTERING


After having optimized the pulses for discrete fixed parameters {θ_i}_i^M, they need to be partitioned in corresponding solution branches, or families. Here, we detail our method of clustering pulses in a fixed number of families using spectral clustering <cit.>. This method sorts the nodes V of a weighted graph G=(V,E,w) into a fixed number of clusters. E are the edges of the graph, and w are the weights on these edges encoding the similarity of two nodes. To apply this method to the discrete pulse sorting, we define a fully-connected graph, where each node i corresponds to one of the optimized pulses found for a specific parameter θ_i, as shown in Fig. <ref>. The weights are set as w(z_1,z_2)=(dist(z_1,z_2)+ϵ)^-1, where dist is some distance on the space of pulses and ϵ=10^-4 is a small regularization parameter. As a basic approach, the L_2 distance <cit.> can be used as

    L_2(z_1,z_2)=∫_0^T |z_1(t)-z_2(t)|^2dt.





This distance function is often implemented as it is efficient to compute. However, it does not fully serve our needs, since it does not capture shifts on the time axis well. The L_2 distance only compares pulse values pointwise in time (see Fig. <ref>, where the highest peak shifts, causing a big L_2 distance between the pulses even though they are quite similar in shape). As long as two pulses do not overlap, a large time shift causes the same maximal L_2 distance as a small shift. To remedy this issue, the Wasserstein-2 distance <cit.> can be used. This distance has its origins in the mathematical framework of optimal transport and defines the cost of moving one distribution onto another, which is more closely related to our intuitive likeness of pulses, see Fig. <ref>. The important point to understand is that the Wasserstein-2 distance treats the pulses as distributions on the plane ℝ^2 and calculates the cost of moving one onto the other and defines this as the distance <cit.>. The exact definition of the distance is rather mathematically involved, and therefore deferred to App. <ref> [For the Wasserstein-2 distance, pulses are discretized as point clouds with points equidistant along the curve, and Sinkhorn-Knopp <cit.> is used to identify the minimizing coupling.].



Using one of the previously defined distances, a similarity matrix consisting of the edge weights w can be constructed, see Fig. <ref>. This serves as the input for the spectral clustering algorithm. As the spectral clustering algorithm requires a fixed number of clusters, we determine the best possible number of clusters by means of the Elbow method <cit.>. This heuristic determines the point at which an additional cluster does not capture the pulse differences better, and over-fitting starts. For this work this value is always found to be 3 clusters, but in more complex problems with high dimensional parameters, could definitely be expected to increase. 



§ PULSE INTERPOLATION


After the originally optimized pulses have been partitioned into families by the clustering algorithm, the following step is to extend the families to the entire parameter space by means of interpolation. In this work, the parameter space will always be θ∈[0,π] so that we can refer to angles. For multiparameter gates, an analogous approach can be pursued.


For each family, we want to first construct pulses on all of {θ_i}_i^M, instead of only the subset assigned to it in the clustering. Consider one of the families, there will be a lowest and highest angle assigned to this family in the clustering. Note that a family does not necessarily include only subsequent angles, but might include gaps between disjoint regions. If an angle θ_i is not assigned to this family and is in such as gap, we take a linear interpolation as an ansatz and optimize the pulses using Boulder Opal. For fixed angles θ_i outside this range, we simply use the lowest or highest angle pulse as an ansatz and optimize. Employing these ansatzes, hopefully a pulse is found for that parameter belonging to the same family (reminiscent of Tikhonov regularization <cit.>). This leaves us with several extended families of pulses Z_j on all predefined parameters {θ_i}^M_i (see Fig. <ref>).


Using linear interpolation within a single family, new pulses for all parameters in the parameter space can be constructed, finalizing our construction of a continuous gate set for each family. Linear interpolation is computationally very efficient, especially when compared to constructing pulses from the ground up. By interpolating within a family, we expect higher fidelities than if we were to interpolate on the original set of pulses.


.





§ RESULTS





 §.§ Distance function comparison


This section contrasts the effectiveness of clustering families for the Wasserstein-2 and L_2 distances. For arbitrarily optimized pulses, there is no established family structure to validate against. In order to straightforwardly compare the distance measures, we construct J mock families of pulses Z_j={z_θ_j,i}_i^M. The pulses within one family are constructed in such a way that pulses z_θ_i and z_θ_j look alike for i close to j, as they would in actual applications. The exact details of this are left for App. <ref>. For each angle θ_i, a corresponding pulse z_true,i from a family Z_true,i∈{Z_1,...,Z_J} is randomly picked for the final pulses. This mimics the pulses found by the Boulder Opal optimization procedure and accrues a set of pulses as in Fig. <ref>b, of which we have knowledge on the family structure.

On these mock pulses, the distance metrics are compared by constructing a distance matrix as in Fig. <ref>, and having the spectral clustering algorithm assign to each pulse i a family Z_dist,i∈{Z_1,...Z_J}. The conditional probability ℙ_dist of finding a correct match is approximated as 


    ℙ(cor.|fnd.)≈∑_i,j1[Z_dist,i=Z_dist,j]1[Z_true,i=Z_true,j]/∑_i,j1[Z_dist,i=Z_dist,j].


The reason for choosing this figure of merit is that when two pulses are found to be matching, they only lead to high fidelity as long as they belong to the same family. Missing a match because two pulses from the same family are clustered in different families will not lead to faulty interpolations, and is thus less important from a fidelity perspective (nevertheless, the Wasserstein-2 distance also outperforms the L_2 distance in terms of missed matches). Figure <ref> shows the comparison between the two distances. Wasserstein-2 can be seen to largely outperform the standard L_2 distance, as it is able to capture the pulse family characteristics better due to its transport based nature. In the rest of this work, the Wasserstein-2 distance is employed for clustering.



    [b]
            
    




 §.§ Interpolation of gates


To test our clustering method for the interpolation of pulses, we first optimize for M=20 equidistant angles θ_i∈[0,π] for R_X(θ), R_Y(θ), R_Z(θ) and R_ZZ(θ) for both Rydberg and cat qubits. This allows us to gather results on a universal set of gates on two vastly different systems, showing the versatility and robustness of our method.

The pulses are clustered using our spectral clustering method from Sec. <ref>, where the Elbow method indicates that for all considered applications 3 clusters is the optimal number. The families found are extended to the entire M=20 predefined angles {θ_i}_i^M using the methods from Sec. <ref>. Lastly, for each family, we interpolate the pulses halfway the predefined angles, and determine their fidelity using (<ref>).

Figure <ref> shows the results for the R_Y(θ) gate on cat qubits. Generally, the R_Y gate is the most interesting of the three qubit gates because it requires both amplitude and phase control. In Fig. <ref>a, we see that when optimizing using Boulder Opal, both the original pulses and the clustered families reach high fidelities, but the interpolation on the original pulses is very bad for almost all angles. For the clustered families, on the other hand, those with good interpolations are found for the entire parameter space. This is highlighted in Fig. <ref>b, where the parameter space is partitioned according to where each family performs best, leading to great improvements in the  interpolation fidelities compared to the original pulses. Figures. <ref>c and  <ref>d further illustrate this  by showing that the original pulses are dissimilar for subsequent angles, resulting in low fidelities, whereas the pulses from the families in the highlighted areas correspond well and thus result in high fidelities.

Similar results can be seen in Fig. <ref> for the 2-qubit R_ZZ(θ) on a Rydberg system. Here we see good interpolations of the original pulses around θ=π/2, which is also retrieved for one of the families. However, for low and high angles the interpolation fidelities become quite low for the original pulses (likely due to the presence of many local minima), but high fidelity pulses ones are constructed for the clustered families.

Figure <ref> shows all fidelities of interpolated pulses for the problems considered. Across all parametrized gates for both qubit types, there is an increase of half up to a full order of magnitude in fidelity. Partitioning the parameter space with specific families (Clustered, pink results) results in a big advantage in both mean and variance of the fidelities compared to individual clusters (purple). Here individual clusters would be comparable to a Tikhonov regularization method as in <cit.>, since ansatzes for pulse optimization are used. Along the same lines, especially note the much smaller variances of the fidelities indicating that our partition clustered method yields good pulses over the entire parameter space instead of only in a specific region. The individual family method definitely results in good interpolations for certain regions of parameter space, but multiple families seem to be necessary to patch together a faithful continuous gate set over the entire parameter space. This hails especially true for the cat qubits, as seen from the fidelities in Fig. <ref>, which is a more complex optimization problem where likely many local minima in parameter/fidelity space exist.


    [b]
            
    



    [b]
            
    


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    [b]

    




§ CONCLUSION


This work discusses and analyzes a new method for constructing continuous parameter gate sets based on the clustering of pulse families. This is an important problem as for virtually all NISQ era applications parametrized gates are required, and optimizing for each individual parameter necessary in a problem is extremely computationally expensive and slow. In our method, spectral clustering using a similarity measure based on Wasserstein-2 distances results in multiple families of pulses on which inter-family interpolation can be performed. The advantage of this method is that multiple families of pulses are found from which the best ones can be selected for a specific regime of parameter space. This method will prove to be useful for experimental realization, where one family is possibly easier to implement than another. On all gates and on both qubit systems considered, our method leads to significantly better fidelities and much more consistent interpolations than would be the case for original pulse optimization or Tikhonov based feedforward methods (individual family). In this, we highlight the great performance in cat qubits system, which shows the importance of our method in increasingly more important complex qubit schemes.


In future work, we hope to extend these methods to multiple parameter gates, which might be implemented in more complex VQA or QAOA type problems. Furthermore, we are curious to see the influence of directly optimizing the fidelity via the quantum channel, thus optimizing more directly for the losses, instead of optimizing the unitary evolution and post-processing the influences of the Lindbladian terms. Different parameters have different optimal gate end times, we wish to extend our methods to accommodate for varying gate times instead of one fixed optimal gate time for the largest parameter value (θ=π in this work). Lastly, we hypothesize that a more sophisticated way of interpolation, potentially based on Wasserstein-2 transport or on the Hamiltonian dynamics, will yield better fidelities for the interpolated pulses. 






§ ACKNOWLEDGEMENTS

We thank Klaas Wijnsma, Jasper Postema, Julian Teske, Kisa Barkemeyer, Kara Maller, and Oliver Tse for fruitful discussions. This research is financially supported by the Dutch Ministry of Economic Affairs and Climate Policy (EZK), as part of the Quantum Delta NL program,  the Horizon Europe programme HORIZON-CL4-2021-DIGITAL-EMERGING-01-30 via the project 101070144 (EuRyQa), and by the Netherlands Organisation for Scientific Research (NWO) under Grant No. 680.92.18.05.



§ COMPETING INTERESTS

The authors declare no competing interests.



§ DATA AVAILABILITY

The data supporting the findings  are available from the corresponding author upon reasonable request.



§ CODE AVAILABILITY

The code supporting the findings are available from the corresponding author upon reasonable request.





apsrev4-1





§ QUBIT SYSTEMS


In this appendix we describe the qubit systems resulting in the system Hamiltonians H_sys, jump operators V_k and control pulses z(t). Furthermore, we describe the pulse optimization procedure in more detail.


In order to restrict the parameter space and consequently enhance the likelihood of yielding pulses in the same family, we set the pulses for qubit 1 equal to those of qubit 0 in the R_ZZ two qubit case.  All pulses are discretized as 30 segment piecewise constant functions, which are smoothened via a Gaussian kernel and are equated to 0 at their beginning and end values, lastly being put through a low-pass filter. For each problem analyzed, the pulse end time T is determined as the end time for which the problem at θ=π has the highest fidelity. The existence of such an optimal time follows from the reasoning that for too large gate times, decoherence effects dominate and decrease fidelity. On the other hand, if the gate time is too small, the quantum speed limit is not reached <cit.> and there is no way of constructing a minimizing pulse, leading to low fidelities. The quantum speed limit likely has the highest value for θ=π because it least resembles the identity. To ensure we reach the quantum speed limit for all θ, we choose T for all problems as the optimal time for θ=π. 



 §.§ Rydberg neutral atoms


This section introduces basic Rydberg physics to identify what control pulses can look like for this system, as discussed in Sec. <ref>. In general, this will yield the Hamiltonian and possible Lindblad operators for the systems, as in (<ref>). A Rydberg system is a system of individual neutral atoms trapped in laser optical tweezers, where the electronic states encode for the qubit manifold <cit.>. For this work, we assume gg qubits <cit.> such that for each qubit we have the states |0⟩, |1⟩ and |r⟩, leading to find d=3^N as in (<ref>).



To perform single qubit transitions on qubit j between states |a⟩ and |b⟩, a laser interacts with the atom to realize the Hamiltonian <cit.>

    H_j^ab=Ω_ab,j(t)/2(e^i φ_ab,j(t)|a⟩_j⟨ b|_j+e^-i φ_ab,j(t)|b⟩_j⟨ a|_j)-Δ_b,j(t)|b⟩_j⟨ b|_j.

Here, Ω_ab,j(t) denotes the coupling strength, φ_ab,j(t) the phase of the laser coupled to atom j, and Δ_b,j(t) = ω_ab,j(t)-ω̃_ab the detuning of the laser frequency ω_ab,j(t) from the energy level difference ω̃_ab. In current Rydberg systems, one has exquisite control over Ω and Δ, and less over φ <cit.>, thus we set φ=0. For our systems, we assume transitions |0⟩↔|1⟩ and |1⟩↔|r⟩. This renders control pulses z(t)∈{Ω_01,j(t),Δ_1,j(t),Ω_1r,j(t),Δ_r,j}. Notice that having both coupling and detuning allows for full control on the Bloch sphere of each individual qubit, allowing for rotational control <cit.>. For this reason (and to restrict parameter space), we only optimize for Ω_01 and Δ_1 for the single qubit gates and set the other pulses equal to z(t)=0.  



The Rydberg states |r⟩ are high-energy excited states that have a passive `always-on' interaction, which is described by a non-controllable Hamiltonian H_V <cit.> as a Van der Waals interaction (VdW) <cit.>

    H_V   =∑_i=1^m∑_j=1^mC_6/R_ij^6|rr⟩_ij⟨ rr|_ij,

where R_ij is the interatomic distance and C_6 is an interaction coefficient. For the total Hamiltonian H_sys

    H_sys=∑_j H_j^01+H_j^1r+H_V.


Furthermore, the Rydberg states have a finite lifetime before decaying to |1⟩. For completion, we also introduce decay from |1⟩ to |0⟩, which results in the jump operators

    V_1,j=|0⟩_j⟨ 1|_j,   V_r,j=|1⟩_j⟨ r|_j.


In the optimization procedure, Ω_max=10MHz defines the timescale, and the pulses are constrained as |z(t)|≤Ω_max. Furthermore, we set γ_1,γ_r=0.01Ω_max which will incur fidelity losses. Lastly,  C_6/R^6=10^3·Ω_max is taken (we only consider up to two qubits in this work). The choices for physical parameters are inspired by <cit.> and summarized in Table <ref>. 








 §.§ Cat qubit Hamiltonians

Our discussion of cat qubits largely follows the explanations from <cit.>. The cat qubit can be realized in a Kerr parametric oscillator with a two-photon pump <cit.>. In the rotating frame of the two-photon pump frequency ω_p, the main system is described by 

    Ĥ_1=-K â^† 2â^2+G_j(t)(â^† 2+â^2)

with K the Kerr non-linearity which is constant and chosen as our timescale, G_j the two-photon drive amplitude, â the annihilation operator and â^† the creation operator. Rewriting the system Hamiltonian gives

    H_1,j=-K(â^† 2-G_j(t)/K)(â^2-G_j(t)/K)+G_j(t)^2/K.

With the knowledge that â|±α⟩ = ±α |±α⟩ one can see that the degenerate eigenstates of the Hamiltonian are the coherent states |±α⟩ with eigenenergy G^2/K. Since these eigenstates are degenerate, a linear combination 

    |C_α^+⟩=N_±(|α⟩±|-α⟩)

of these states is also an eigenstate. The states from (<ref>) are the so-called cat states, with N_± a normalization constant. The orthogonality of the cat states from (<ref>) allows for the following encoding

    |0⟩ = |C^+_α⟩ + |C^-_α⟩/√(2),    |1⟩ = |C^+_α⟩ - |C^-_α⟩/√(2),

as our computational basis, with |0⟩ and |1⟩ denoting the computational basis states. The corresponding Pauli matrices are defined through the computational basis states, e.g. Y=i|0⟩⟨1|-i| 1⟩⟨0|. The computational basis is chosen this way, foremost because single photon-loss (the dominant error mechanism in the system) is exponentially suppressed in α. Because we can not simulate the entire Fock space, we truncate its dimension at 20 states for single qubit gates and at 16 for R_ZZ.


The control of these systems is described through the following Hamiltonians

    H_2,j=-Δ_j(t) â_j^†â_j+E_j(t)(â_je^-iθ_j + â_j^† e^iθ_j),    
        H_3=g(t)(â_1â_2^† + â_1^†â_2),

where a detuning Δ_j(t) is created between the two-photon drive and the resonator Δ = ω_r - 2 ω_p, with ω_r and ω_p the resonator and two-photon pump frequency respectively. θ_j the phase of the drive, which is chosen at θ_j=0 (except for R_Y). E_j(t) is the amplitude of the single-photon drive. g(t) is a two-photon exchange between the two resonators. When |g(t)|,|E_j(t)|,|Δ_j(t)|≪ G the computational states are approximately kept in the computational basis. Together with single photon loss, dynamics to outside the qubit manifold is the main source of error. This leaves us with the pulses z(t)∈{E_j(t),Δ_j(t),G_j(t)}. The total system Hamiltonian takes the form

    H_sys=∑_j H_1,j+H_2,j+H_3.

Single photon losses on qubit j are obviously modelled by an annihilation operator a_j, resulting in a jump operator of the form V_j=a_j.






§ WASSERSTEIN-2 NORM


The Wasserstein-2 distance W_2(z_1,z_2) calculates the cost of moving the pulse z_1 onto z_2, and is more closely related to our intuitive understanding of similarity between pulses than the L_2 distance <cit.>. To understand this distance, we first define a distribution on a pulse μ[z] as


    μ[z](A)=1/c[z]∫_(t,y)∈ Aδ(y=z[t])d(t,y),   A⊂ℝ^2.

Thus, the measure is over ℝ^2 and only puts weight on the points of the pulse (t,z(t)). Here c[z] is a normalization constant s.t. μ[z](ℝ^2)=1 and is simply equal to the length of the curve, e.g. 

    c[z]=∫_0^T √(1+(∂_tz)^2)dt.

We now want to transport the distribution μ[z_1] to the distribution μ[z_2], and for that purpose define a coupling γ∈Π(μ[z_1],μ[z_2]), which is a joint probability measure on ℝ^2×ℝ^2 whose marginals are μ[z_1] and μ[z_2] <cit.> (see Fig. <ref>), e.g.

    ∫_ℝ^2γ(x,y)dy=μ[z_1](x),  ∫_ℝ^2γ(x,y)dx=μ[z_2](y).

Here, Π(μ[z_1],ν[z_1]) is the set of all transport couplings between  distributions μ[z_1] and μ[z_2]. The Wasserstein-2 distance W_2(z_1,z_2) is then defined as the minimal coupling cost

    W_2(z_1, z_2)^2:=min_γ∈Π(μ[z_1],μ[z_2])∫_ℝ^2×ℝ^2|x-y|^2  dγ(x,y).




§ PULSE CONSTRUCTION


In order to construct a mock family of pulses, we define for θ_0=0 a sum of G Gaussians where each Gaussian g has a random amplitude a_0,g,1, center a_0,g,2 and variance a_0,g,3 chosen according to some uniform random distributions. For subsequent θ_i+1, we then have these coefficients evolve according to

    a_i+1,g,k=a_i,g,k+μ_g,kΔθ+σ_g,k𝒩√(Δθ),

where μ_g,k and σ_g,k are predefined drifts and variances, respectively. 𝒩 is a normally distributed random variable and Δθ is the step size. These kinds of pulses mimic the pulse optimization solutions we get when constructing pulses for a continuously parametrized gate U(θ). When the step size, drift and variances are high, subsequent pulses look less alike and the clustering problem becomes harder to solve, leading to lesser results for both Wasserstein-2 and L_2 distances. Nevertheless, in all tried out configurations we saw Wasserstein-2 outperform L_2, especially when subsequent pulses are more alike, which can always be achieved by lowering Δθ.



 Paper Title: Operator on Operator Regression in Quantum Probability.


eqnarray
t1Theorem[section]
d1Definition[section]
c1Corollary[section]
l1Lemma[section] r1Remark[section]
e1Example[section]
p1Proposition[section]
a1Result[section]







Operator on Operator Regression in Quantum Probability
    Suprio Bhar 

	IIT Kanpur

	Department 
 of Mathematics and Statistics 

	Kanpur 208016, India

	Email: suprio@iitk.ac.in 

	Subhra Sankar Dhar 

	IIT Kanpur

	Department of Mathematics and Statistics 

	Kanpur 208106, India

	Email: subhra@iitk.ac.in

        Soumalya Joardar

        IISER Kolkata

        Department of Mathematics and Statistics

        Mohanpur 741246, India

        Email : Soumalya@iiserkol.ac.in

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



    Abstarct 

 

This article introduces operator on operator regression in quantum probability. Here in the regression model, the response and the independent variables are certain operator valued observables, and they are linearly associated with unknown scalar coefficient (denoted by β), and the error is a random operator. In the course of this study, we propose a quantum version of a  class of estimators (denoted by M estimator) of β, and the large sample behaviour of those quantum version of the estimators are derived, given the fact that the true model is also linear and the samples are observed eigenvalue pairs of the operator valued observables. 



§ INTRODUCTION

Regression analysis is one of the most well-known toolkit in statistical modelling. More specifically, regression analysis studies the “relationship" between the dependent/response variable and one or more independent/explanatory variables. This “relationship" can be linear, non-linear or even non-parametric as well (see, e.g., <cit.> and a few relevant references therein), and using various statistical techniques, one may estimate the unknown parameters involved in the “relationship". Recently, the concepts of regression have been extended to functional valued random elements also; for instance, the interested reader may look at <cit.> and a few relevant references therein regarding function on function regression. However, to the best of our knowledge, regression analysis is an area unexplored in the setting of quantum probability. This work aims to investigate various statistical issues involved in operator on operator regression in quantum probability. We first provide a literature review and then brief the research problem without much technical details in the following. 

In the context of other statistical concepts studied in the language of quantum probability theory, recently, <cit.> extended the concepts of sufficient statistic and Rao-Blackwellization in the framework of quantum probability, and a few years before, <cit.> overviewed and reviewed the problem of statistical sufficiency in the setting of quantum statistics. They provided sufficient and necessary conditions for sufficiency and established many results from traditional statistical theory in the language of noncommutative geometry, and at the same time, <cit.> derived the noncommutative version of well-known factorization theorem in traditional Statistics (see, e.g, <cit.>). 

Let us now consider a pair of operator valued observables (X, Y) (see, e.g., <cit.> for details on observables) with observed n pairs of eigen values (λ_1, μ_1), …, (λ_n, μ_n). Observe that here in the setting of quantum probability, we are observing the eigen values of (X, Y), unlike the observations on the random element (X, Y) as in traditional Statistics/classical Probability. Suppose that unobserved operators Y and X are linearly associated in true sense with scalar valued coefficient (see (<ref>)), and in the sense of regression modelling, the operator valued observables are assumed to be linear with scalar valued coefficient having some operator valued error (see (<ref>)). In this work, based on (λ_1, μ_1), …, (λ_n, μ_n), we estimate (denoted by β̂_n as an estimator) the unknown parameter β involved in the model described in (<ref>). In the course of this study, we establish the connection between the concepts in quantum probability and classical probability (see Section <ref>), and in view of this relation, we investigate the large sample properties β̂_n (see Section <ref>).  

The rest of the article is structured as follows. Section <ref> describes the preliminaries of the model in terms of quantum probability, and for this problem, a key connection between the quantum probability and classical probability is investigated in Section <ref>, and the problem is reformulated in terms of classical probability in Section <ref>. The large sample properties of the proposed estimator is studied in Section <ref>, and Section <ref> contains a few relevant concluding remarks. Finally, necessary technical details are provided in the Appendix in Section <ref>.








§ PRELIMINARIES : MODEL


Let  be a real separable Hilbert space with inner-product ·· and norm ·_. If T and S are two bounded linear operators on  related by T = β S for some non-zero real scalar β, then given any eigen-value λ of S, we have an eigen-value μ = βλ of T with the same eigen vector of unit norm. Again, given any eigen-value μ of T, we have an eigen-value λ = μ/β of S with the same eigen vector of unit norm. In light of this fact, one may expect that the linear regression model associating two operator valued observables can be represented by their corresponding eigen values. The research problem is foramlly described in the following.  

Let X and Y be two compact self-adjoint operator (on ) valued observables related by 

    Y = β_0 X,
 where β_0∈ℝ is unknown. We now consider the operator on operator regression model

    Y = β X + ϵ,
 where ϵ is a random operator valued error, and β∈ℝ is an unknown constant. In this work, our objective is to propose an estimator (denoted by β̂_n) of the unknown β based on the observed eigenvalues of (X, Y). 
The technical assumptions are stated in Section <ref>, where the large sample properties of β̂_n are studied.   


Suppose that (λ_1, μ_1), …, (λ_n, μ_n) are n pairs of observed eigen-values of (X, Y). Motivated by our earlier observation on deterministic operators T and S, we assume that each pair (λ_j, μ_j) appear with a common eigen-vector v_j of unit norm. Here, as mentioned earlier, using the observed (λ_j, μ_j)_j = 1^n, we would like to estimate the unknown parameter β. The estimation procedure of β is described in the following. 

In order to carry our the estimation of β, we first reformulate the problem, from originally stated in terms of X and Y in the quantum probability setup to their eigen-value pairs (λ, μ) in the classical probability setup. Next, using the classical probability model thus obtained the problem is reduced to a standard linear regression problem and we can then use classical statistical tools in order to estimate β. Section <ref> studies the eigen decomposition of operator valued observables X and Y, which is the key concept in reformulating the problem in the setup of classical probability. 



 §.§ Distribution of eigen decomposition of X and Y : From quantum to classical probability


Assume that X and Y are compact self-adjoint operators on  with discrete spectrum given by σ(X) and σ(Y), respectively. As per our discussion above, the eigenspaces corresponding to each feasible eigen-value pairs are the same. Consequently, we have the following spectral decompositions (see <cit.>):

    X = ∑_α∈σ(X)α P_α,    Y = ∑_α∈σ(Y)α P_α,

where the sum is countable, and P_α denotes the projection onto the eigen-space of any eigen-value α. Here, the eigenvalues are all real and the eigen-spaces corresponding to 0 ≠α∈σ(X) are finite-dimensional. Now, given any state ϕ (unit trace), we have the expectations in the setting of quantum probability as



    E X = tr(ϕ X) = ∑_α∈σ(X)α  tr(ϕ P_α),    E Y = tr(ϕ Y) = ∑_α∈σ(Y)α  tr (ϕ P_α).


Note that tr(ϕ P_α) ≥ 0, ∀α∈σ(X). Moreover, ∑_α∈σ(X) tr(ϕ P_α) = tr(ρ) = 1. A similar statement is true when X is replaced by Y. 


Observe that one can think of λ_j's drawn independently from a discrete probability distribution supported on σ(X) and with the probability mass function 
    f_λ(α) = tr(ϕ P_α) 1_(α∈σ(X)).
 Similarly, μ_j's are drawn independently from a discrete probability distribution supported on σ(Y) and with the probability mass function 
    f_μ(α) = tr(ϕ P_α)1_(α∈σ(Y)).
 This fact enables us to reformulate the problem in the set up of traditional statistics.




 §.§ A reformulation : estimation of β


Given n pairs (λ_1, μ_1), …, (λ_n, μ_n) of observed eigen-values of (X, Y), assume that v_1, v_2, ⋯, v_n be the corresponding eigen-vectors of unit norm. Then,

    X v_j = λ_j v_j ,
 where for j = 1, …, n,  ||v_j||_ = 1, and similarly, we have 

    Y v_j = μ_j v_j .

Next, it follows from (<ref>), (<ref>) and (<ref>) that for all j = 1, …, n, Y v_j = β X v_j + ϵ v_j, which yields

    μ_jv_j = βλ_jv_j + ϵ v_j,








and consequently,

    μ_j = v_jμ_jv_j = v_jβλ_jv_j + ϵ v_j = βλ_j + v_jϵ v_j, ∀ j = 1, …, n.


We now estimate β, involved in (<ref>), based on (λ_j, μ_j)_j = 1^n in various ways, by assuming conditions on the error terms v_jϵ v_j,  j = 1, …, n. Now, recall Remark <ref> that λ_j's are i.i.d. random variables following a discrete probability distribution supported on σ(X) and with the probability mass function 
    f_λ(α) = tr(ϕ P_α) 1_(α∈σ(X)),
 and μ_j's are i.i.d. random variables following a discrete probability distribution supported on σ(Y) and with the probability mass function 
    f_μ(α) = tr(ϕ P_α)1_(α∈σ(Y)).


 For the data (λ_j, μ_j) (j = 1, …, n), the M-estimator of β (see (<ref>)), which is denoted by β̂_n, is defined as 

    β̂_n   =    argmin_β∑_j = 1^nρ(v_jϵ v_j)
        =     argmin_β∑_j = 1^nρ(μ_j - βλ_j) ,
 where ρ(.) is a certain convex function, and few more assumptions will be stated in the subsequent section. Observe that for ρ(x) = x^2, β̂_n coincides with the well-known least squares estimator (see, e.g., <cit.>, pp. 42–43) based on the data (λ_j, μ_j)_j = 1^n, and when ρ (x) = |x|, β̂_n coincides with the well-known least absolute deviation estimator (see, e.g., <cit.>, pp. 42–43) based on the data (λ_j, μ_j)_j = 1^n. Besides, it also includes Huber's estimator with ρ(x) = x^21_(|x|≤ c)/2 + (c|x| - c^2/2)1_(|x| > c), where c> 0 (see, e.g., <cit.>, p. 43), L^q regression estimate with ρ(x) = |x|^q, where q∈ [1, 2] (see, e.g., <cit.>) and well-known α-th regression quantiles with ρ(x) = |x| + (2α - 1) x, where α∈ (0, 1) (see, e.g., <cit.>, p. 43, equation (5.5)).  This is one of the reasons to consider the technique of M-estimation, which includes many well-known estimators of the unknown regression parameter. 



§ LARGE SAMPLE PROPERTIES OF Β̂_N


In order to study the large sample properties of β̂_n defined in (<ref>),  one needs to assume the following technical conditions. 

(A1) ρ(.) is a strictly convex function. 

(A2) The error random variables v_jϵ v_j (j = 1, …, n) are i.i.d., and the common distribution of v_jϵ v_j, which is denoted by F (depends on joint distribution of (λ, μ)), is such that F (D) = 0. Here D is the set of discontinuity points of ρ'(.), where ρ'(.) denotes the derivative of ρ(.).

(A3) E[ρ' (v_1ϵ v_1 + c)] = ac + o(|c|) as |c|→ 0. Here a and c are two some constants. 

(A4) g(c) : = E[ρ' (v_1ϵ v_1 + c) - ρ' (v_1ϵ v_1)] exists for some c∈ (-δ, δ), where δ > 0 is a some constant. Moreover, g(.) is a continuous function at 0.

(A5) E[{ρ' (v_1ϵ v_1)}^2] : = D > 0.

(A6) S_n : = ∑_j = 1^nλ_j^2 > 0 for some n≥ n_0 and 

    d_n^2 : = max_1≤ j ≤ nλ_j^2/S_n→ 0 as  n→∞ .



Condition (A1) is required to have unique minima of the equation described in (<ref>), and it is a common assumption across the literature of M-estimation (see, e.g., <cit.>) in traditional Statistics. Condition (A2) indicates certain smoothness of the function ρ along with the fact that the error terms (in the forms of inner product) are i.i.d. random variables. Conditions (A3) and (A4) imply the existence of the derivative of the criterion function of M estimation in some neighbourhood of 0, which is necessary to solve minimization problem involved in M estimation. Next, Condition (A5) imply that the M-estimator β̂ (see (<ref>)), after appropriate normalization, will weakly converge to a non-degenerate random variable (see the statement of Theorem <ref> below, and finally, Condition (A6) is required to have the asymptotic normality of β̂_n, and from statistical point of view, one can interpret as the variation explained by any eigen value will not be a dominating factor.


Now, we state the consistency and the asymptotic normality results associated with 
β̂_n.


Under (A1)–(A6), β̂_np→β as n→∞. Here 
p→ denotes the convergence in probability, and β̂_n and β are the same as defined in (<ref>) and (<ref>), respectively.  



Under (A1)–(A6), 
    T_n^-1/2K_n (β̂_n - β)w→ Z,
 where Z is a random variable associated with standard normal distribution, and w→ denotes the convergence in distribution. Here T_n = D S_n and K_n = a S_n, where D, a and S_n are the same as defined in (A5), (A3) and (A6), respectively. 



In the set up of quantum probability, the observations (λ_j, μ_j)_j = 1^n are n eigen value pairs of (X, Y) unlike the classical probability, where the observations are (X_j, Y_j)_j = 1^n obtained from the distribution of (X, Y). Despite this issue, Theorems <ref> and <ref> assert that β̂_n is a consistent estimator of β, and after appropriate normalization, (β̂_n - β) converges weakly to a standard normal random variable. 




















 


§ CONCLUDING REMARKS

In this work, observe that the true model (see (<ref>)) is also linear with scalar valued coefficient, i.e., in other words, we work on specified model. However, the study will be entirely different if the model becomes mis-specified, which is of great interest in many statistical problem (see, e.g., <cit.> and a few relevant references therein). Investigating this issue may be of interest to future research. Besides, we have assumed our operators to be compact and self-adjoint. The classical analog would be to assume that the real random variables are discrete. It will be of interest when the involved operators are just self-adjoint. Then the spectrum may be continuous and in this case, our technique would not suffice. Moreover, in this work as the parameter β is scalar, the operators involved here are apriori simultaneously disgonalizable and therefore they can be measured simultaneously.  



In the course of this study, it is assumed that the error random variables v_jϵ v_j (j = 1, …, n) involved in (<ref>) are i.i.d., and the assertions in the main results (i.e., Theorems <ref> and <ref>) rely on this assumption. However, this assumption can be weakened by following the approach considered in <cit.>. In this work, we did not dig in this issue as establishing the large sample properties of β̂_n is not the main theme of the work, and therefore, we kept the assumption simpler so that the readers can appreciate the notion on quantum probability conveniently for operator on operator regression.  

Acknowledgement: Subhra Sankar Dhar gratefully acknowledges his core research grant (CRG/2022/001489), Government of India. Suprio Bhar gratefully acknowledges the Matrics grant MTR/2021/000517 from the Science and Engineering Research Board (Department of Science & Technology, Government of India). Soumalya Joardar acknowledges support from the SERB MATRICS grant (MTR 2022/000515). 



apalike 




§ APPENDIX : TECHNICAL DETAILS


Proof of Theorem <ref>: The arguments in the proof are parallel to the proof of Theorem 2.2 in <cit.>. Without loss of generality, we assume that β = 0. It is now enough to show that 

    P[|β̂_n| ≥ a_n]→ 0
 as n→∞, where {a_n}_n≥ 1 is such that a_n→∞ as n→∞. Observe that it follows from (3.11) in <cit.> that there exists a sequence a_n^' such that a_n^'→∞ as n→∞ and a_n^'≤ a_n for all n∈ℕ. Moreover, in view of it, we also have 

    sup_|β|≤ a_n^'|∑_j = 1^n[ρ(v_jϵ v_j - βλ_j) - ρ(v_jϵ v_j) + βλ_jρ'(v_jϵ v_j)] - β^2K_n/2|p→ 0
 as n→∞, where K_n is the same as defined in the statement of Theorem <ref>. 

Further, when |β| = a_n^' for some n∈ℕ, then observe that 

    β^2K_n/2≥a_n^'2max_1≤ j≤ nλ_j/2,
 which follows from the definition of of K_n defined in the statement of Theorem <ref> and S_n defined in (A6) before Remark <ref>. Moreover, observe that using (A4) on (<ref>), we have 

    ∑_j = 1^nλ_jρ'(v_jϵ v_j) = O_p(1) ,
 and hence, 

    sup_|β|≤ a_n^'|β∑_j = 1^nλ_jρ'(v_jϵ v_j)| = O_p(a_n^').


Therefore, using (<ref>), (<ref>), (<ref>) and (<ref>), we have 

    P(inf_|β| = a_n^'∑_j = 1^nρ(v_jϵ v_j - βλ_j) - ρ(v_jϵ v_j) ≤ 0)→ 0
 as n→∞, and next, using (A1), we have 

    P(inf_|β| ≥ a_n^'∑_j = 1^nρ(v_jϵ v_j - βλ_j) - ρ(v_jϵ v_j) ≤ 0)→ 0
 as n→∞. 

Finally, by the definition of β̂_n in (<ref>), we have 

    P(|β̂_n|≥ a_n^')→ 0
 as n→∞. As a_n^'≤ a_n for all n∈ℕ, the proof follows.  


Proof of Theorem <ref>: Without loss of generality, we here consider β = 0. First of all, using (A6) and in view of Lindeberg Feller CLT (see <cit.>), we have 

    ∑_j = 1^nλ_jρ'(v_jϵ v_j) : = K_nβ_nw→ Z,
 as n→∞, where Z is a random variable associated with standard normal distribution, K_n is the same as defined in the statement of Theorem <ref>, and `w→' denotes weak convergence.
 Hence, in order to prove this theorem, it is enough to show that 

    β̂_n -β_np→ 0
 as n→∞, 
where β_n and β̂_n are the same as defined in (<ref>) and (<ref>), respectively. Observe that using (<ref>) and the definition of β_n in (<ref>), we 
have 

    ∑_j = 1^n[ρ(v_jϵ v_j - β_nλ_j) - ρ(v_jϵ v_j)] + ββ_n K_n/2p→ 0
 as n→∞, and for some sequence {b_n}_n≥ 1 such that b_n→∞ as n→∞ and for some arbitrary constant δ > 0, we have 

    sup_|β|≤ b_n + δ|∑_j = 1^n[ρ(v_jϵ v_j - βλ_j) - ρ(v_jϵ v_j)] + ββ_n K_n - β^2K_n/2|p→ 0
 as n→∞. Now, it follows from (<ref>) and (<ref>), we have 

    sup_|β - β_n| = δ|∑_j = 1^n[ρ(v_jϵ v_j - βλ_j) - ρ(v_jϵ v_j - β_nλ_j)]  - (β - β_n)^2K_n/2|p→ 0
 as n→∞. Moreover, observe that when |β - β_n| = δ, we have 

    (β - β_n)^2K_n≥(min_1≤ j ≤ nλ_j)δ^2.
 Therefore, using (<ref>) and (<ref>), we have 

    P(|β - β_n|> δ)→ 0
 as n→∞, where δ > 0 is an arbitrary constant. It completes the proof. 


 Paper Title: Highly-efficient quantum Fourier transformations for some nonabelian groups.


FERMILAB-PUB-24-0241-SQMS-T













emm712@gwu.edu



Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Quantum Artificial Intelligence Laboratory (QuAIL),
NASA Ames Research Center, Moffett Field, CA, 94035, USA
USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA

Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Fermi National Accelerator Laboratory, Batavia,  Illinois, 60510, USA



Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Quantum Artificial Intelligence Laboratory (QuAIL),
NASA Ames Research Center, Moffett Field, CA, 94035, USA
USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA


egustafson@usra.edu
Superconducting and Quantum Materials System Center (SQMS), Batavia, Illinois, 60510, USA
Quantum Artificial Intelligence Laboratory (QuAIL),
NASA Ames Research Center, Moffett Field, CA, 94035, USA
USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA


§ ABSTRACT  
    Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. 
While the standard abelian QFT is well-studied, important variants corresponding to  nonabelian groups of interest have seen less development.
    In particular, fast nonabelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the nonabelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of nonabelian groups of interest for high energy physics, 𝔹𝕋, 𝔹𝕆, Δ(27), Δ(54), and Σ(36×3). 
   For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations.  
    Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by up to three orders of magnitude for the finite groups that we have investigated.

Highly-efficient quantum Fourier transformations for certain nonabelian groups
    Erik Gustafson
    February 2024
==============================================================================





§ INTRODUCTION


Fourier transforms have numerous applications in both classical and quantum computing.   
Classical applications range from signal analysis and compressed sensing, to coding theory, physics, and number theory 
<cit.>.  
In terms of quantum algorithms, it forms an important subroutine for Shor's algorithm <cit.>, the hidden subgroup <cit.> and hidden shift problems <cit.>, phase estimation <cit.>, and – of particular interest here – quantum simulation of high energy physics (HEP) <cit.>. 


Shor's algorithm and its extensions for 
prime factoring and discrete logarithms rely on the ability of quantum computers to efficiently solve the hidden subgroup problem for finite abelian groups using the quantum Fourier transform (QFT).
Beyond special cases, 
it is unknown if efficient QFT circuits imply an efficient solutions to the related hidden subgroup problem, a question connected to numerous deep open problems in quantum computing <cit.>. Indeed, the known nonabelian fast Fourier transforms are often more complex <cit.> than their abelian counterparts and developing efficient algorithms remains an active area of research both classically and quantumly <cit.>. Furthermore, even for particular classes of groups where asymptotically efficient quantum circuit are known, detailed resource analysis of specific cases can yield dramatic improvements over the general techniques. 

The need for fast QFTs for nonabelian groups is 
particularly acute in the quantum simulations of HEP, where previous work found that the naive Fourier transform represented ≳ 95% of the total gate cost <cit.>. Important targets for HEP applications are the QFTs for crystallographic subgroups of SU(2) <cit.> and SU(3) <cit.>, and beyond those cases SU(N) <cit.> more generally. 
As we will see below, the reduction in resource costs for HEP simulations from fast QFTs provide multiple orders of magnitude improvement. While we focus on groups of interest to HEP, we anticipate our work to have broader algorithmic applications. A recent example <cit.> utilized the QFT over the symmetric group <cit.> within generalized quantum phase estimation <cit.> to derive quantum algorithms and complexity results concerning computing structural properties of the group known as Kronecker coefficients. 


Mathematically, Fourier transforms are defined over a specific group – either abelian or nonabelian, continuous or finite – and correspond to a unitary transformation from the group algebra to a complex vector
space whose basis vectors are the matrix elements of the group's irreducible representations (irreps). For a finite group G, the discrete Fourier transform (DFT) can be defined as

    f̂(ρ) = √(d_ρ/|G|)∑_g ∈ G f(g) ρ(g),

where | G | is the size of the group, d_ρ is the dimensionality of the representation ρ, and f is a function over G. Implemented naively, the abelian DFT over ℤ_n has a classical complexity of O(n^2). Through repeated application of a subgroup-decomposition, it is possible to reduce the classical complexity to O(n log(n)) with a fast Fourier transform (FFT) using the Cooley-Tukey method <cit.>, though a similar algorithm was also developed much earlier by Gauss <cit.>.

A generalization of the Cooley-Tukey method to non-abelian groups which constructs the FFT through subgroup-decomposition was first identified by Diaconis and Rockmore <cit.>. A naive approach, that would compute Eq. (<ref>) directly would yield an asymptotic scaling of O(| G |^2). On the other hand, the Diaconis-Rockmore algorithm, like the Cooley-Tukey algorithm, relies on the fact that one could write any irrep of a group element as ρ(g) = ρ(ℓ) ρ(h) whenever g =ℓ h where h ∈ H 
belongs to a subgroup of H ⊂ G, and ℓ is an element of the left transversal. With this decomposition, the problem essentially reduces to computing the Fourier transform of the subgroup H. In this way, Diaconis and Rockmore showed that one can reduce the runtime of computing the Fourier transform of the symmetric group S_N from O(| S_N |^2) to O(| S_N |log| S_N |)[where |S_N|=N!].

Given the unitarity of the Fourier transform, it is possible to implement the DFT and FFT as quantum algorithms. The simplest example is that of simultaneously applying a Hadamard gate to each qubit corresponds to the QFT over the group ℤ^⊗ n_2. The usual quantum Fourier transform (QFT) circuit over the abelian group ℤ_N essentially uses this idea with the chain of subgroups ℤ/2^nℤ⊃ℤ/2^n-1ℤ⊃…⊃ℤ/2ℤ⊃{1} <cit.>. Similar QFT circuits can be constructed over other non-abelian groups, as in the early work by Hoyer <cit.>, Beals <cit.>, and Pueschel et al. <cit.>. The term quantum Fourier transform (QFT) is sometimes used to refer only to the quantum implementation of the FFT for the group ℤ_2^n but in this work we will use QFT to refer to any algorithm for performing Fourier transformations over general finite groups, G. For the quantum circuits, we will use 𝔘^G_FT and 𝔘^G_FFT to represent the naive and fast Fourier transforms respectively. In this case we consider “naive" to mean explicitly building the unitary matrix corresponding to the Fourier transform and then compiling it using available software libraries, such as Qiskit <cit.>.

In this work we will construct 𝔘^G_FFT for specific discrete subgroups of SU(2) and SU(3) relying on the methods of <cit.>. We briefly review the general construction of fast fourier transforms in Sec. <ref>.  Sec. <ref> discusses the basic gates for qubits and qutrits we will use to construct 𝔘^G_FFT.  The next two sections cover specific 𝔘^G_FFTs. Sec. <ref> is devoted to the crystallike subgroups of SU(2), binary tetrahedral[It is worth noting that 𝔹𝕋 is isomorphic to the Clifford group] (𝔹𝕋) and binary octahedral (𝔹𝕆), and Sec. <ref> for the SU(3) subgroups Δ(27), Δ(54), and Σ(36×3). Comparison of using 𝔘^G_FFT vs 𝔘^G_FT in fiducial quantum simulations are presented in Sec. <ref>, following by concluding remarks and discussion of future research directions 
in Sec. <ref>. 



§ FAST FOURIER TRANSFORMATION ALGORITHM



Before beginning the discussion of the fast Fourier transformation it is crucial to define several technical terms that will be used throughout this work: left (right) transversal, inner conjugate, and left (right) regular representations.
Let G be a group, and H a subgroup. For any g ∈ G, the left (right) cosets of H in G are the sets obtained by left (right) multiplication by g. The left (right) transversal for a subgroup H of some group G is then the set of representative elements of each left (right) coset such that exactly one element of each coset appears in the transversal.
An example of this would be numbers { 1, i} form the right transversal of the group ℤ_2 (with elements { 1, -1}) in ℤ_4.

A subgroup H ⊂ G is said to be normal when conjugating any element of H by an element of G returns another element of H. In particular, this is true of any element t in the transversal,

    t^-1 h t = h', with  h,h'∈ H

The property of possessing a normal subgroup is a crucial feature of the method of Pueschel et al. <cit.>, which allows us to build the Fourier transform for the parent group using this method. An important operation in this regard is the inner conjugation by an element of the transversal t for a given representation, ϕ, of a subgroup H,

    ϕ^t(h) = ϕ(t^-1 h t).

Representations for which ϕ^t(h) = ϕ(h) for all t are said to be extendable. When building the Fourier transform for a group G from a normal subgroup H, one must build up the irreps of the group G from those of its subgroup H, and in the process, the extendable representations incur trivial phase factors.

The final operation we need to consider is the left regular representation. This left regular representation is a functional operator L(g'), which takes as input a group element, g, and performs the permutation:

    L(g')|g⟩ = |g'g⟩,

where g,g'∈ G. The right regular representation can be similarly defined. The regular representations are important operations as they must be block diagonalized by the QFT. In fact, Pueschel et al. <cit.> define a QFT as a unitary change of basis of the Hilbert space that block diagonalizes the regular representations.


Deriving 𝔘^G_FFT requires an algorithmic construction through a systematic extension of Fourier transforms along a series of nested subgroups <cit.>.There are numerous paths, given by the lattice of subgroups, one could take to develop FFT for non-abelian groups as shown in Fig. <ref> for the example of 𝔹𝕆. Regardless of the path through group space chosen, the general procedure is to find and construct a quantum circuit for a unitary operator which naturally extends the FFT from one finite group in the series to the next <cit.>.




While multiple methods for constructing these Fourier transforms exist, we will use the method 
proposed in Ref. <cit.>.
The algorithm works by appending a list of elements, denoted a right transversal, to an existing group whose QFT is known, e.g. ℤ_4, and then using the effects of the right transversal to extend the irreps of the subgroup to those of a parent group, e.g. ℚ_8 or ℤ_8.  The structure for these circuits is shown in Fig. <ref>.



In this method, the quantum circuit for a given Fourier transformation can be broken up into six pieces that form a recursive algorithm: the subgroup fast Fourier transform (FFT), two irrep permutations (P and C), a twiddle matrix (T), a cyclic fast Fourier transform over the transversal (DFT), and a phase-kickback operation (Φ).

The FFT operation is performed first on the quantum register corresponding to the subgroup.
This operation takes the group element basis into a mixed basis; the subgroup operations exist in the irrep basis while the transversal elements are still in the group element basis. This is the portion of the algorithm that allows for a recursive generation as each successive subgroup can be broken down in this way.

The P and C operations are operations that permute the subgroup irreps such that the book-keeping and quantum circuits are easily manipulated. P takes irreps and permutes them in such a way that they are identifiable for easy bookkeeping.
The C operation then takes irreps which are not invariant under conjugation by the transversal, i.e. ρ^a(t g t^-1) ≠ρ^a, and adjusts them so they are nested together.
This will be important later for the phase-kickback operation, Φ. 

The twiddle operation, T is an operation controlled on the transversal register that either mixes the conjugate sets to make a larger dimensional irrep or transforms an existing n-dimensional irrep under the explicit representation. This is immediately followed by DFT on the transversal register which mixes the irreps and ensures that the transversal component is moved to the irrep basis. The final step involves a phase kickback that applies certain phases to the given irreps so that they are the actual irreps and not an isomorphism. This is important as even though irreps are really only meaningful within the Fourier transform up to unitary equivalence, in the block diagonalized regular representation, the same irrep must appear as many times as its dimensionality along the block-diagonal. These pieces together form the recursive structure necessary to build a nonabelian quantum Fourier transform. 
    

In this work we construct 𝔘_FFT^G for five cases not currently provided in the literature.  These are 𝔹𝕋,𝔹𝕆 and Δ(27),Δ(54),Σ(36×3) which correspond to subgroups of SU(3) and SU(2) respectively.  The nested subgroup paths we will construct are:

    1→Z_3×Z_3 →Δ(27)→Δ(54)→Σ(36×3)
       1→Z_2 →Z_4 →Q_8 →BT→BO.

The 𝔘_FFT^G for 𝔹𝕋, 𝔹𝕆, and Σ(36×3) provide the resource cost reductions for quantum simulations of lattice gauge theories for these respective groups <cit.>.  





§ PRIMITIVE GATES


Throughout this work we consider two quantum architectures: the standard qubit-based and a heterogeneous qubit-qutrit one.  Our consideration of qubit-qutrit devices is motivated by the non-abelian groups studied here, which contain parts that are naturally trinary-valued. In this section, we discuss the primitive gates we will use to construct the FFTs.  

The qubit gates include the usual arbitrary U(2) rotations (including X,Y,Z), CNOT, and the 3-qubit Toffoli. While CNOT+U(2) forms a universal set <cit.>, we use Toffoli for compression of circuits in presenting them.  Ultimately, all these circuits will be decomposed into a fault-tolerant gate set. For most cases, we will decompose all the operations to the lowest possible form; the primary exception will be arbitrary two-qubit rotations for which efficient decompositions are readily available <cit.>. 

For qubit-qutrit gates, we use the set of gates provided in Ref. <cit.>. The one qutrit gates include the generalization of the Pauli gates to qutrits,

    X_a,b   = |a⟩⟨ b| + |b⟩⟨ a| + ∑_c≠ a,b |c⟩⟨ c|
    
        Z_a   = ∑_c=0^2 (1 - 2 δ_c, a) |c⟩⟨ c|,

and the qutrit equivalent of the Hadamard gate,

    H_3 = 1/√(3)[     1     1     1;     1   ω_3 ω_3^2;     1 ω_3^2   ω_3;       ],

where ω_3=e^2π i / 3. Similar to the qubit Hadamard, this corresponding to a FFT on ℤ_3. Another important single qutrit operation is implemented by the clock-shift gate and its inverse

    χ = [ 0 1 0; 0 0 1; 1 0 0;   ],    χ^-1 = [ 0 0 1; 1 0 0; 0 1 0;   ] .


The final qutrit gates are the T_3 and S_3=T_3^3 which are the generalizations of the T and S gates for qubits that are important for fault-tolerant compilation:

    T_3=Diag(1,ω_9,ω_9^8),   S_3=Diag(1,ω_3,ω_3^2)

where ω_n=e^2π i/n is the n-th root of unity.

For entangling gates, we consider the controlled sum (controlled increment) gate which performs modular addition on the qutrits and is denoted as a controlled χ. For each of these qutrit operators, we also desire a qubit decomposition. 

Because of the mismatch in dimensionality between qutrits and qubits, there is some freedom in how to embed a single qutrit in two qubits. We choose to assign states according to their decimal integer values, such that 

    |0⟩_3 ≡ |00⟩_2,     |1⟩_3 ≡ |01⟩_2 ,     |2⟩_3 ≡ |10⟩_2.

and the |11⟩_2 state is forbidden.  This embedding is nice because it allows for easy construction of the two-qubit unitary from the qutrit one by simply direct summing the qutrit unitary with 1_1.  For example, the qubit H_3 is



    H^ qubit_3 = H_3^ qutrit⊕1_1=1/√(3)[     1     1     1     0;     1   ω_3 ω_3^2     0;     1 ω_3^2   ω_3     0;     0     0     0  √(3) ].

This unitary can be decomposed using available transpilers such as Qiskit to 3 CNOTs and 14 R_z gates. We also denote an additional H_3 gate mapping for qubits which appears in the derivation of the 𝔹𝕋 circuits; this operator is

    (H_3^ qubit)' = 1⊕ H_3^ qutrit = 1/√(3)[  √(3)     0     0     0;     0     1     1     1;     0     1   ω_3 ω_3^2;     0     1 ω_3^2   ω_3 ].


The qubit version of the permutation gate χ defined in Eq. (<ref>) is provided in Fig. <ref>. If this operator needs to be controlled then one applies this operator once provided the product of the controls is 1 modulus 3, the inverse operator is applied if the product of the controls is 2 modulus 3, and left alone if the product of controls is 0 modulus 3. The most efficient way to do this for more than one control is to compute the modulo 1 and modulo 2 values into two ancillae and then apply χ or χ^† controlled on the respective ancilla. 



The two remaining circuit operators that need decomposition are the phase operation Z_2 and the permuation operator X_1,2. The phase operation is given by

    Z_2 = Diag(1,1,-1,1),
 with its circuit found in Fig. <ref>. The X_1,2 gate is

    X_1,2 = [ 1 0 0 0; 0 0 1 0; 0 1 0 0; 0 0 0 1;   ],

and is identifiable as the usual SWAP gate (See Fig. <ref>).

A final crucial component for transforming qutrit gates to qubit ones is how to implement control on the resulting multi-qubit states. In theory, this just involves on controlling on the states |00⟩_2, |01⟩_2, and |10⟩_2 respectively. However, because the |11⟩_2 state is forbidden, some control operations can be simplified. Specifically, for the |01⟩_2 and |10⟩_2 states, one only needs to control on the second or first register respectively because the |11⟩ state is never occupied. When necessary, we also use the qudit |d⟩ notation when considering pure matrix representations and the qubit |ab⟩ notation when mapping to quantum circuits. Now that we have cleared this ambiguity.

These rules are sufficient for decomposing the circuits provided in the main text. 





§ QFT FOR SUBGROUPS OF SU(2)


As discussed above, there are numerous paths to construct an FFT for 𝔹𝕋 and 𝔹𝕆.  We will pursue the path 

    1 →ℤ_2→ℤ_4 →ℚ_8 →𝔹𝕋→𝔹𝕆

as it correspond most closely to the results from <cit.>. 
The consequence of this choice will become apparent following the discussion of the group structures, their presentations in terms of generating elements, and irreps.


 §.§ Structure and irreps of 



Before delving into the Fourier transformations, it is important to list out the structure of the principal finite groups of interest for SU(2): ℚ_8, 𝔹𝕋, and 𝔹𝕆. 
Each of these groups forms a subgroup in a nested series which extend from one subgroup to the next via a right transversal.
The groups ℚ_8, 𝔹𝕋, and 𝔹𝕆 can respectively be represented with the following ordered products

    f =    (-1)^a 𝐣^b 𝐤^c 
    
    g =    (-1)^a 𝐣^b 𝐤^c 𝐮^d
    
     h =    (-1)^a𝐣^b𝐤^c𝐮^d𝐭^e.

The generators which appear in these above equations have the following matrix presentation in the faithful representation:

    𝐣   = [  0  1; -1  0;    ], 𝐤 = [  i  0;  0 -i;    ],
    𝐮   =[   -η   -η;  η^* -η^* ], 𝐭 = 1/√(2)[  1 -i; -i  1;    ]

where η = 1 + i/2, 0≤ a, b, c, e≤ 1, and 0≤ d ≤ 2.     

Additional details about 𝔹𝕋 and 𝔹𝕆 can be found in Refs. <cit.> however we quote here the important information regarding the irreps, denoted ξ^m for ℚ_8, ρ^m for 𝔹𝕋 and ρ̅^m for 𝔹𝕆. The irreps for all three groups are listed in Tabs. <ref>, <ref>, and <ref> with ω_3 = e^2π i / 3. 






    




 §.§ Fourier Transformation for 


 

The construction of 𝔘_FFT^𝔹𝕋 requires the identification of the six constituent gates: ℚ_8 FFT, P, C, T, DFT, and Φ.
The first, the FFT_ℚ_8 was derived in <cit.> and is presented in  Fig. <ref>. Given the choice of implementation for 𝒯_𝔹𝕋 that will be taken below, we will see that the C and P transformations are unnecessary.

We thus need to connect the irrep vectors of ℚ_8, |ξ^a⟩, with their corresponding computational basis states.
These states are explicitly

    |0⟩_𝐤|0⟩_𝐣|0⟩_-1 =  |ξ^1⟩,    |1⟩_𝐤|0⟩_𝐣|0⟩_-1 = |ξ^3⟩
     
    |0⟩_𝐤|1⟩_𝐣|0⟩_-1 =  |ξ^2⟩,  |1⟩_𝐤|1⟩_𝐣|0⟩_-1 = |ξ^4⟩
    { |a⟩_𝐤|b⟩_𝐣|1⟩_-1: 0≤ a,b≤ 1} =  |ξ^5⟩,

where the corresponding irreps are enumerated in Table <ref>. The only remaining step to generate the other gates is to identify the sets of inner conjugates. For 𝔹𝕋, we can identify the inner conjugacy transformations using the results from Refs. <cit.>:

    𝐮 (-1)^a 𝐣^b 𝐤^c 𝐮^2 =     (-1)^a 𝐣^c𝐤^b + c
    𝐮^2 (-1)^a 𝐣^b 𝐤^c 𝐮 =     (-1)^a 𝐣^b + c𝐤^b.

These transformation rules immediately tell us what the pairs of inner conjugates are; ξ^1 and ξ^5 are left invariant under this transformation, i.e. these irreps are extendable. 
Meanwhile, we can identify that (ξ^2)^𝐮(f) = ξ^4(f) and (ξ^2)^𝐮^2(f) = ξ^3(f). This implies that {ξ^2, ξ^4, ξ^3} form a set of inner conjugates and are not extendable representations.

The irreps {ξ^2, ξ^3, ξ^4} when combined with the transversal, { 1, 𝐮, 𝐮^2}, make the 3d irrep of 𝔹𝕋, ρ^7. 
This allows us to write down the twiddle operation T. This operation will end up being a controlled operation on the ℚ_8 register of the form 

    T_𝔹𝕋 = ∑_d=0^2 |d⟩⟨ d| ⊗𝒯_𝔹𝕋^d

where

    𝒯_𝔹𝕋 = 1_1⊕χ⊕ (𝐮_2⊗1_2).

The quantum circuit which implements this transformation is highlighted in green in Fig. <ref>. The first two Toffoli gates perform the cyclic permutations, while the controlled 𝐮 operation performs a controlled rotation on the |⟩_𝐣 register corresponding to the presentation of 𝐮 in the ρ^4 irrep show in Eq. <ref>.
 This leaves the implementation of Φ which can be broken down similar to T_𝔹𝕋 in Eq. (<ref>) as

    Φ_𝔹𝕋 = ∑_d=0^2 |d⟩⟨ d| ⊗ϕ_𝔹𝕋^d,

where

    ϕ_𝔹𝕋 = Diag(1,1, ω, ω^2, 1, 1, 1, 1).

The phase kickback here needs to be applied to the irreps which form a conjugate set, in this case ξ^2, ξ^3, ξ^4 but leaves the other irreps unchanged. 
The complete quantum Fourier transformation for  then strings these gates together and is shown in Fig. <ref> with the phase kickback highlighted in blue. It is important to note the following, the H_3' gates do not need to be controlled on the |⟩_𝐮 register since it forms a basis transformation taking the permutations of the Toffoli gates to phases. 





 §.§ Fourier Transformation for 


The construction of the Fourier transformation for  will follow that of , with the subgroup FFT being that of .  Next, we can identify the irreps after 𝔘_FFT^𝔹𝕋 with their corresponding basis vectors. 
The 1d irreps correspond to the basis vectors

    |0⟩_𝐮|0⟩_𝐤|0⟩_𝐣|0⟩_-1   = |ρ^1⟩
    
     |1⟩_𝐮|0⟩_𝐤|0⟩_𝐣|0⟩_-1   = |ρ^2⟩
    
     |2⟩_𝐮|0⟩_𝐤|0⟩_𝐣|0⟩_-1   = |ρ^3⟩.

The 2d irreps corresponding to sets of quadruplets

    { |0⟩_𝐮 |a⟩_𝐤|b⟩_𝐣|1⟩_-1}   = |ρ^4⟩
    { |1⟩_𝐮 |a⟩_𝐤|b⟩_𝐣|1⟩_-1}   = |ρ^5⟩
    { |2⟩_𝐮 |a⟩_𝐤|b⟩_𝐣|1⟩_-1}   = |ρ^6⟩
 
where 0≤ a,b≤ 1. 
Finally the 3d irrep will correspond to the nonaplet

    { |d⟩_𝐮 |1⟩_𝐤|0⟩_𝐣|0⟩_-1,    
    
           |d⟩_𝐮 |0⟩_𝐤|1⟩_𝐣|0⟩_-1,    
    
           |d⟩_𝐮 |1⟩_𝐤|1⟩_𝐣|0⟩_-1}    = |ρ^7⟩

for 0≤ d ≤ 2. We next identify the inner conjugates under 𝐭 from the expression

    𝐭 (-1)^a 𝐣^b 𝐤^c 𝐮^d 𝐭^-1 = 
    (-1)^a'𝐣^c + δ_d,1 + δ_d,2𝐤^b + δ_d, 1𝐮^2d

where a' is an unimportant polynomial of the c, b, δ_d,1, and δ_d,2. 
We note that the nontrivial effect of inner conjugation by 𝐭 is 𝐮↔𝐮^2.
It then follows that  

    (ρ^1)^𝐭 = ρ^1,   (ρ^4)^𝐭   = ρ^4,   (ρ^7)^𝐭 = ρ^7
    
    (ρ^2)^𝐭   = ρ^3
    
    (ρ^5)^𝐭   = ρ^6

and thus the inner conjugates are {ρ^2, ρ^3} and {ρ^5,ρ^6}. This implies that ρ^2 and ρ^3 will mix to create the 2d irrep of 𝔹𝕆, ρ̅_3. Meanwhile the 2d irreps ρ^5 and ρ^6 will combine to generate the 4d irrep ρ̅_8.

Everything is now in place to construct the twiddle operator, T_𝔹𝕆, and the phase kickback Φ_𝔹𝕆.
Similar to , we can write decompose the twiddle matrix

    T_𝔹𝕆 = ∑_e=0^1 |e⟩⟨ e| ⊗𝒯_𝔹𝕆^e.

The only remaining issue is to identify what the matrix 𝒯_𝔹𝕆 is. We can identify this by evaluating the effects on the basis states for 𝔹𝕋:


    𝒯_𝔹𝕆   = 1_1⊕ X⊕ (1_3⊗ [-(𝐮_2⊗1_1)])⊕
       [  1/√(2)       0       0       0       0       0 -i/√(2)       0       0       0       0       0;       0       0       0       0       0       0       0       0       0       0       1       0;       0       0       0       0       0       0       0       0       0       0       0       1;       0       0       0  1/√(2)       0       0       0       0       0 -i/√(2)       0       0;       0       0       0       0       0       0       0      -1       0       0       0       0;       0       0       0       0       0       0       0       0      -1       0       0       0; -i/√(2)       0       0       0       0       0  1/√(2)       0       0       0       0       0;       0       1       0       0       0       0       0       0       0       0       0       0;       0       0       1       0       0       0       0       0       0       0       0       0;       0       0       0 -i/√(2)       0       0       0       0       0  1/√(2)       0       0;       0       0       0       0       1       0       0       0       0       0       0       0;       0       0       0       0       0       1       0       0       0       0       0       0;         ]
 
This operator can be translated to a quantum circuit with a modicum of work into the one shown in the green highlighted portion of Fig. <ref>. In most cases the ancilla present in the quantum circuit will be unentangled either through running the Fourier transformation in reverse or by otherwise uncomputing the values they hold.

The only remaining component necessary is the phase-kickback

    Φ_𝔹𝕆 = ∑_e=0^1|e⟩⟨ e|⊗ϕ^d_𝔹𝕆

The phase kickback gets applied to the original states corresponding to |ρ^3⟩ and |ρ^6⟩ irreps.
This yields the operator 

    ϕ_𝔹𝕆 =    Diag(1, 1, -1) ⊕1_9⊕
       Diag(1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1).

We provide the full quantum Fourier transformation for 𝔹𝕆 in Fig. <ref>. This circuit includes 2 ancilla which are eventually uncomputed when the transformation is run in reverse. 





§ QFT FOR SUBGROUPS OF SU(3)


For the case of SU(3), there are two motivations for the particular groups we investigate. So-called flavour groups are important for model-building in particle physics to understand the mass splittings between different flavor generations, e.g. see Ref. <cit.>. Δ(3n^2) and Δ(6n^2) are prime examples of flavour groups. Other groups, for example Σ(ϕ×3), have found interest for simulations in quantum simulations for lattice gauge theorys in HEP <cit.>. For the work here, it is nice that Δ(3n^2) form normal subgroups of SU(3)



 §.§ Structure and irreps of 



The group elements, g ∈Δ(27), h ∈Δ(54), and f ∈Σ(3× 36), can be written as

    g    = ω_3^p 𝐂^q 𝐄^r
    
        h    = ω_3^p 𝐂^q 𝐄^r 𝐕^2s
    
        f    = ω_3^p 𝐂^q 𝐄^r 𝐕^2s + t

respectively, where 0≤ p, q, r ≤ 2 and 0≤ s, t≤ 1 <cit.>. 
Each of these subgroups is extended to a larger subgroup of SU(3) according to

    1→Z_3×Z_3 Δ(27)Δ(54)Σ(36×3)


where the elements above the arrows indicate the right transveral used to extend the group.
We also provide the irreps for Δ(27), Δ(54), and Σ(3× 36) are denoted ξ^a, ρ̅^a, and ρ^a respectively and provided in Tabs. <ref>, <ref>, and <ref>.
In addition, we also need to enumerate the irreps of ℤ_3×ℤ_3. We denote these irreps as χ^3a + b + 1 with 0≤ a, b≤ 2. These irreps have the following presentation
    
    χ^3a + b + 1(ω_3^p 𝐂^q) = e^2π i / 3 (a p + q b)

    where p and q correspond to the generators ω_3 and 𝐂 in Eqs. (<ref>) to (<ref>).
    

    
    
    
    


 §.§ Fourier transformation for 


We need to first identify the operator FFT which generates the Fourier transformation for ℤ_3×ℤ_3. 
This quantum circuit is simply the tensor product of a discrete abelian quantum Fourier transformation across each of the individual ℤ_3 groups.
This leaves us with the irreps corresponding to the following basis vectors:

    |00⟩ =    |χ^1⟩    |01⟩ =    |χ^2⟩    |02⟩ =    |χ^3⟩
    
        |10⟩ =    |χ^4⟩    |11⟩ =    |χ^5⟩   |12⟩ =    |χ^6⟩
    
        |20⟩ =    |χ^7⟩    |21⟩ =    |χ^8⟩    |22⟩ =    |χ^9⟩


Now we need to identify which irreps form the conjugate pairs from <cit.> and which are trivially extended.
This involves calculating what χ^3a + b + 1(𝐄^r ω_3^p 𝐂^q 𝐄^-r) is equal to. 
We find that {χ^1, χ^2, χ^3} are trivial, i.e., χ^m(𝐄^r ω_3^p 𝐂^q 𝐄^-r) = χ^m for m=1,2,3, while {χ^4, χ^5, χ^6} and {χ^7, χ^8, χ^9} form inner conjugates.
We can now write down the operator for T_Δ(27),

    T_Δ(27) = ∑_r=0^2 |r⟩⟨ r|⊗𝒯_Δ(27)^r

where,

    𝒯_Δ(27) = 1_3⊕χ^-1⊕χ

Additionally we can write down the phase kickback matrix,

    Φ_Δ(27) = ∑_r=0^r |r⟩⟨ r| ⊗ϕ_Δ(27)^r

with

    ϕ_Δ(27) = 1_3⊕Diag(1, ω_3^2, ω_3) ⊕Diag(1, ω_3, ω_3^2).
 
These pieces in turn give us the necessary unitaries to construct the 𝔘_FFT^Δ(27). The operator in Eq. (<ref>) is given by a inverted controlled sum gate, while the operator in Eq. (<ref>) is a inverted controlled phase gate. We provide the full quantum circuit for qutrits in left panel of Fig. <ref> with each of the components highlighted and labeled. 
After this Fourier transformation we can identify the computational basis states with their corresponding irreps:

    |000⟩ =     |ξ^1⟩,     |001⟩ =     |ξ^2⟩,     |002⟩ =     |ξ^3⟩,
    
        |010⟩ =     |ξ^4⟩,     |011⟩ =     |ξ^5⟩,     |012⟩ =     |ξ^6⟩,
    
        |020⟩ =     |ξ^7⟩,     |021⟩ =     |ξ^8⟩,    |022⟩ =     |ξ^9⟩,


    {|1ab⟩: 0≤ a,b≤ 2 } =     |ξ^10⟩
    {|2ab⟩: 0≤ a,b ≤ 2} =     |ξ^11⟩



While this qutrit decomposition for the fast Fourier transformation is the most straightforward to right down, it may not necessarily be the most efficient way as fault tolerant hardware is developed. In this context it is important to have a purely qubit based version of this fast Fourier transformation.




We provide the quantum circuit for 𝔘_FFT^Δ(27) using a homogenous qubit register in the right panel of Fig. <ref>. This circuit is the functional equivalent of the one provided in Fig. <ref>. Each of the three qutrits is broken down into a two qubit subregister. 

The main non-trivial component of the twiddle operation is how the multi-controlled shift operation is applied. We use the same rule as discussed in the previous section to map onto the first ancilla.
We compute the operators that correspond to a product equal to 2 modulo 3 on the first ancilla and a product equal to 1 modulo 3 on the second register. 
This reclaimable scratch register computation is done to minimizes the T-gate count for the quantum simulation. 
This operator ends up being repeated again for Φ_Δ(27). 
In total this circuit requires 24 Toffoli operations.  
The remaining T-gates come from the decomposition of the H_3 gates discussed in the prior subsection. As mentioned above each of these operations requires 14 R_z rotations and 3 transversal CNOTs. 
This gives us a total T-gate count of:

    168 + 80.5 log_2(1 / ϵ),

where ϵ is the target synthesis error. The circuit has a maximal T-gate width of 4 and requires 2 ancilla; which corresponds to a temporary 33% qubit memory overhead increase for the scratch registers. 







 §.§ Fourier transformation for 


As the derivation follows similarly to the previous ones, we will abbreviate it going further. We first need to identify the irreps of Δ(27) which form conjugate pairs.
We can identify this by observing how the operator 𝐕^2 transfroms an arbitrary element of Δ(27); we find that

    𝐕^2 ω_3^p 𝐂^q 𝐄^r 𝐕^2 = ω_3^p 𝐂^2q𝐄^2r.

This result implies that we have the following conjugate pairs from the Δ(27) irreps:

    {ξ^2, ξ^3}, {ξ^4, ξ^7}, {ξ^5, ξ^9}, {ξ^6, ξ^8}, {ξ^10, ξ^11}

These conjugate pairs allow us to identify the operator Φ_Δ(54) which is given by

    Φ_Δ(54) =    ∑_s=0^1 |s⟩⟨ s| ⊗
    (
       1_2 ⊕ -1 ⊕1_3 ⊕ -1_3⊕1_9⊕-1_9)^s.






The twiddle matrix, T_Δ(54), is 

    T_Δ(54) = ∑_s=0^1 |s⟩⟨ s|⊗𝒯_Δ(54)

where

    𝒯_Δ(54) =    X_1,2⊕(X⊗ X_1,2) ⊕1_3⊕
       (X⊗1_3)⊕1_3⊕(X⊗1_3).

The first two terms of matrix sum in 𝒯_Δ(54), correspond to the trivial irrep and the mixing of the other 1d irreps. The last terms in the sum are the mixing introduced by 𝐕^2 to the 3d irreps ξ^10 and ξ^11.

The total fast Fourier transformation is provided in on the left panel of Fig. <ref>. In this figure both the twiddle and phase kickback operations highlighted as well as the Fourier transformation across the transversal. We now proceed to the details of the qubit decomposition for this fast Fourier transformation. Because the 𝐕^2 register only requires a qubit, there is only the decomposition of the qutrit operations that are important. 

The decomposition to qubit gates is substantively simpler than in the target case for Δ(27). The controlled X_1,2 operations are mapped to controlled swaps while the Z_2 operations are mapped to controlled CZ gates. This decomposition using the results of the appendix provides us with the circuit in the right panel of Fig. <ref>. This circuit uses the same 2 ancilla as in the Δ(27) circuit. The decomposition requires 6 Toffoli gates for T_Δ(54) and 12 for Φ_Δ(54) giving a total of 126 T gates on top of the Δ(27) costs. This yields a total T gate cost of 

    294 + 80.5 log_2(1 / ϵ).

This requires 5 ancilla qubits, the majority of which occur in the phase kickback operation. 



 §.§ Fourier transformation for 

In order to progress from  Δ(54) to Σ(36×3), we need to construct a basis for the irreps of Δ(54).  We take the following elements:

    |ρ̅^1⟩   =  |0000⟩      |ρ̅^2⟩=|1000⟩
    
        |ρ̅^3⟩   ={ |0100⟩,|0200⟩, |0210⟩,|0020⟩}
    
        |ρ̅^4⟩   ={ |1100⟩,|1200⟩, |1210⟩,|1020⟩}
    
        |ρ̅^5⟩   ={ |0010⟩,|0020⟩,|0120⟩,|0220⟩}
    
        |ρ̅^6⟩   ={ |1000⟩,|1110⟩, |1120⟩,|1220⟩}
    
        |ρ̅^7⟩   ={ |0ab1⟩: 0≤ a,b ≤ 2}
    
        |ρ̅^8⟩   ={ |1ab1⟩: 0≤ a,b ≤ 2}
    
        |ρ̅^9⟩   ={ |0ab2⟩:  0≤ a,b ≤ 2}
    
        |ρ̅^10   =⟩{ |1ab2⟩:  0≤ a,b ≤ 2}.

We also need the following result from inner conjugation of Δ(54) by 𝐕:

    𝐕 (ω_3^p 𝐂^q 𝐄^r 𝐕^2s)𝐕^-1   = ω_3^p + 2qr𝐂^2r𝐄^q𝐕^2s.

This tells us that the inner conjugates are {ρ̅^3,ρ̅^4}, {ρ̅^5,ρ̅^6}, {ρ̅^7,ρ̅^8}, {ρ̅^9,ρ̅^10}
This immediately allows us to identify that shuffling and transformation between the states needs to occur according to the representations in Tab. <ref>. In total this gives us the twiddle operations

    T_Σ(36×3) = ∑_t=0^1 |t⟩⟨ t| ⊗𝒯_Σ(36× 3)

with

    𝒯_Σ(3× 36) = [ 1 0; 0 i;   ]⊕(1_4 ⊗χ_4)⊕𝒱⊕ (𝒱^*)

where χ_4 is the 4d generalization of the clock-shift operation

    χ_4=[ 0 1 0 0; 0 0 1 0; 0 0 0 1; 1 0 0 0;   ]

and

    𝒱 = 1_6⊗ (-iH_3).

Addtionally we can immediately identify the phase kickback operation as

    Φ_Σ(36× 3) = ∑_t |t⟩⟨ t| ⊗(⊕_j=0^27[  1  0;  0 -1 ])^t.

The quantum circuit using a mixed qutrit-qubit system is provided in Fig. <ref>. The pure qubit decomposition into a fault tolerant cirucit is provided in the remainder of this subsection.
The Σ(36×3) fast Fourier transformation shown in Fig. <ref> has the decomposition to a 8 qubit register as shown in Fig. <ref>. 
Unlike the other two circuits where the twiddle matrix and phase kick back were approximately equal in T-gate costs. The twiddle matrix is the most expensive portion of the Σ(36× 3) fast Fourier transformation. The first controlled S matrix requires 8 clean ancilla qubits for the scratch register. This scratch register bloat is only necessary at one small point for the controlled-S gate. The explicit C^nNOTs that appear in the circuit require 20 Toffoli operations in total. There is an implicit set of 14 Toffoli gates within the controlled-U operation which is shown explicitly in Fig. <ref>.  

The final place where T-gate synthesis presents is in the controlled-V and controlled-V̅ case which follows from the derivation of Ref. <cit.>. The first step is to diagonalize the V operator, which will require 32 R_z gates when dressing both sides of the diagonalizing operator. When diagonalized, the operator V has in the qubit space, the diagonalized matrix,

    V_diag. = Diag(-i,i,1,1),

while V̅ has the diagonalized form

    V̅_ diag. = Diag(i,-i,1,1).

These controlled operations together form a diagonal matrix which can be implementation with 4 R_z and 6 CNOT gates using the methods of Ref. <cit.>. Therefore, in total the controlled V and V̅ incured a cost of 36 R_z gates and no Toffoli gate. 
This all together brings the total number of additional T-gates on top of the Δ(54) gates to 238 + 36.8 log_2(1 / ϵ) for a total T-gate count of

    532 + 117.3 log_2(1 / ϵ).







        
    


§ OUTLOOK FOR QUANTUM SIMULATIONS FOR QFTS


The results from this work allow us to make both concrete predictions for existing groups as well as extrapolations about the costs of simulations for various other nonabelian groups of import. The cost comparison for various decompositions are provided in Tab. <ref>.

For one method of quantum simulation <cit.>, there are four primitive gates that must be constructed: the inverse, multiplication, trace and Fourier tranform gate;  which the Fourier transform dominating the resource costs. The main cost for quantum simulations involves mapping a total target synthesis error ϵ̅_T^I to a total number of T-gates N_T^fid. This comes from a target cost function C_T^I. Greater details about these methods and the finer nuances of these calculations can be found in Refs. <cit.>. 
The T-gate cost provided in this work can be considered quasi-optimal and an approximate lower bound for the total resources necessary for implementation.  For the groups 𝔹𝕋,𝔹𝕆, and Σ36×3, the total simulation cost are presented in Tab. <ref>.

This reduction in cost for the 𝔹𝕋 Fourier transformation still leaves it as the most costly operation but nevertheless reduces the overall simulations costs for pure gauge viscosity simulations proposed in Ref. <cit.> by an additional order of magnitude and by 11 orders of magnitude compared to Ref. <cit.>. 
Meanwhile 𝔘_FFT^𝔹𝕆 is nearly two orders of magnitude cheaper than 𝔘_FT^𝔹𝕆. This brings the cost down for Fourier transformation to be on par with the trace and multiplication gates in terms of T-gate cost, bringing the total simulation cost down by 2 orders of magnitude. Even greater reductions are found with 𝔘_FFT^Σ(36×3) which reduces the cost of the gate by a factor of nearly three orders of magnitude at the expense of doubling the qubit overhead; resulting in a total reduction in simulation cost by a factor of 580. 









§ DISCUSSION


In this work, we have derived fast quantum Fourier transforms for a number of nonabelian groups of interest in high energy physics and beyond. A potentially interesting next step relevant to a wide variety of systems would be the exploring applications and tradeoffs involving qudits or mixed qubit and qudit systems. For example the symmetric and alternating groups, 𝒮_n and 𝒜_n, accelerate in size quite quickly with the number of elements being n! and n!/2 where n is the number of elements being “permuted". This factorial scaling indicates certain qubit memory may be wasted. For example with the symmetric group of the amount of “wasted" states can be between 20-45% for n ≤ 20. This might demonstrate that there are hardware advantages to the use of qudit based systems for these sorts of problems.

The general structure of these fast Fourier transformations for smaller groups is paramount for the construction of fast Fourier transformations for larger ones. Hence results of this paper are applicable to further (super)groups in the same way. In particular the results of Σ(36×3) will be crucially important for the groups, Σ(72×3), Σ(216×3), and Σ(360×3), which are crucial for simulations of quantum chromodynamics. Additionally, it is important to note that our formulation of the quantum Fourier transform relied on a particular encoding of the discrete groups, and therefore modifications may be required for other encodings <cit.>.  For quantum simulations of lattice gauge theories which use other digitizations, the implementation of fast Fourier transforms requires more investigation.

It worth also noting that the notions of approximations to the standard ℤ_n quantum Fourier transformation exist <cit.>. An fascinating direction is to develop analogous constructions for so-called approximate quantum Fourier transformations for nonabelian groups, as well as identify potential applications where such approximate implementations yield potential advantages over exact implementations. 

One may wish to further explore applications of novel insights related to nonabelian QFTs to the HSP. Various papers have considered quantum algorithms for the nonabelian HSP, see Refs. <cit.>,  and this work has demonstrated explicitly how to constructions the fast Fourier transformation for nontrivial example groups; a crucial step forward toward efficient algorithms for HSP. The research performed in this work demonstrates that nonabelian fast Fourier transformations can be constructed efficiently for fault tolerant qubit based hardware and can offer orders-of-magnitude reduction in T-gate cost for quantum simulation. 

    The authors would like to thank Namit Anand, Aaron Lott, and Norman Tubman for helpful comments and suggestions. We are grateful for support from NASA Ames Research Center. Erik Gustafson, Stuart Hadfield, and M. Sohaib Alam were supported by the NASA Academic Mission Services, Contract No. NNA16BD14C and NASA-DOE interagency agreement SAA2-403602, and through the Superconducting Quantum and Materials System Center (SQMS) under contract number DE-AC02-07CH11359. This material is based on work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center 
    under contract number DE-AC02-07CH11359 (Henry Lamm). Edison Murairi is supported under the grant number DE-FG02-95ER40907. Fermilab is operated by Fermi Research Alliance, LLC under contract number DE-AC02-07CH11359 with the United States Department of Energy.






 Paper Title: Thermodynamic Roles of Quantum Environments: From Heat Baths to Work Reservoirs.











Institute of Physics, University of Freiburg, 
Hermann-Herder-Straße 3, D-79104 Freiburg, Germany
Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy



Institute of Physics, University of Freiburg, 
Hermann-Herder-Straße 3, D-79104 Freiburg, Germany

EUCOR Centre for Quantum Science and Quantum Computing,
University of Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany


§ ABSTRACT 
Environments in quantum thermodynamics usually take the role of heat baths. These baths are Markovian, weakly coupled to the system, and initialized in a thermal state. Whenever one of these properties is missing, standard quantum thermodynamics is no longer suitable to treat the thermodynamic properties of the system that result from the interaction with the environment. 
Using a recently proposed framework for open system quantum thermodynamics which is valid for arbitrary couplings and non-Markovian effects, we show that within the very same model, described by a Fano-Anderson Hamiltonian, the environment can take three different thermodynamic roles: a standard heat bath, exchanging only heat with the system, a work reservoir, exchanging only work, and a hybrid environment, providing both types of energy exchange. The exact role of the environment is determined by the strength and structure of the coupling, and by its initial state. The latter also dictates the long time behaviour of the open system, leading to thermal equilibrium for an initial thermal state and to a nonequilibrium steady state when there are displaced environmental modes.


 


Thermodynamic Roles of Quantum Environments: From Heat Baths to Work Reservoirs
    Heinz-Peter Breuer
    August 5, 2024
===============================================================================





§ INTRODUCTION


To model the nonequilibrium dynamics of small quantum systems in contact with heat baths, and to assess their thermodynamic properties, one makes use of the theory of open quantum systems <cit.>. This allows to link dynamical processes such as dissipation and decoherence to the exchange of heat and work between system and bath, and to the observation of irreversible behaviour in terms of positive entropy production rates <cit.>.
A heat bath interacting with the system is in these cases phenomenologically modelled through a Lindblad master equation describing the evolution of the quantum system of interest, which typically leads to its thermalization to an equilibrium state described by a Gibbs state at the temperature of the bath.
The heat bath thus exchanges heat with the system while external work protocols – represented as a time-dependent bare system Hamiltonian – describe work performed on the system.
The employment of the Lindblad master equation to describe the dynamics stands on heavy assumptions regarding the nature of the environment and its relationship to the system: the bath must be weakly coupled to the system, it must interact so that its two-point correlation functions decay sufficiently fast with respect to the typical system relaxation time and such that the secular approximation is allowed. More so, only specific Lindblad master equations, like the weak-coupling quantum optical master equation, satisfy the detailed balance condition required for thermalization <cit.> – which, furthermore, requires neglecting the Lamb shift term in the system Hamiltonian.

By now, the field of quantum thermodynamics has evolved so rapidly that more and more strong coupling effects are taken into account, leading to, e.g., studies on dynamical renormalization of thermodynamic quantities <cit.> and on extensions of the famous Jarzynski equality to this regime <cit.>. To understand the actual role of environments for regimes where the Lindblad approach is prohibited, one needs more refined open system tools, along with a microscopic modelling of the quantum environment. 

While one expects that infinitely large environments initialized in a thermal state are proper microscopic models of a heat bath – as can be usually argued in the Markovian regime –, it is still unclear whether strong coupling or non-Markovian effects can spoil this assumption.
For the sake of generality, one might suppose that a general environment, particularly strongly coupled and/or not memoryless, will not simply exchange heat with the system: without further assumptions, the interaction with the environment may lead to effective driving on the system, and thus to work exchange. 

In this work, we define as extreme cases a “quantum heat bath” as an environment which exchanges only heat with the quantum system of interest, while we term “work reservoir” any environment exchanging energy only in the form of work. We investigate the thermodynamic role of an infinitely large environment in the strong coupling, non-Markovian regime by examining a paradigmatic Hamiltonian – sometimes called the Fano-Anderson model – which is exactly solvable for the reduced system and at the same time allows a continuum of modes as the environment. The model, which is physically relevant and has several applications in condensed matter physics, nuclear, atomic and molecular physics, as well as quantum optics (see, e.g., <cit.>), is also suitable to represent a large heat bath from a microscopic perspective, and has all the required properties – e.g., approach to a unique equilibrium steady state – to describe a proper thermodynamical situation. By evaluating work and heat in the strong coupling regime according to the renormalized system Hamiltonian that emerges form the exact interaction with the environment (as suggested in Ref.<cit.> and, for the Fano-Anderson model, also in Ref. <cit.>), we will see that the shape and strength of the coupling, as well as the possible initial displacement of the bath modes, play a decisive role in determining the main thermodynamic function of the environment, which in the most cases exchanges both heat and work with the system.

The structure of this work is the following. In Sec. <ref> we recall the definitions of thermodynamic quantities according to the approach developed in <cit.>. These definitions are based on the canonical form of the exact time-local master equation for the system degrees of freedom.
In Sec. <ref> we show the exact master equation for the Fano-Anderson model in terms of a general, initially uncorrelated, Gaussian state of the environment. We show that, while the model is completely relaxing to a final steady state (or to a nonequilibrium steady state (NESS) in the case of initial displacement in the environment), the environment will in general exchange both work and heat with the system even when initialized in a thermal state. In Sec. <ref> we show instances where the two extremes of perfect heat bath (Sec. <ref>) and work reservoir (Sec. <ref>) are reached, 
showcasing how the approach of <cit.> describes within the same formalism the loss of information due to the contact with a heat bath and the emergence of coherent driving of a system from a microscopic perspective. In Sec. <ref> we show and give an interpretation to a specific case where the environment acts as a hybrid agent, exchanging both work and heat with the reduced system.

In all that follows, the initial state of the system is assumed to be a completely arbitrary state. We will, however, care a great deal about the initial state of the environment; for ease of exposition, we might therefore denote the initial environmental state as simply “the initial state” throughout this work.



§ DYNAMICALLY EMERGENT QUANTUM THERMODYNAMICS


The framework for the description of the thermodynamics of an open system in the regime of general couplings, following <cit.>, assumes that the open system S and the environment E, which represents a general environment, are coupled through the total microscopic Hamiltonian 

    H(t) = H_S(t) + H_E + H_I(t),

where H_S and H_E are the system and the environmental Hamiltonian, respectively, and
H_I denotes the interaction between them. The time-dependency of system and interaction Hamiltonian describe possible work protocols on the system and modulation of the interaction – including switching on and off –, while the environmental Hamiltonian is assumed to be time-independent in agreement with the assumption that we do not directly act on the environment. The total system thus undergoes a unitary evolution generated by (<ref>), which conventionally starts with an uncorrelated state between system and environment, i.e. ρ_SE(0)= ρ_S(0)⊗ρ_E(0).

In the strong coupling regime, the interaction energy associated to the Hamiltonian H_I is non-negligible with respect to the bare system energy associated to H_S. This energy surplus results in a renormalization of the system energy levels – as it happens, for example, with the Lamb-shift induced by the electromagnetic vacuum in the weak-coupling regime. In the general case, as developed in <cit.>, the renormalized Hamiltonian for the system emerges from the dynamics and is found as the unitary part of the generator for the open system, i.e. the operator K_S in the canonical form of the exact time-local master equation for the open system S (also known as time-convolutionless master equation) <cit.>:

    d/dtρ_S(t) = _t[ρ_S(t)] = -i [K_S(t),ρ_S(t)] + 𝒟_t  [ρ_S(t)].

The second part of the master equation, also called dissipator, is of the following form

    𝒟_t [ρ_S] = ∑_kγ_k(t)[L_k(t)
      ρ_S L_k^†(t) - 1/2{L_k^†(t)L_k(t),ρ_S}]

with decay rates γ_k(t)∈ℝ and corresponding Lindblad operators L_k(t). The above general structure of the time-local master equation can be
derived from the requirements of the preservation of Hermiticity and trace of the density matrix 
<cit.>. 
This differential equation, which, being exact, is given in general by an explicitly time-dependent 
generator _t, completely describes the dynamics of the system and thus how all strong coupling and memory effects influence it <cit.>. While it in principle exists for any open system <cit.>, its exact derivation, namely its explicit expression in terms of Hamiltonian parameters and initial environmental state,
is often challenging. Algorithmic methods when the exact derivation is not feasible include perturbation expansions in the system-environment coupling through the time-convolutionless projection operator technique <cit.>, and numerical techniques such as HEOM <cit.>.
In the present paper, the model under study is exactly solvable and the master equation can be derived exactly, see <cit.> and Sec. <ref>.

The separation of the master equation (<ref>) into a dissipator and a part generating unitary evolution is not unique <cit.>, but can be made so by imposing tracelessness of the Lindblad operators L_k(t) <cit.>. This requirement is equivalent to minimizing the dissipator part as a superoperator according to a set of norms <cit.>, including one that is averaging over input and output pure random states following the Haar measure on the unitary group, giving equal weight to all pure states of the open system. Alternatively, it means that all the non-Hermitian Lindblad operators are taken to describe coupling between orthogonal states, so that their dissipator terms identify processes linked to heat exchange.

Via the identification of the emergent Hamiltonian K_S(t), which can also in principle be time-dependent, as a renormalized energy operator for the system, the internal energy of the system is defined as

    U_S (t) = {K_S(t)ρ_S(t)}.

The first law of thermodynamics is then given by

    Δ U_S (t) ≡ U_S(t) - U_S(0) = W_S(t) + Q_S(t),

where the work and heat contributions arise from the change in energy levels and in the system state, respectively:

    W_S(t)    =   ∫_0^t dτ {K̇_S(τ) ρ_S(τ) },  
    
     Q_S(t)    =   ∫_0^t dτ { K_S(τ) ρ̇_S(τ)  }.

Because of the structure of (<ref>), the heat contribution Q_S(t) is determined by the change in the system state due to the dissipative evolution, as it can be expressed as the following

    Q_S(t) = ∫_0^t dτ { K_S(τ) 𝒟_τ[ρ_S(τ)] }.


Assuming furthermore that the bath is sufficiently large such that it does not suffer noticeable temperature changes during the evolution, entropy production is defined in analogy with the Clausius inequality as

    Σ_S (t) = Δ S_S(t) - βδ Q_S(t)  ,

with Δ S_S(t)=S(ρ_S(t))-S(ρ_S(0)) the change of the von Neumann entropy of the reduced system and β the inverse initial temperature of the environment.
Under the condition that the instantaneous Gibbs state of the reduced system ρ_S^G(t) = e^-β K_S(t)/Z_S(t) suffers no dissipation, i.e.  𝒟_t [ρ_S^G(t)]=0, one can prove <cit.> that negative entropy production rates imply the presence of non-Markovian effects in the sense of information backflow <cit.>

    σ_S(t) ≡Σ̇_S(t) ≤ 0  non-Markovianity ,

and one may therefore understand information backflow as a necessary condition for breaking the second law of quantum thermodynamics.




§ THE FANO-ANDERSON MODEL


We consider the integrable model of a bosonic mode, our reduced system, linearly coupled to a continuum of bosonic modes, the environment, within rotating wave approximation. This model, which can be equivalently formulated in terms of fermionic modes, is known as the Fano-Anderson model <cit.> and describes quantum systems coupled to environment through particle exchange. The Hamiltonian of the total system reads

    H^FA = ω_0 a^† a_H_S +  ∑_j ω_j c_j^† c_j_H_E +  ∑_j (g_j a^† c_j + g_j^* a c_j^†)_H_I  ,

where a^†, a are the bosonic creation and annihilation operators on the central system, whose main frequency is ω_0, and c_j^†, c_j are the creation and annihilation operators of the environmental modes with frequencies ω_j. The parameters g_j represent the coupling constants between the central mode a and each bath mode c_j.



 §.§ Exact time-local master equation

For a Gaussian initial state of the environment, the reduced dynamics is Gaussianity preserving and can be solved exactly by solving the Heisenberg equations of motion <cit.>. Then, the exact TCL master equation can be found by making an ansatz for its generator, and comparing the coefficients with the exact evolution of first and second moments of the system <cit.>.
Assuming that the modes in the environment are initially prepared uncorrelated and each in a Gaussian state, the exact master equation for the central mode in the Fano-Anderson model reads (see Appendix <ref> for the derivation):

    d/dtρ_S(t) =    -i [ω_r(t) a^†a +i f(t) a^† -i f^*(t)a,ρ_S(t)] 
         + γ(t)(N(t)+1) [aρ_Sa^†-1/2{a^†a,ρ_S}] 
     
        + γ(t) N(t) [a^†ρ_Sa-1/2{aa^†,ρ_S}] 
         + δ^*(t)[a^†ρ_Sa^†-1/2{a^†a^†,ρ_S}] 
         + δ(t) [aρ_Sa-1/2{aa,ρ_S}] 
      .

All the coefficients of the master equation are dependent on the coupling strength and structure, and in general on the initial temperature, displacement and squeezing of the environment. In the above, the parameters ω_r(t) and γ(t)  depend exclusively on the coupling through the Green function G(t) solving 

    Ġ(t) +i ω_0 G(t) +∫_0^t dτ𝒦(t-τ)G(τ) =0   ,

with G(0)=1, where 𝒦(t-τ) = ∑_j |g_j|^2 e^-iω_j (t-τ) is the memory kernel due to the coupling landscape, i.e. depending on the spectral density J(ω) in the continuum limit 

    𝒦(t -τ) = ∫_0^∞ dω J(ω) e^-iω (t-τ)  .

The two parameters are then given explicitly by

    ω_r(t) = - {Ġ(t)/G(t)}  ,   γ(t) = - 2{Ġ(t)/G(t)}  .


All other parameters in the master equation depend on the initial state of the environment. For example, the parameter δ(t) is due to initial squeezing of the environment and reads

    δ(t) = J̇(t) - 2Ġ(t)/G(t)J(t)    ,

with

    J(t) = ∑_j g_j^2 ⟨⟨ c_j c_j ⟩⟩_0 [ ∫_0^t dτ G(t-τ)e^-iω_jτ]^2   ,

where we have defined the notation

    ⟨⟨ AB ⟩⟩_0 := ⟨ AB ⟩_0 - ⟨ A⟩_0⟨ B⟩_0   .

The other coefficients are also affected by initial squeezing; however, for zero squeezing we have that δ(t) vanishes (and with it the last two terms in (<ref>)), and that the other parameters take on specific meanings.

The coefficient N(t), for example, depends then only on the initial temperature, describes the available number of excitations in the environment as seen from the system (see Sec. <ref>), and reads

    N(t) = I(t) + İ(t)/γ(t)  ,

with noise integral

    I(t) = ∑_j |g_j|^2 ⟨⟨ c_j^† c_j ⟩⟩_0 | ∫_0^t dτ G(t-τ)e^-iω_jτ|^2   .

It vanishes for the zero temperature case.

The parameter f(t) represents instead the emergence of coherent driving and depends only on the initial displacement of the modes

    f(t) = Ḟ(t) - Ġ(t)/G(t)F(t)

with

    F(t) := -i ∑_j g_j ⟨ c_j ⟩_0 ∫_0^t dτ G(t-τ) e^-iω_jτ  .

The driving force f(t) thus vanishes in the case of no initial displacement of the environmental modes. We remark that no external driving of this form was imposed on the bare system Hamiltonian H_S, and that the linear driving in the master equation is entirely emerging from tracing out the environmental degrees of freedom. It is useful to note that, were we to impose such an external linear driving on the bare system, it would in principle undergo renormalization due to coupling with the (thermal) environment, unless the latter is in some sense inert, see Appendix <ref>.

The master equation is already in generalized Lindblad form and satisfies the principle of minimal dissipation, as the Lindblad operators a and a^† can be considered traceless even if the Hilbert space is infinite (they are traceless in any truncation of the Hilbert space to any number of Fock states). In the case of a thermal state, it had been already obtained in <cit.> using path integral techniques, and in <cit.> using the present technique. 



 §.§ Effective Hamiltonian, heat and work

According to Sec. <ref> we can thus identify the emergent Hamiltonian for the Fano-Anderson model from the coherent part of the master equation, which gives

    K_S(t) = ω_r(t) a^†a -i f(t) a^† +i f^*(t)a  .

It represents a bosonic mode with a renormalized, time-dependent frequency, and a time-dependent non-adiabatic driving term with force f(t) (which is not periodic in the general case). The driving term arises exclusively in the presence of initial displacement in the bath, while the time-dependent renormalization of the frequency is a general feature, appearing also at zero temperature. 
Notice that since ω_r(t) depends only on the Green function G(t), then it depends exclusively on the bare central frequency ω_0 and on the spectral density J(ω). We remark that this is not a typical case in open quantum systems as K_S(t) depends in general also on, e.g, the environmental temperature (see the case of the Jaynes-Cummings model <cit.>). We ascribe this effect as due to the Gaussianity of the evolution.

For the rest of this work, we focus on the case where the bath is in general thermal, and possibly displaced, such that δ(t)=0. This allows us to explore different roles of the environment as a heat bath or a work reservoir. Before we do that, though, we argue that the Hamiltonian (<ref>) is in principle suitable as a microscopic model of a mode coupled to a thermal bath; we do so by looking at the general expression for first law quantities – namely work, heat and internal energy – in the case of a thermal initial state of the environment, and showing that the open system relaxes to an equilibrium state in the long time limit (in the thermal case) or to a nonequilibrium steady state (NESS) in the case of additional initial displacement of the environment.

Considering the initial environmental state to be thermal, ρ_E(0)= e^-β H_E/Z_E, leads to vanishing parameters in the master equation (<ref>), namely f(t)=δ(t) ≡ 0. Then, the effective Hamiltonian is K_S(t)= ω_r(t)a^† a and the internal energy of the central mode follows simply as

    U_S(t) = ω_r(t) n(t)  ,

where we denoted with n(t):= ⟨ a^† a ⟩_t the evolved average excitation number of the system. Consequently, the net work exchanged up to time t is given by

    δ W_S(t) = ∫_0^t d τω̇_r(τ) n(τ)  ,

with a work exchange rate ω̇_r(t) n(t) that is present only for a renormalized frequency ω_r that varies in time. The net heat exchanged is instead given by

    δ Q_S(t) = ∫_0^t d τω_r(τ) ṅ(τ)  .

Using the master equation (<ref>) we find that ṅ(t) = γ(t) [N(t) - n(t)], and we can divide heat into two distinct contributions

    δ Q_S(t)    = ∫_0^t d τ [ Q̇_S^in(τ) - Q̇_S^out(τ)  ] 
       = ∫_0^t d τω_r(τ) γ(τ) [N(τ) - n(τ)]  .

Here,  Q̇_S^out(t) =  ω_r(t) γ(t) n(t) describes the outgoing heat rate, and is given by the energy gain (gained, in this case, by the environment) ω_r(t) times the renormalized system-bath transition rate γ(t) and the average excitations n(t) in the system; this part of the heat therefore describes how outgoing heat exchange arises from excitations in the central mode spontaneously decaying to the environment. In a similar way, Q̇_S^in(t) =  ω_r(t) γ(t) N(t) describes the incoming heat rate (this time gained by the system) where we find the same renormalized frequency ω_r(t) and the same transition rate γ(t). What changes is the available number of excitations, given by N(t),  which as mentioned previously assumes the role of a renormalized average excitation number of the bath, representing the available excitations present in the environment as seen from the system. Note that this does not mean that N(t) is the actual average number of excitations in the bath (this will in general depend on the initial state of the system), but it represents in a way the number of excitations that are “available” to enter into the system from the environment. In the Born-Markov limit <cit.>, for example, N(t) becomes time-independent and approaches the Planck distribution at the initial temperature of the environment.



 §.§ Relaxation to equilibrium


Let us show that, in the continuum limit, the open system in the Fano-Anderson model (with a thermal initial environmental state) reaches a unique equilibrium state.
To witness complete relaxation of the reduced system – namely that all initial states converge to a unique steady state at long times – we simply assume that the Green function G(t) decays to zero at infinity

    G(t)  0   .

This leads to the disappearance of any initial condition contribution to all the relevant moments, see eqs. (<ref>)-(<ref>). In particular we find vanishing expectation values for a and aa, and a final excitation number given by the noise integral:

    ⟨ a ⟩_t   0 ,
       ⟨ aa ⟩_t   0 , 
       ⟨ a^†a ⟩_t     I(∞) =: n.

The above implies that the unique final steady state is thermal,

    ρ_S = e^-X a^† a/{e^-X a^† a}  ,   with   X= ln( 1+n/n)  .

The entire noise integral I(∞), namely the final excitation number, can be written explicitly in terms of the spectral density J(ω) (see Appendix <ref> for the proof):

    n = ∫_0^∞ω J(ω) 1/e^βω-11/[ω_0+Δ(ω)-ω]^2+π^2 J^2(ω)  ,

where we defined the principal value integral

    Δ(ω) = 𝒫∫_0^∞ω' J(ω')/ω - ω'  .

The final steady state of the central mode in the case of a thermal environment is thus determined by both the initial temperature and the spectral density.

Using this result, it is possible to prove that the final steady state also coincides with the expected mean-force equilibrium state, namely the partial trace over the environment of a global thermal state at the initial temperature of the bath:

    ρ_S^*= _E {e^-β H_SE/Z_SE} .

It is important to note that such a statement should always be proven and can never be assumed: since the total system is a closed system undergoing unitary evolution, and since it was initialized in a product state, it is impossible for it to reach a global state e^-β H_SE/Z_SE. The fact that it looks this way from the open system perspective as a result of information loss must be shown for the specific model. The full proof makes use of the fluctuation dissipation theorem – and the assumption that all poles of Ĝ(z) (with Ĝ the Laplace transform of G(t)) have negative real part – and is reported in Appendix <ref>.

Under these assumptions, and that in the long-time limit also the parameters ω_r and γ reach a unique final value, the effective Hamiltonian converges to K_S= ω_r a^† a. A natural question is how the steady state emergent Hamiltonian relates to the Hamiltonian of mean force <cit.>, namely the temperature-dependent Hamiltonian H^*_S such that

    e^-β H^*_S(β)/Z^*_S= _E {e^-β H_SE/Z_SE}  .

While they are indeed related, it turns out that they are not the same; the Hamiltonian of mean force depends on temperature, while K_S, in this model, does not. They are instead connected by the relation

    β H^*_S = β_r K_S + const.  ,

where β_r is the steady state value of a renormalized temperature β_r(t), which connects the instantaneous equilibrium state of the master equation (<ref>) with a Gibbs state relative to K_S(t). The instantaneous renormalized temperature can also help in formulating a second law of quantum thermodynamics for the Fano-Anderson model, whose violations have a clear-cut connection to the presence of information backflow. We will illustrate this next.




 §.§ Instantaneous Gibbs state at the renormalized temperature


We define the time-dependent renormalized temperature β_r(t) directly from the parameters in the master equation (<ref>) via

    γ (t) N(t) = γ (t) (N(t)+1) e^β_r(t) ω_r(t)  ,

which leads to 

    β_r(t) = 1/ω_r(t)ln(N(t)+1/N(t))   .

Notice that this also means that the coefficient N(t), which we interpreted in Sec. <ref> as the number of excitations available in the environment for the system to absorb, is given in terms of the renormalized temperature by

    N(t) = 1/e^β_r(t) ω_r(t) -1  ,

namely by a Planck distribution for the renormalized frequency ω_r(t) and temperature β_r(t). This leads to the interpretation of β_r(t) as an effective temperature of the environment, as perceived by the system through the interaction. 

Using this renormalized temperature, we define the following instantaneous Gibbs state 

    ρ_S^G_r(t) = e^-β_r(t) ω_r(t) a^† a/{e^-β_r(t) ω_r(t) a^† a}  .

It is easy to prove that, due to the structure of the master equation in the thermal case, this is a fixed point of the evolution, in parallel with Markovian master equations satisfying detailed balance <cit.>. Namely, _t [ρ_S^G_r(t)] = 0 for all times, with _t the time local generator describing the master equation (<ref>). 

We exploit this property to formulate a fully renormalized second law of quantum thermodynamics for the Fano-Anderson model, by defining the entropy production for the system as

    Σ_S(t) = Δ S(ρ_S(t)) - ∫_0^t dτβ_r(τ) Q̇_S(τ)   ,

i.e. we have modified Eq. (<ref>) in order to take into account a varying temperature of the environment, as it is seen from the perspective of the open system through the interaction. 
We do this in the same way as in <cit.>, though with a different definition of bath temperature.
This leads to a form of entropy production rate

    σ_S(t) = -d/dt|_τ=0S(ρ_S(t+τ)||ρ_S^G_r(t))  ,

where, however, the new instantaneous Gibbs state is automatically, at each point in time, a fixed point of the evolution. This form of entropy production rate, thus, is such that violations of the second law of thermodynamics (namely, σ_S(t)< 0 for some t), imply the presence of information backflow, as shown in <cit.>. The difference here is that, using the renormalized temperature, no additional condition is needed to establish the link between information flow and entropy production rates. 


Furthermore, regarding approach to equilibrium in the long time limit, we find automatically that the system steady state coincides with the renormalized Gibbs state 

    ρ_S^G_r = ρ_S   .

From this it follows that the renormalized temperature in the long time limit is also such that 

    ⟨a^† a|_⟩∞ = 1/e^β_r ω_r -1  ,

showing that the average occupation number of the system oscillator approaches in the long time limit a Planck distribution with renormalized frequency and temperature. Thus, β_r also takes the role as the effective temperature of the system, indicating how system and environment have reached equilibrium. The renormalized temperature in the relaxation limit indeed coincides with the system temperature (at infinite times) that was defined for the Fano-Anderson model in <cit.>.

Moreover, recalling the fact that the final steady state is the mean force state ρ_S^*, it is clear that the long time limit of the emergent Hamiltonian K_S in this model is linked to the Hamiltonian of mean force via relation (<ref>).



 §.§ Strong coupling nonequilibrium steady state


In the case where there is some initial displacement of the environmental modes, the system does not reach thermal equilibrium, as it is continuously driven out of it by the driving term in the Hamiltonian. However, under the same assumption (<ref>) of vanishing Green function in the long time limit, the system asymptotically approaches a unique NESS described by the long time limit of the moments

    ⟨ a ⟩_t  ∼F(t)  ,
       ⟨ aa ⟩_t  ∼F^2(t)   , 
       ⟨ a^†a ⟩_t ∼n + |F(t)|^2 
       ,

i.e. the state evolves as the thermal state ρ_S^G_r from the last section, but with a time-dependent displacement (see Appendix <ref> for details)

    F(t) = -i ∑_j φ_j e^-iω_j t

with

    φ_j=  α_j g_j Ĝ(-iω_j)    ,

where α_j is the initial displacement of the j-th mode and Ĝ is the Laplace transform of G. We can therefore denote the nonequilibrium steady state of the open system with a displaced equilibrium state

    ρ_S(t) = D_t ρ_S^G_r D_t^†  ,

with 

    D_t:= D[F(t)] = e^F(t)a^† - F^*(t)a

the displacement operator for the time-dependent displacement F(t). The master equation generator in the long time limit indeed reads 

    ℒ_t[X] = -i [K_S(t), X] + 𝒟[X]   ,

where 𝒟 is the dissipator with rates given by γN and γ(N+1) and the (still time-dependent) effective Hamiltonian reads

    K_S(t) = ω_r a^†a +i f(t) a^† -i f^*(t)a  ,

with driving force

    f(t) = ∑_j [ω_r - ω_j -i/2γ]φ_j e^-iω_j t  .

Then, the long-time dynamics of the state ρ_S(t) becomes a unitary evolution given by

    d/dtρ_S(t) = ℒ_t[ρ_S(t)] ≡Ḋ_t D_t^†ρ_S(t) + ρ_S(t)D_tḊ_t^†  .


We call the state ρ_S(t) a NESS because the state of the open system tends asymptotically to this unique, unitarily evolved state after the function G(t) has decayed, independently of its initial conditions. We remark, however, that it does not represent a NESS in the usual sense <cit.>, as, in general, the state is not periodic in time and its energy fluxes are not constant. The latter can be seen from the fact that its internal energy

    U_S(t) = ω_r (n + |F(t)|^2) +i (f(t) F^*(t) + h.c.)

is a non-trivial function of time. The internal energy of the state in the long time evolution thus keeps changing, along with heat and work fluxes. 

A particular case where the asymptotic state is also a NESS in the usual sense is when there is only one displaced mode in the environment. Let us denote the mode with the index d. Then, the displacement of the steady state is periodic in time (with period T=2π/ω_d) and given by

    F(t) = -i φ_d e^-iω_d t = -iα_d g_d Ĝ(-iω_d) e^-iω_d t  .

The generator ℒ_t of the master equation then assumes the form of the Floquet-Lindblad master equation <cit.> which describes the dynamics of periodically driven dissipative quantum systems.
We note at this point that the result (<ref>) has been obtained earlier in <cit.> by different methods. 

In the case of a single-mode driving, the internal energy of the system approaches the constant value

    U_S(t) = ω_r (n - |φ_d|^2) + 2ω_d |φ_d|^2   ,

which is in general a function of renormalized temperature β_r, of the frequency, displacement and coupling of the driving mode (ω_d, α_d, g_d), but also of the entire spectral density J(ω) describing the coupling to the rest of the environment.

Since there is no variation of internal energy in the long-time limit, the fluxes of work and heat (which are themselves constant) balance each other out:

    Q̇_S =    - Ẇ_S 
     
      =    -γω_d|φ_d|^2   ,

so that there is work consistently being performed on the system to keep it out of equilibrium at a constant energy. Moreover, since the long time evolution is unitary, the von Neumann entropy of the system also reaches the constant value of S(ρ_S^G_r). The entropy production rate is thus determined solely by the heat rate and gives a measure of how far the NESS is from thermal equilibrium:

    σ̇ =    -β_rQ̇_S(t)
     
     =   β_r γω_d|α_d|^2|g_d|^2|Ĝ(-iω_d)|^2   .

Here the Laplace transform of the function G reads, in the continuum limit (see Appendix <ref>)

    Ĝ(-iω_d) = 1/i(ω_0+Δ(ω_d)-ω_d)+π J(ω_d)  ,

where we defined the principal value integral

    Δ(ω_d) = 𝒫∫_0^∞ω' J(ω')/ω_d - ω'  .

The denominator of Ĝ(-iω_d) shows a generalized resonance condition for the system and the driving mode. 
Therefore, the entropy production for the NESS becomes stronger when the displaced mode in the environment and the system are closer to resonance. Furthermore, typically, the higher the driving mode coupling and displacement, the farther the NESS from equilibrium.




§ EXTREME ROLES OF THE ENVIRONMENT: HEAT BATH VS. WORK RESERVOIR


Now that we have seen the general solution, master equation and thermodynamic features of the Fano-Anderson model (<ref>), we want to discuss, in this section, the impact of details like coupling strength, spectral density shape, and initial environmental state on the thermodynamic properties of the central mode. In particular, since we have seen in Sec. <ref> that arbitrary coupling to the external environment leads in general to both heat and work exchange, we 
want to define special cases that induce extreme thermodynamic regimes – namely, situations in which the environment acts as either a proper heat bath, exchanging only heat with the central mode, or as a work reservoir, changing only the energy levels of the system in time with negligible dissipation, thus exchanging only work-like energy.

We start in Sec. <ref> by considering the Born-Markov limit of the master equation, in the weak coupling regime, and showing that there is no extra work contribution given by the environment, while it is in general responsible for non-zero heat exchange.  We see that this is also the case of a purely Markovian bath modelled through a completely flat spectral density, without the need for perturbation theory, showcasing how purely white-noise, Markovian environments take the role of heat baths. On the contrary, we present in Sec. <ref> a situation where the displaced initial state of the environment induces driving – thus, an emergent work protocol – on the system while the contribution coming from dissipation, i.e. heat exchange, is negligible with respect to the driving in the semiclassical limit. This is a case where we see clearly that tracing out a special environment – even an infinite one endowed with a temperature – can still give rise to an effective unitary evolution of the system which is governed by a time-dependent Hamiltonian. 




 §.§ Perfect heat bath: Born-Markov limit and white noise


We enter the Born-Markov regime first by assuming weak system-environment coupling. We thus isolate a coupling (perturbation) parameter λ from the interaction Hamiltonian, which then becomes λ H_I(t). We then determine the second order expansion of the parameters γ(t) and ω_r(t) (see Appendix <ref>). This last quantity, in particular, is still time dependent, showcasing that a second order expansion in the coupling still leads to time dependency in the emergent Hamiltonian, and thus to work exchange in general. As a consequence, weak coupling – in the sense of small coupling parameters – is not enough to justify the assumption that the environment acts as a thermal bath. However, if we now also assume to be in the regime where the Markovian approximation is valid, then one obtains the following time-independent values for the renormalized frequency and rate:

    γ^M =    2 λ^2 π J(ω_0)    ,  
    ω_r^M =   ω_0 + λ^2 δω =  ω_0 + λ^2 𝒫∫_0^∞ωJ(ω)/(ω_0-ω)  .


The shift λ^2 δω is the usual Lamb shift of the Lindblad master equation. Especially due to the time-independence of the renormalized frequency ω_r^M, there is no extra work contribution from the bath, therefore zero work on the reduced system δ W_S(t) =0, and the environment acts as a true heat bath. In the current case, the heat rate inherits an additional term due to the shift of the renormalized frequency, so that Q̇_S(t) = {(ω_0 + λ^2 δω) a^† a 𝒟[ρ_S(t)]}; nonetheless, since the dissipator is already in second order in λ, this contribution is neglected and we obtain results that are compatible with the weak coupling formulation of quantum thermodynamics <cit.>. Moreover, this limit is recovered also if an external linear driving is imposed on the system (see Appendix <ref>), as the approximations are such that the work protocol does not undergo renormalization due to the coupling to the bath (we remark, however, that this is a special case).

Another simple case in which the environment does not perform any work on the system, but for which 
we can avoid a perturbative treatment, is that of a completely flat spectral density describing a fully Markovian bath, namely J(ω) = γ_0/2π. Note that the rate γ_0 need not be small with respect to other scales in the system, if we allow the extension of the integral in the memory kernel (<ref>) to the whole real axis. Then this gives

    𝒦(t-τ) =  γ_0 δ(t-τ)   ,

such that the differential equation (<ref>) becomes, under the convention ∫_0^ϵδ(x) dx = 1/2, the following

    Ġ(t) +i ω_0 G(t) + γ_0/2 G(t) = 0   ,

with simple solution G(t) = e^-iω_0 t e^-γ_0 t/2. Then, the renormalized frequency coincides with the bare value, ω_r(t)≡ω_0. This means that there is no time-dependent renormalization of the system Hamiltonian, and thus no work contribution in the non-driven case of the Fano-Anderson model. Moreover, the transition rate is equivalent to the rate in the spectral density, i.e. γ(t) ≡γ_0. Since this rate is time independent, it also holds that a linear driving protocol also does not get modified by the presence of the bath (see Appendix <ref>). From the arguments above, it is also likely that there is no kind of driving protocol on the system which would undergo renormalization in this case of a purely Markovian bath, and we therefore consider this idealized case as the most paradigmatic case of a “proper” heat bath. 




 §.§ Perfect work reservoir: displaced initial state

We now study a particular case where the environment assumes a completely different function for the system, namely that of a work reservoir performing a protocol on the central mode. 

Assume that, instead of a purely thermal state, the environment is initialized in a displaced thermal state. 

It is well known from quantum optics that a displaced electromagnetic field vacuum (zero temperature) is equivalent to a classical driving term in the system Hamiltonian <cit.>. Here we explore this property to construct a work reservoir at any finite temperature.

Let the displacement be arbitrary for now, and let α⃗= {α_j}_j denote the set of displacements that each mode in the environment c_j undergoes, so that the collective displacement operator is given by 

    D(α⃗) = ⊗_j D_j(α_j)   ,   with   D_j(α_j) = e^α_j c_j^† - α_j^* c_j   .

Furthermore, we denote by |α_max| the maximum displacement magnitude, which we assume can in principle be performed on different modes of the environment. Let us label the modes with maximum displacement with the index k, such that α_k = |α_max|e^iθ_k.

As we did in the previous section, we also assume weak coupling between the system and the environment by isolating the small coupling parameter λ from the interaction Hamiltonian, such that each coupling strength between the j-th mode and the central oscillator is given by λ g_j.

The limit of the Fano-Anderson model as a system coupled to a work reservoir is now obtained if we take the semiclassical limit, namely vanishing coupling λ→ 0 and infinite displacement |α_max| →∞ such that the parameter ϵ := λ |α_max| is finite. 

Since the Green function G(t) depends only on the coupling strength and not on the initial state of the environment, all the parameters in the master equation depending only on G(t) are easily found in the weak coupling limit as follows:

    G(t)   e^-iω_0t , 
       ω_r(t)  ω_0 , 
       γ(t)   0 ,

while the noise integral I(t) depends on the initial temperature of the environment but not on the initial displacement, such that

    I(t)   0 .

This already shows that, in the limit of vanishing coupling, the dissipator in the master equation (<ref>) vanishes completely, as expected. What survives, however, is the linear driving term. 
Indeed, rewriting the term F(t) in terms of ϵ, namely

    F(t) = -i ϵ∑_j g_j α_j/|α_max|∫_0^t dτ G(t-τ) e^-iω_j τ  ,

we can take the limit of weak coupling and large displacement, so that only the modes we previously labelled with k survive, giving

    F_cl(t) = - ϵ∑_k g_k e^iθ_k e^-iω_k t - e^-iω_0t/ω_0-ω_k  .

It leads to a residual driving in the emergent Hamiltonian described by

    f_cl(t) = - i ϵ∑_k g_k e^iθ_k e^-iω_k t  .


The dynamics of the open system in the semiclassical limit is therefore described by a master equation for a closed, driven bosonic mode, namely 

    ρ̇^cl_S(t) = - i [ ω_0 a^† a +  f_cl(t) a^† - f_cl^*(t) a,ρ^cl_S(t)]   .

The infinitely large environment of the Fano-Anderson model is then responsible for a work protocol on the system with negligible dissipation, and can thus be identified as an ideal work reservoir. 
If there is only one maximally displaced mode, we recover the equation for a harmonic oscillator driven by coherent light. 

We remark here that a similar conclusion for the Caldeira-Legget model – i.e., that an initially displaced environment leads to an effective work contribution – has been reported in <cit.> while this manuscript was in preparation.
Additionally, a link between the emergence of effective work and the presence of coherences in the initial state of the environment is identified in <cit.>, although this connection is established through a different framework.



§ HYBRID ENVIRONMENT

In the previous sections, we have looked at two examples for which the environment in the Fano-Anderson model has extreme thermodynamic functions, either exchanging only work or only heat with the central mode. These examples required strong assumptions – like that of fully Markovian behaviour or of large environmental mode displacement – and also assumed a small coupling parameter. Deviating from these particular assumption results in the environment acting in a hybrid regime, namely exchanging both work and heat with the central mode. Especially at stronger couplings, this represents the most general case. Nonetheless, it is beneficial to look explicitly at a specific example of a hybrid reservoir, and to try to interpret the mechanisms behind this behaviour, especially when the initial state is thermal (such that the environment would traditionally be seen as a heat bath).

We start with a thermal environmental state but choose the coupling with the environment in such a way that non-Markovian behaviour is present. This is done, in our case study, by picking a spectral density with a well defined peak; the structure of choice is that of a Lorentzian, as it makes most of the calculation completely analytic as reported also in <cit.>, where the fact that the environment performs work on the system is exploited to obtain enhanced efficiencies for the Otto cycle.
We use this here instead to give an interpretation of the emergent work by understanding the main peak of the spectral density as an effective driving mode through a reaction coordinate mapping of the model. This hints at the conclusion that structured couplings – by means of peaked spectral densities – are responsible for non-Markovian behaviour which, in turn, leads to driving (time-dependent) contributions in the effective Hamiltonian, even when starting in a thermal state of the environment.




 §.§ Lorentzian spectral density

Let us recall from <cit.> the more explicit results for the parameters in the master equation by assuming the spectral density to be a Lorentzian, i.e.

    J(ω) = γ_0/2 πη^2/(ω_c - ω)^2 + η^2  .

Here, η represents the spectral width – as a guiding intuitive principle, therefore, the smaller η, the more non-Markovian effects will be present – and ω_c=ω_0-Δ describes the position of the peak, so that Δ gives the detuning between the central mode and the spectral density peak. The parameter γ_0, instead, gives a measure for the strength of the coupling and corresponds to the Markovian decay rate when at resonance (Δ=0):

    γ^M = 2π J(ω_0) = γ_0 η^2/Δ^2+η^2  ,     γ^M |_Δ=0 = γ_0   .


With the spectral density (<ref>), the memory kernel (<ref>) can be found analytically by extending the range of the integral to the entire real axis and assuming that we can neglect the effects from the fact that J(ω)≠ 0 at negative frequencies (for a discussion on this point see <cit.>). It is given by

    𝒦(t) = γ_0 η/2 e^-η |t| e^-i ω_c t  ,

such that the Green's function G(t) can be found via Laplace transform of its integro-differential equation (<ref>) as

    G(t) = e^-i ω_0 t/μ_2 - μ_1(μ_2 e^μ_1 t - μ_1 e^μ_2 t)  ,

where μ_1,2 are the roots of the quadratic equation 

    μ^2 + (η - i Δ) μ + γ_0 η/2 = 0  .

Notice that, in this case, the Green's function (<ref>) has the wanted properties mentioned in Sec. <ref>, which are sufficient to prove that the system always relaxes to a unique steady state, and that this state is exactly the reduced thermal equilibrium state at the initial temperature of the environment.

Then, the coupling to the bath induces a time-dependent renormalization of the oscillator frequency, which in this case reads

    ω_r(t)  = ω_0 - Im{μ_1 μ_2 e^μ_1 t - e^μ_2 t/μ_2 e^μ_1 t - μ_1 e^μ_2 t}  ,

and leads to work contributions to the reduced system's energy due to the interaction with the environment. The time-dependent transition rate γ(t) is instead given by

    γ(t)  =  Re{μ_1 μ_2 e^μ_1 t - e^μ_2 t/μ_2 e^μ_1 t - μ_1 e^μ_2 t}  .


From these considerations, and from the fact that both a well-defined time-dependent frequency renormalization and a non-zero transition rate exist, we conclude as expected that the environment exchanges both heat and work with the reduced system. 



  §.§.§ Second order parameters

Evaluating main parameters of the Fano-Anderson master equation, like ω_r or γ, at second order of approximation is useful to understand how spectral density parameters like width and height influence the renormalization and dynamical properties of the central mode. In the case of a Lorentzian spectral density, we find (see Appendix <ref> for the second order expansion)

    Ġ^(2)(t)/G^(2)(t) = - i ω_0 -γ_0 η/2e^(iΔ-η)t-1/iΔ -η  ,

which leads to the following renormalized frequency and transition rate (see also <cit.>):

    ω_r^(2) (t) =   ω_0 + γ_0 ηΔ / 2/η^2 + Δ^2[1 - e^- η t( cosΔ t + η/ΔsinΔ t ) ]   ,
    γ^(2) (t) =   γ_0 η^2/η^2 + Δ^2[1 - e^- η t( cosΔ t - Δ/ηsinΔ t ) ]   .

The terms feature oscillations of frequency given by the detuning Δ. This is true for the second order expansion, while higher orders, as well as the exact result,  might feature instead more complex frequency spectra. From the above we conclude that, at second order, more peaked spectral densities (smaller η) have longer lived oscillations in the parameters, while wider spectral densities (larger η) converge faster to the long time limit values. For the renormalized frequency, the long time (Markovian) limit is given by

    lim_t→∞ω_r^(2) (t) =  ω_0 + γ_0 ηΔ / 2/η^2 + Δ^2 .

At small detuning with respect to spectral width (Δ/η≪ 1), the renormalized frequency is higher for more peaked spectral densities, while the opposite is true for larger detuning with respect to the width (Δ/η≫ 1); thus, the magnitude of the renormalized frequency – at least at low orders in the coupling strength – seems to be dictated by the slope of the spectral density at the value of the bare central frequency, as was also observed for thermodynamic quantities in <cit.>. Analogously, the renormalized transition rate at infinite times, i.e.

    lim_t→∞γ^(2) (t) =   γ_0 η^2/η^2 + Δ^2  ,

is instead determined by the height of the spectral density at the bare central frequency, and gives simply γ_0 at resonance, or in the limit of large η (Markovian limit).



 §.§ Spectral density peaks do work


The intuition that the peak of the Lorentzian spectral density is responsible for the emergent work protocol, coming from the previous observation that flat spectral densities lead to a perfect heat bath (Sec.<ref>) and from the study of second order parameters, can be better understood and interpreted by performing a reaction coordinate mapping <cit.>. This method consists in a unitary normal mode transformation of the environment, which allows to incorporate the most important environmental degrees of freedom into a single resulting collective mode, called the reaction coordinate (RC).
The RC couples directly to the reduced system – here, what the reduced system actually is is not very important; the method works in several different scenarios – and, at the same time, interacts with the residual environment. The transformed residual environment couples only to the RC and not to the original reduced system, and the hope is that this coupling is weak and such that it is possible to apply second-order Born-Markov approximations and to obtain a standard Lindblad master equation for the enlarged reduced system – namely, the original reduced system plus the RC. 


This method is efficient and widely used – particularly also in quantum thermodynamics, see e.g. <cit.> and <cit.> –, and is general to the point that it can also be applied to a fermionic bath <cit.>. In our context we apply it to the Fano-Anderson bosonic bath, leading to a Hamiltonian for the enlarged reduced system that describes the interaction between the central mode and the RC <cit.>, namely

    H_S+RC = ω_0 a^† a +  ω_RC b^† b + g a^† b + g^* a b^†  ,

with b denoting the bosonic creation and annihilation operators for the reaction coordinate and |g| the coupling strength between RC and central mode, while the RC is also weakly coupled to the residual Markovian reservoir, now identified by a bath of bosonic modes c̃_j:

    H_RC = H_S+RC + ∑_j ω_j c̃_j^†c̃_j + ∑_j[ g̃_j b^†c̃_j  + g̃_j^* b c̃_j^†]    ,

with a modified coupling strength |g̃_j|. In the continuum limit, this corresponds to a different spectral density J̃(ω) with respect to the original one. 
Usually, if the original spectral density is peaked enough (such that there is one distinct main frequency in the bath that can be extracted with the RC mapping), then the residual spectral density is expected to allow a standard Born-Markov treatment of the residual bath. We will see next that this is indeed the case for a Lorentzian spectral density, as long as the bandwidth η is not too large.



  §.§.§ Reaction coordinate mapping

The method behind the reaction coordinate mapping is well known and we will not report it here. The mapping depends on the original coupling (which must be linear) and on the desired one as outcome of the procedure; it is possible to obtain the final Hamiltonian (<ref>) from the Fano-Anderson Hamiltonian, and the relevant mapping – namely, the derivation of formulas for |g|, ω_RC and J̃(ω) – can be found in <cit.>. 

Our choice of a Lorentzian spectral density can lead to technical problems, e.g. diverging integrals, due to its inherent unphysical nature as a spectral density. However, this choice often leads to straightforward analytical results if we extend the frequency integrals to the whole real axis, and assume that the contributions due to this are negligible. We apply the same prescription to the reaction coordinate mapping, so that values converge and are consistent. Then, the coupling parameter between system and RC is given by

    |g|^2 = ∫_-∞^+∞ J(ω) dω = 𝒦(0) = γ_0 η/2  ,

and can be made arbitrarily large by increasing γ_0. The width η also influences the coupling, but we will assume next that it represents a small parameter.
The reaction coordinate frequency ω_RC is given by

    ω_RC = 1/|g|^2∫_-∞^+∞ dωω J(ω) = η/π∫_-∞^+∞ dωω+ω_c/ω^2 + η^2  = ω_c   ,

where we have evaluated the integral symmetrically between values -Ω and Ω and then taken the limit Ω→ + ∞.  Thus, the RC frequency is in this case given by the frequency at which the spectral density peaks.

The new spectral density describing the coupling between the RC and the rest of the modes is then given by

    J̃(ω) =  |g|^2 J(ω)/[ 𝒫∫_-∞^+∞ dω' J(ω')/ω' - ω]^2 + [ π J(ω)]^2 = |g|^2 J(ω)/|𝒦̂(-i ω)|^2  ,

where we used eq. (<ref>). With expression (<ref>) for the memory kernel of a Lorentzian spectral density, we find that |𝒦̂(-i ω)|^2 = πγ_0 J(ω)/2, so that the new spectral density is flat and simply given by

    J̃(ω) = γ̃_0/2π  ,     with  γ̃_0 = 2η  .

This corresponds to a flat spectral density with coupling strength γ̃_0 proportional to the width of the original Lorentzian spectral density. This means that the more peaked the spectral density is, the lower is the resulting coupling between the RC and the residual bath. From this we conclude that for small width η we are in the regime where the Born-Markov approximation is allowed, and we can derive a Lindblad master equation for the system+RC which fits very well the exact dynamics and reads <cit.>

    d/dtρ_S+RC(t) = -i [H_S+RC, ρ_S+RC] +𝒟[ρ_S+RC]   ,

where the dissipator is given by two terms with b and b^† operators for the RC as Lindblad operators, i.e.

    𝒟[ρ] = W_- [b ρ b^† -1/2{ b^† b , ρ}] + W_+ [b^†ρ b -1/2{ b b^† , ρ}]   ,

and where the corresponding rates are such that 

    W_- - W_+ = γ̃_0 = 2η  .


The described procedure tells us that the exact evolution of the central mode can either be obtained by solving for the system using the entire non-Markovian bath, or by tracing out the RC in the reaction coordinate mapping using the master equation (<ref>).

What we see therefore is that a Fano-Anderson model with a sufficiently peaked Lorentzian spectral density is equivalent to a reaction-coordinate model where the central mode is linearly coupled to a second mode, which is in turn connected to a Markovian bath. Since the master equation for system+RC is in Lindblad form with a time-independent Hamiltonian, we know that the residual bath is not performing any work and only providing heat to the two modes. The actual work protocol on the central mode must therefore emerge from tracing out the RC. As the RC mode comes out of a well defined narrow peak in the spectral density, we obtain from this the intuition that peaks in environmental spectral densities are responsible for the emergence of driving (work) on the system.

We remark, nonetheless, that this does not mean that spectral density peaks lead only to work. Indeed, coupling a system with a single mode leads to both heat and work contributions; by taking the limit η→ 0 with |g|^2= γ_0 η/2 fixed, we obtain the limiting case of two coupled bosonic modes, as

    lim_η→ 0 J(ω) = |g|^2 δ(ω - ω_c)   ,

while the residual bath disappears. 
Then, even assuming weak coupling such that we can look at second order quantities, we find

    lim_η→ 0ω_r^(2) (t)    = ω_0 + γ_0 η/ 2Δ[1 - cosΔ t ]    , 
    lim_η→ 0γ^(2) (t)    =  γ_0 η/ΔsinΔ t    ,

which implies both heat and work contributions. 




§ CONCLUSIONS

In this paper, we examined the diverse roles that the same quantum environment can take by analyzing the Fano-Anderson model, for different structures of spectral density and different initial environmental states. The model, while exactly solvable, is entirely suitable for thermodynamic studies since, as we showed, it leads to relaxation to equilibrium (when the environment is initially in a thermal state) or to the approach to a NESS (if there is initial displacement in the environmental modes).
By leveraging a recently proposed framework for open system quantum thermodynamics, we were able to make considerations that go beyond standard weak-coupling quantum thermodynamics.
Crucial aspects we found for the model are the renormalization of the system frequency and the driving amplitude induced by the interaction with the environment, and the thermodynamic properties of the nonequilibrium steady state emerging in the case of periodic driving, which can be described by a
master equation in Floquet-Lindblad form.
Furthermore, we demonstrated that a single model could reveal an environment acting as a heat bath, work reservoir, or a hybrid of both. 

We found that in the regime of weak-coupling and Markovian dynamics, the environment starting from a thermal state typically exchanges only heat with the system, and thus takes the role of a heat bath as it is traditionally assumed. Outside of weak coupling, a flat spectral density is still required to obtain a similar result, showcasing how Markovian noise is related to heat exchange. On the contrary, when the environment is initialized in a displaced thermal state, it can exert coherent driving forces on the system. In the semiclassical limit of weak coupling and large displacement, the heat exchange becomes negligible and the environment exchanges only work with the system, behaving as a work reservoir. Finally, we explored the case of a hybrid environment starting from a thermal state, where yet both heat and work exchanges occur simultaneously. Choosing a Lorentzian spectral density and understanding the dynamics through a reaction-coordinate mapping, we attribute the emergence of work protocols to peaks in the spectral density. 

From this study we conclude that the actual definition of a quantum heat bath needs to be stated with caution: interactions at a quantum level, even to an infinite number of degrees of freedom, can be highly sensitive to details (like the initial state of the environment) and can turn the same physical environment from a heat bath to an effective work reservoir. In particular, we identify two sources of effective work on quantum systems: initial coherences in the environment (displaced environmental modes), and colored noise, i.e. structured spectral densities leading to non-Markovian effects.

Learning how to properly engineer a quantum environment (and accordingly, its coupling to the system) is therefore crucial for the employment of quantum thermodynamics outside of weak coupling in practical applications. Emergent work protocols can, for example, be exploited for the purpose of autonomous quantum heat engines <cit.>, or to enhance standard thermodynamic cycles <cit.>. 



We would like to thank Andrea Smirne and Franklin Rodrigues for useful feedback on the project and the manuscript. This project has received funding from the European Union's Framework Programme 
for Research and Innovation Horizon 2020 (2014-2020) under the 
Marie Skłodowska-Curie Grant Agreement No. 847471. We acknowledge support by the Open Access Publication Fund of the University of Freiburg. This work has also been supported by the German Research Foundation (DFG) through FOR 5099.







§ EXACT TCL MASTER EQUATION

We derive here the exact master equation (<ref>) for the Fano-Anderson model in an initial Gaussian state.
From the commutation relations
[c_j,c^†_k]=δ_jk, [a,a^†]=1, and [a,c^†_j]=0, we write the exact coupled Heisenberg equations of motion for all system and bath modes according to the total Hamiltonian (<ref>):

    ȧ(t)    = -i ω_0 a(t) -i ∑_j g_j c_j(t)   , 
    ċ_j(t)    = -i ω_j c_j(t) -i  g_j^* a(t)   ,

which, formally solving for c_j(t), lead to

    c_j(t) = e^iω_jc_j(0) - i g_j^* ∫_0^t τ e^-iω_j(t-τ) a(τ)   ,

and thus to the integro-differential equation for the system mode

    d/d ta(t)  +i ω_0 a(t) + ∫_0^t τ𝒦(t-τ) a(τ) = c(t)   .

The dynamics of the reduced system is thus determined by the coupling with the environmental modes and their initial condition as encoded in the above defined memory kernel and inhomogeneity, which read

    𝒦(t,τ)    ≡𝒦(t-τ) 
     := ∑_j |g_j|^2 e^-iω_j (t-τ)  , 
    
    c(t)    := -i ∑_j g_j e^-iω_j t c_j(0)   .

We take the continuum limit in the environment by introducing a spectral density J(ω) whose role is to describe both the density of bath modes present at a certain frequency, and how strongly these modes couple to the central system. Therefore we identify

    J(ω) = ∑_j |g_j|^2 δ(ω-ω_j)  ,

which can be replaced, as in all practical applications, by a continuous function. The choice of this function therefore determines the properties of the environment and its coupling with the system. In terms of the spectral density the memory kernel is then written as

    𝒦(t,τ) = ∫_0^∞ dω J(ω) e^-iω (t-τ)  .


To solve the integro-differential equation (<ref>), we define the Green function G(t) as the solution of the homogeneous integro-differential equation

    d/dtG(t) + iω_0 G(t) +  ∫_0^t τ𝒦(t-τ) G(τ) = 0

corresponding to the initial value G(0)=1. From this it follows the exact Heisenberg evolution for a, namely

    a(t) = G(t) a(0) + ∫_0^t τ G(t-τ) c(τ)   ,

which then gives all relevant information on the dynamics of the reduced system, as a function of initial conditions a(0) and c_j(0) only.  We assume a factorizing total initial state ρ_SE(0) = ρ_S(0) ⊗ρ_E(0),  where the initial environmental state is for now a general Gaussian state of decoupled modes

    ρ_E(0) = ⊗_jρ_j(0)   .

so that

    ⟨ c(t) ⟩   = -i ∑_j g_j e^-iω_j t⟨ c_j(0) ⟩ , 
    ⟨ a(0) c(t) ⟩   = -i ∑_j g_j e^-iω_j t⟨ a(0)⟩⟨ c_j(0)⟩  , 
    ⟨ a(0) c^†(t) ⟩   = -i ∑_j g_j e^iω_j t⟨ a(0)⟩⟨ c_j^†(0)⟩ , 
    ⟨ c(t_1) c(t_2) ⟩   = - ∑_j k g_j g_k e^-iω_j t_1 e^-iω_k t_2⟨ c_j(0) c_k(0)⟩  , 
    ⟨ c^†(t_1) c(t_2) ⟩   =  ∑_j k g^*_j g_k e^iω_j t_1 e^-iω_k t_2⟨ c^†_j(0) c_k(0)⟩  .


From the Heisenberg evolution (<ref>) we can find expressions for the first and second moments of the open system mode, which represent all relevant moments of the system,  in terms of the Green function G(t) and the initial environmental conditions:

    ⟨ a ⟩_t =     G(t) ⟨ a ⟩_0 + F(t)  , 
    ⟨ aa ⟩_t =     G^2(t) ⟨ aa ⟩_0 + 2 G(t) F(t) ⟨a|_⟩0 
       +F^2(t) - J(t), 
    ⟨ a^†a ⟩_t =     |G(t)|^2 ⟨ a^†a ⟩_0 + G(t)F^*(t)⟨a|_⟩0 
       + G^*(t)F(t)⟨a^†|_⟩0 + |F(t)|^2 + I(t),

where we have defined, using the notation ⟨⟨AB|⟩|_⟩0 = ⟨AB|_⟩0-⟨A|_⟩0⟨B|_⟩0,

    F(t):=    -i ∑_j g_j ∫_0^t dτ  G(t-τ)  e^-iω_j τ⟨c_j|_⟩0   ,
    
     I(t) :=   ∑_j |g_j|^2 ∫_0^t dτ_1  ∫_0^t dτ_2  G^*(t-τ_1) G(t-τ_2) ×
           × e^i ω_j(τ_1-τ_2)⟨⟨c_j^† c_j|⟩|_⟩0  , 
    
     J(t) :=   ∑_j g_j^2 ∫_0^t dτ_1  ∫_0^t dτ_2 G(t-τ_1) G(t-τ_2) ×
           × e^-i ω_j(τ_1+τ_2)⟨⟨c_j c_j|⟩|_⟩0  .


To derive the master equation in time local form, we first write down the exact equations of motion for the system moments, so that we can use them to match a suitable master equation ansatz:

    d/dt⟨ a ⟩_t =   Ġ(t)/G(t)⟨ a ⟩_t
     +f(t), 
    d/dt⟨ aa ⟩_t =    2 Ġ(t)/G(t)⟨ aa ⟩_t
     +2 f(t)⟨a|_⟩t +2 Ġ(t)/G(t) J(t) - J̇(t) , 
    d/dt⟨ a^†a ⟩_t =     2 Re(Ġ(t)/G(t)) ⟨ a^†a ⟩_t 
     +f(t) ⟨a^†|_⟩t
      +f^*(t) ⟨a|_⟩t 
       - 2 Re(Ġ(t)/G(t)) I(t) + İ(t).

where

    f(t) = Ḟ(t)- Ġ(t)/G(t) F(t)

Based on the fact that the microscopic Hamiltonian is quadratic – thus that the dynamics preserves the Gaussianity of states – we make the ansatz that the master equation must be quadratic in the creation and annihilation operators a^†, a of the central mode. Namely we already assume that the master equation is in the following time-local, generalized Lindblad form

    d/dtρ_S(t) = -i [K̃_S(t),ρ_S(t)] + 𝒟̃_t  [ρ_S(t)],

where the effective Hamiltonian is at most quadratic

    K̃_S(t) = ω_r(t) a^†a + if_r(t)a^† - if_r^*(t)a,

(there could be in principle also terms in aa and a^† a^†, but they actually drop out) and the dissipator is given in terms of creation and annihilation operators as Lindblad operators, though in non-diagonal form

    𝒟̃_t[ρ_S] =     
     d_1(t) [aρ_Sa^†-1/2{a^†a,ρ_S}] 
       +d_2(t) [a^†ρ_Sa-1/2{aa^†,ρ_S}] 
         - d_3(t) [aρ_Sa-1/2{aa,ρ_S}] 
       - d^*_3(t) [a^†ρ_Sa^†-1/2{a^†a^†,ρ_S}]   .

This dissipator can be understood as already satisfying minimal dissipation according to Sec. <ref>. In the ansatz we have given five time dependent coefficients ω_r(t), f_r(t), d_1(t), d_2(t) and d_3(t); notice that the coefficients ω_r(t), d_1(t) and d_2(t) must be real as a consequence of Hermiticity preservation of the generator. According to the conjectured master equation, we find the following equations of motion for the moments in terms of the five coefficients:

    d/dt⟨ a ⟩_t =   [ -iω_r(t) + d_2(t)-d_1(t)/2] ⟨ a ⟩_t + f_r(t),  
    d/dt⟨ aa ⟩_t =   [-2iω_r(t) + d_2(t) - d_1(t)] ⟨ a a ⟩_t  
       +2  f_r(t) ⟨ a ⟩_t  + d_3(t)
    d/dt⟨ a^†a ⟩_t =   [d_2(t)-d_1(t)] ⟨ a^† a ⟩_t + d_2(t)
       + f_r(t) ⟨ a ⟩_t^* + f_r^*(t) ⟨ a ⟩_t  .

Comparing these equations to (<ref>)-(<ref>), we find that 

    ω_r(t) =  -(Ġ(t)/G(t))   ,

while the complex parameters are

    f_r(t)    = f(t) 
    
    d_3 (t)    = J̇(t) - 2 Ġ(t)/G(t) J(t)

By defining now a generalized transition rate γ(t) and a generalized average bath excitation number N(t)

    γ(t) := - 2 ( Ġ(t)/G(t))   ,    
     N(t) := I(t) + İ(t)/γ(t)  ,

we get, for the remaining coefficients

    d_1(t) = d_2(t) + γ(t) = γ(t)(N(t)+1)  .


Combining these results leads to the exact master equation for the central mode in the Fano-Anderson model (<ref>).





§ RENORMALIZED DRIVING

In the context of the Fano-Anderson model we can also see, exactly and explicitly, what happens when we impose an external driving protocol on the reduced system while it is connected to the environment. In light of the frequency renormalization of the bare frequency of the system Hamiltonian as a consequence of interaction, a good question to ask is whether an arbitrary coupling to an environment modifies the effect of the external work protocol in terms of work contributions. We will see that this is in general the case in the Fano-Anderson model, where the driving function undergoes renormalization, similarly to the central mode frequency. 

We start with the same assumptions as in the main text, namely factorizing total initial state with a thermal initial environmental state, and modify the global Hamiltonian of the model (<ref>) to allow for external driving on the central mode, i.e. substituting H_S with

    H_S(t) = ω_0 a^† a +i l(t) a^† -i l^*(t) a   ,

where l(t) is taken to be a completely arbitrary complex function of time. Then, the solution of the model follows just as in Appendix <ref>, with modified Heisenberg equations of motion

    ȧ(t)    = -i ω_0 a(t) +l(t) -i ∑_j g_j c_j(t)   , 
    ċ_j(t)    = -i ω_j c_j(t) -i  g_j^* a(t)   ,

where the equation for the bath modes is identical to the non-driven case, due to the fact that the driving protocol only acts on the system. As before, we find an integro-differential equation for the system mode

    / ta(t)  +i ω_0 a(t) + ∫_0^t τ𝒦(t-τ) a(τ) = c(t) +l(t)   .

with the same memory kernel (<ref>) as before, but where the driving function l(t) is added to the inhomogeneity (<ref>). Since the homogeneous part of the differential equation is equivalent to the undriven case, we can use the exact same Green's function G(t) as in Appendix <ref>, see eq. (<ref>). This will make it much easier to compare upcoming results to the undriven case; notice, however, that this is not true for all driving protocols — it wouldn't be the case, for example, if the time dependency of the system Hamiltonian would have been realized by changing directly the central mode frequency with time. For our chosen driving terms, though, the exact Heisenberg evolution for a is given by

    a(t) = G(t) a(0) + ∫_0^t τ G(t-τ) c(τ) + F(t)   ,

where we have defined

    F(t) = ∫_0^t τ G(t-τ) l(τ)   .


We proceed exactly as we did in the undriven case,  deriving from the Heisenberg evolution (<ref>) exact expressions for the first and second moments of the open system mode, which are sufficient to describe the whole dynamics, and which are now given also in terms of the driving function l(t):

    ⟨ a ⟩_t =    G(t) ⟨ a ⟩_0 +F(t), 
    ⟨ aa ⟩_t =    G^2(t) ⟨ aa ⟩_0 +2F(t)G(t)  ⟨ a ⟩_0 + F^2(t)  
    ⟨ a^†a ⟩_t =    |G(t)|^2 ⟨ a^†a ⟩_0 + I(t) + F(t)G^*(t)  ⟨ a ⟩_0^* 
       + F^*(t)G(t)  ⟨ a ⟩_0 + |F(t)|^2

where the noise integral I(t) is still given by (<ref>). Again from the above we derive the exact equations of motion for the relevant system moments, with the goal to compare their terms with those given by a master equation ansatz. They read:

    d/dt⟨ a ⟩_t =   Ġ(t)/G(t)⟨ a ⟩_t +Ḟ(t) - Ġ(t)/G(t) F(t)   , 
    d/dt⟨ aa ⟩_t =    2 Ġ(t)/G(t)⟨ aa ⟩_t + 2[ Ḟ(t) - Ġ(t)/G(t) F(t) ]  ⟨ a ⟩_t     , 
    d/dt⟨ a^†a ⟩_t =     2 Re(Ġ(t)/G(t)) ⟨ a^†a ⟩_t - 2 Re(Ġ(t)/G(t)) I(t) 
        + İ(t)
      + [ Ḟ(t) - Ġ(t)/G(t) F(t) ]  ⟨ a ⟩_t^* 
        + [ Ḟ^*(t) - Ġ^*(t)/G^*(t) F^*(t) ]  ⟨ a ⟩_t   .


Now we make again an ansatz for the master equation. Since the new driven Hamiltonian is still at most quadratic, with a shape that fits our previous ansatz (<ref>), we keep the exact same ansatz for both parts of the master equation, namely eq. (<ref>) for K_S and eq.  (<ref>) for the dissipator. The equations for the moments ⟨ a ⟩_t, ⟨ aa ⟩_t and ⟨ a^† a ⟩_t are then still given by eqns. (<ref>)-(<ref>). Comparing their coefficients ω_r, f_r, and d_1,2,3 with the new equations (<ref>), (<ref>) and (<ref>), we find a perfect correspondence with the earlier coefficients ω_r(t), γ(t), d_1(t), d_2(t) and N(t) (d_3(t)=0 because the initial state of the environment is thermal).

The master equation is therefore identical to the undriven case except for the driving force parameter

    f_r(t) =   Ḟ(t) -Ġ(t)/G(t) F(t) 
    
     =    l(t) + ∫_0^t τ[  Ġ(t-τ)/G(t-τ) - Ġ(t)/G(t)] G(t-τ) l(τ)   .


This equation shows that there is a time-dependent shift in the original driving force that depends on the shape of G(t). This means that, in general, a driving protocol acting on an open system may be affected by the coupling with the environment, thus one should be careful when considering such scenarios. Here, for example, the driving force gets renormalized and reshaped in the effective Hamiltonian (we expect that similar effects appear in classical systems <cit.>). We can also observe, by writing the new driving force in terms of other renormalized parameters, namely

    f_r(t) =    l(t) -i∫_0^t τ  [ω_r(t-τ) - ω_r(t)] G(t-τ) l(τ) 
       - ∫_0^t τ1/2[γ(t-τ) - γ(t)] G(t-τ) l(τ)   ,

that if both ω_r and γ are time independent, then the force in the Hamiltonian is not renormalized, and the driving protocol remains unaltered. However, even if the renormalized frequency ω_r is time independent – which would lead to zero work contribution in the undriven case – it is still possible that the eventual time-dependency of γ affects the driving renormalization non-trivially. The take-home message is: in arbitrarily coupled open systems, even if in the absence of external work protocols the environment does not perform work on the system, it does not follow that an additional external driving would be left unmodified. Thus, extra care should always be taken to ensure that the real effective work protocol, with possible modifications emerging from the interaction with the bath, is adequately treated.



§ RELAXATION TO EQUILIBRIUM AND TO NONEQUILIBRIUM STEADY STATES

In this section, we report several details needed to understand the relaxation of the open system to a unique steady state in the long time limit.



 §.§ Final excitation number

To evaluate in general the final steady state for a thermal case, we take the limit to infinity of the noise integral (<ref>). We can also perform a change of variables and exploit the fact that the correlation function ⟨ c^†(τ_1) c(τ_2) ⟩ depends only on the time difference (τ_1-τ_2) to find

    I(∞) = ∫_0^∞ d t_1  ∫_0^∞ d t_2 G^*(t_1) G(t_2) ⟨ c^†(t_2) c(t_1) ⟩  .

The correlation function reads

    ⟨ c^†(t_2) c(t_1) ⟩ =   ∑_i,j g_i^* g_j e^iω_i t_2 e^-iω_jt_1⟨ c_i^†(0) c_j(0) ⟩
    =   ∑_j |g_j |^2e^iω_j (t_2-t_1)1/e^βω_j-1  ,

where we have used ⟨ c_i^†(0) c_j(0) ⟩ = δ_ijn_j(0) with n_j(0) the Planck distribution for energy ω_j and inverse temperature β. Taking the continuum limit we find

    ⟨ c^†(t_2) c(t_1) ⟩ = ∫_0^∞ω J(ω) e^iω (t_2-t_1)1/e^βω-1  ,

which, inserted into (<ref>) gives the final steady state excitation number in terms of the spectral density and the Laplace transform of G(t), namely

    ⟨ a^† a ⟩_∞ = ∫_0^∞ω J(ω) 1/e^βω-1  |Ĝ(-iω)|^2    .

The Laplace transform Ĝ is given by eq. (<ref>) in terms of the Laplace transform of the memory kernel (<ref>):

    Ĝ(-iω) = 1/i(ω_0-ω)+𝒦̂(-iω)  ,

with

    𝒦̂(z) = ∫_0^∞ω' J(ω')/z+iω'  .

In turn, this is given in terms of a principal value integral

    𝒦̂(-iω)= π J(ω) + i Δ(ω)   ,


    Δ(ω) = 𝒫∫_0^∞ω' J(ω')/ω - ω'  ,

giving finally

    Ĝ(-iω) = 1/i(ω_0+Δ(ω)-ω)+π J(ω)  ,

and the steady state excitation number

    ⟨ a^† a ⟩_∞ = ∫_0^∞ω J(ω)   1/e^βω-1×
       ×1/[ω_0+Δ(ω)-ω]^2+π^2 J^2(ω)  .

We will use this result in the next section to prove that the above is equivalent to the equilibrium expectation value — namely considering system and environment in a global Gibbs state at the initial bath temperature.



 §.§ Proof of thermal equilibrium at bath temperature

We have already proven that the model relaxes to a unique final steady state; we will now show that this state is equivalent to the mean-force state obtained by tracing out a global thermal state:

    ρ_S^*= _E {e^-β H_SE/Z_SE} ,

with β still the initial inverse temperature of the environment. To prove this it is sufficient to show that the average values of all relevant moments at long times, namely eqs. (<ref>)-(<ref>), coincide with the average taken over a global Gibbs state. The expectation value of a and aa are zero for a Gibbs state of the total Hamiltonian, so the only value left to check is that of 

    ⟨ a^† a ⟩_eq := { a^† a ⊗𝕀e^-β H_SE/Z_SE}  .

For this purpose we make use of the equilibrium fluctuation dissipation theorem, for which the expectation value of the position operator X = (a^†+a)/√(2) squared is given in terms of the Fourier transform of the response function 

    χ(t) := i θ(t)⟨ [X(t), X(0)] ⟩ = i/2θ(t) ( G(t)- G^*(t) )   ,

namely

    ⟨ X^2 ⟩_eq =  1/2π∫_-∞^+∞ω(βω/2){χ̃(ω)}  ,

where χ̃(ω) denotes the Fourier transform of χ(t). Evaluating the above is useful because of the easily proven relation

    ⟨ a^† a ⟩_eq = ⟨ X^2 ⟩_eq - 1/2 =1/π∫_0^∞ω (2 n(ω)+1){χ̃(ω)}  ,

where we have used the fact that the integrand in (<ref>) is symmetric in ω and have inserted (βω/2) = 2 n(ω)+1.
Using (<ref>), the Fourier transform of χ is given by 

    χ̃(ω) =  ∫_-∞^+∞ t e^i ω tχ(t) =  i/2[ Ĝ(-i ω) - Ĝ^*(i ω) ]   ,

such that its imaginary part is given by 

    {χ̃(ω)} =  1/2[ {Ĝ(-i ω)} - {Ĝ(i ω)}]   .
 Here, the second term vanishes and, recalling (<ref>), we are left with

    {χ̃(ω)} = 1/2π J(ω)/[ω_0+Δ(ω)-ω]^2+π^2 J^2(ω)  .

Notice now that the following integral, whenever all poles of Ĝ(z) have negative real part, is equal to one:

    ∫_0^∞ω J(ω)/[ω_0+Δ(ω)-ω]^2+π^2 J^2(ω) =1   ;

using all the above in (<ref>) we finally arrive at

    ⟨ a^† a ⟩_eq =∫_0^∞ω J(ω)n(ω) 1/[ω_0+Δ(ω)-ω]^2+π^2 J^2(ω)  ,

which is identical to the steady state value (<ref>). This shows that whenever G(t) has negative Laplace transform poles (which in turn implies that G(t) vanishes in the long time limit, as already assumed) then the final steady state is equivalent to the state which shows the system's perspective of global thermal equilibrium.



 §.§ Approach to a nonequilibrium steady state


From the exact evolution of the moments (<ref>)-(<ref>), the limit of the Green function G(t) decaying to zero at long times leads to the disappearance of all dependence on the initial system state just like in the thermal case, and the noise integral I(t) converges to the same value I(∞) of (<ref>). 

To understand what happens to the displacement term F(t) in the long time limit, we rewrite it as

    F(t) =    -i ∑_j g_j α_j e^-iω_j t∫_0^t dτ G(τ)e^iω_j τ .

Then, in the limit where G has decayed we can take the limit to infinity in the integral to obtain

    F(t) =    -i ∑_j g_j α_j e^-iω_j t∫_0^∞ dτ G(τ)e^iω_j τ = 
    
    =   -i ∑_j g_j Ĝ(-iω_j) α_j e^-iω_j t  ,

which gives the final expression for the nonequilibrium steady state displacement. 

Since the long time limit of Ġ(t)/G(t) is given by the steady values of γ and ω_r in the undriven case,

    lim_t→∞Ġ(t)/G(t) = -(γ/2 + i ω_r)

one obtains the long time limit driving force (<ref>) simply from

    f(t) = Ḟ(t) + (γ/2 + i ω_r)F(t)   .


To check that the unitary evolution of the state ρ_S(t) is also described by the long-time limit master equation (<ref>), we first rewrite equation (<ref>) as a commutator with a Hamiltonian:

    d/dtρ_S(t) = -i [H(t), ρ_S(t)]   ,

where the Hamiltonian generating the unitary evolution of the NESS is given by H(t)= i Ḋ_t D^†_t, thus in turn by 

    H(t)= i Ḟ(t) a^† - i Ḟ^*(t) a   .

It should hold that ℒ_t[ρ_S(t)] =  -i [H(t), ρ_S(t)]. One can see that, due to the properties of displacement operators on a and a^† and that 𝒟[ρ^G_r_S]= 0, the dissipator part acting on the NESS gives itself a Hamiltonian contribution

    𝒟[ρ_S(t)] = -i [H_D(t), ρ_S(t)]   ,

with 

    H_D(t)= i γ/2(F^*(t) a - F(t) a^†)   ,

which leads to ℒ_t[ρ_S(t)] =-i[K_S(t)+H_D(t),ρ_S(t)]. It is easy to check that 

    K_S(t)+H_D(t) = ω_r (a^† - F^*(t))(a - F(t)) + H(t)   ,

and that 

    [ω_r (a^† - F^*(t))(a - F(t)) ,  ρ_S(t)] = D_t[ω_r a^† a ,  ρ^G_r_S]D^†_t =0  ,

which proves the equivalence between the two evolutions on this state.




§ BORN-MARKOV APPROXIMATION

To obtain the Born-Markov limit of the model in the thermal case, we assume weak system-bath coupling by isolating a coupling parameter λ from the interaction Hamiltonian.
This leads to a spectral density of second order in the coupling, namely λ^2 J(ω). Then, the still exact integro-differential equation (<ref>) for the Green function G(t), where we now explicitly write out the coupling parameter λ, gives

    d/dtG(t) + iω_0 G(t) + λ^2  ∫_0^t τ𝒦(t-τ) G(τ) = 0.

From the above we can evaluate G(t) at different orders of the coupling to obtain a second-order approximation. Because of the structure of the differential equation, we make the ansatz that only terms of even order in λ appear in G(t), so that we can write it as

    G(t) = G_0(t) +λ^2 G_2(t) +λ^4 G_4(t) + ...   .

Proceding order by order in solving the differential equation gives the zeroth order equation for G_0 and its solution:

    d/dtG_0(t) + iω_0 G_0(t) = 0      G_0(t) = e^-i ω_0 t  ,

with G_0(0)=1, and consequently the second order ones for G_2:

    d/dtG_2(t) + iω_0 G_2(t) +  ∫_0^t τ𝒦(t-τ) G_0(τ) = 0   ,

with G_2(0)=0. This implies

    G_2(t) = - e^-i ω_0 t∫_0^t τ∫_0^τ s 𝒦(τ-s) e^-i ω_0 (τ - s)  .


We want to use these results to evaluate the full second order Green function and its differential equation, in order to obtain an approximation of the relevant parameters in the master equation. They are:

    G^(2)(t) = e^-i ω_0 t[ 1- λ^2 ∫_0^t τ∫_0^τ s 𝒦(τ-s) e^-i ω_0 (τ - s)]  ,


    Ġ^(2)(t) =     - iω_0 G^(2)(t)  
        - λ^2 e^-i ω_0 t∫_0^t τ𝒦(t-τ) e^-i ω_0 (t-τ)  .

Inserting the expression of the memory kernel in terms of the spectral density gives the following equation for the central second order quantity

    Ġ^(2)(t)/G^(2)(t) = -i ω_0 -λ^2 ∫_0^∞ω J(ω) ∫_0^t τ    e^-i (ω-ω_0) τ  ,

which determines the second order expansion of the parameters γ(t) and ω_r(t). 

The Markovian approximation is employed by taking the limit t→∞ in the integral to obtain

    Ġ^(2)(t)/G^(2)(t) = -i ω_0 - iλ^2 𝒫∫_0^∞ωJ(ω)/(ω_0-ω) - λ^2π J(ω_0)   ,

where 𝒫 denotes the Cauchy principal value. The real and imaginary part give the Markovian coefficients in the master equation at second order:

    γ_0 =    2 λ^2 π J(ω_0)    ,  
    ω_r =   ω_0 + λ^2 δω =  ω_0 + λ^2 𝒫∫_0^∞ωJ(ω)/(ω_0-ω)  ,

which are time independent as a consequence of the Markov approximation. 

Also in the case of a linear driving force as explored in Appendix <ref>, we find that the driving force does not get renormalized. Indeed, recalling the results of Appendix <ref> for the renormalization of the driving force, namely eq. (<ref>), we see that finding time-independent rate γ and frequency ω_r in the undriven model implies that the driving force is left unaltered. Thus, in the case where the Born-Markov approximation can be applied, there is no modification of a driving protocol of the form (<ref>), such that the work done on the system is fully determined by the external driving force, while the environment is only providing heat exchange. However, we remark again that this result might not hold for all types of driving – for example, it might fail for quadratic driving as opposed to the proposed linear driving. 









