Paper Title: Time-resolved Stochastic Dynamics of Quantum Thermal Machines.





ahegde4@rochester.edu

Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA



patrick.potts@unibas.ch
Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82 CH-4056, Switzerland


glandi@ur.rochester.edu

Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA


§ ABSTRACT 
Steady-state quantum thermal machines are typically characterized by a continuous flow of heat between different reservoirs.
However, at the level of discrete stochastic realizations, heat flow is unraveled as a series of abrupt quantum jumps, each representing the exchange of finite quanta with the environment.
In this work, we present a framework that resolves the dynamics of quantum thermal machines into cycles that are classified as engine-like, cooling-like, or idle.
We explore the statistics of each cycle type and its duration, enabling us to determine both the fraction of cycles useful for thermodynamic tasks and the average waiting time between cycles of the same type.
Our framework presents a novel approach in characterizing thermal machines, with significant relevance to modern experiments, such as mesoscopic transport using quantum dots.


Time-resolved Stochastic Dynamics of Quantum Thermal Machines
    Gabriel T. Landi
    Received ...; accepted ...
=============================================================



Introduction.—
A typical quantum thermal machine consists of a system situated between hot and cold thermal baths, extracting or absorbing energy in the form of work, as depicted in Fig. <ref> <cit.>.
As an engine, it extracts work while transferring heat from hot to cold; as a refrigerator, it absorbs work to move heat from cold to hot.
In autonomous machines, this is usually pictured as a continuous process, where heat and work constantly flow through the system <cit.>.
However, within the microscopic domain, the stochastic nature of system and bath interactions endows an alternative perspective where energy is exchanged with the baths in the form of abrupt jumps. 
This is the basis for stochastic thermodynamics in classical (Pauli) rate equations <cit.>, as well as quantum models in the quantum jumps formalism <cit.>.

The jumps occur at random times and in random “channels.”
Let us broadly classify these channels as either an injection (I) or an extraction (E) of energy into or out of the hot (h) or cold (c) baths, resulting in four distinct types of monitored channels 𝕄 = {I_h, E_h, I_c, E_c}. 
Generalizing to multiple injection and extraction channels per bath is straightforward.
The quantum trajectory of such a machine, in the quantum jump unraveling, appears as a random string of the form

    I_h E_c I_c I_h E_h E_c I_h I_c E_h I_c …,

along with their timestamps t_1, t_2, …, indicating when each jump occurred. This representation of the dynamics is grounded in experimental observations <cit.>.
Note that (<ref>) accounts solely for events representing heat exchange with the environment. 
We assume work-based events are undetectable, as they are often associated with unitary drives in many models <cit.>.



The central question we address in this study is can specific thermodynamic cycles be identified solely from strings like (<ref>) so that their statistics can be explored?
For instance, one might intuitively characterize the sequence I_c E_h as a refrigeration cycle, since an energy quanta was injected from c to the system, and subsequently extracted to h suggesting work consumption.
Similarly, I_h E_c could be seen as an engine-like process (or accelerator <cit.>).
These are both examples of “useful cycles.”
Conversely, pairs such as I_h E_h and I_c E_c are events that fail to peddle quanta of energy overall, and incur no entropy production. We refer to these as “idle cycles".
While a machine might operate as an engine on average, the stochastic nature of these processes manifests in individual realizations yielding different cycles.

Classifying cycles raises several meaningful questions, such as
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  * What is the probability of each type of cycle?
    
  * How are cycles related to steady-state currents?
    
  * What is the time required to complete each cycle?
    
  * How many idle cycles precede a useful one?

These questions relate to the extensive literature on Full Counting Statistics (FCS) <cit.> and Fluctuation Theorems <cit.>.
Addressing them involves exploring time-resolved and cycle-resolved quantities, offering a nuanced understanding of the dynamics.

In attempting to classify cycles this way, a challenge arises when the system can withhold multiple excitations at once. 
For instance, in the string (<ref>), what meaning should be ascribed to the substring I_c I_h E_h E_c?
Because excitations are indistinguishable, it is impossible to infer if this was 
1̧ I_c 2̧ I_h 2̧ E_h 1̧ E_c (two idles) or
1̧ I_c 2̧ I_h 1̧ E_h 2̧ E_c (a refrigeration followed by an engine cycle).
While this is not an issue as far as the average heat and work currents are concerned, it does cause ambiguity in defining time-resolved quantities.
In this letter, we focus on systems that can retain only one excitation at a time; i.e., when injections and extractions alternate (I_∙ E_∙ I_∙ E_∙…) in the trajectory.
This assumption is common in experiments involving single <cit.> or double <cit.> quantum dots in the Coulomb blockade regime, as well as realizations of quantum heat engines <cit.>.

First, we establish the restrictions imposed by the single-excitation hypothesis on a quantum Markovian master equation.
Then, we employ the tools of waiting time distributions of the quantum jump unraveling <cit.> to fully characterize the statistics of cycles. 
Finally, we illustrate our results with a three-level maser example. 

Theory.—
We consider a finite-dimensional system weakly coupled to hot and cold baths.
Work may be performed either by a driven Hamiltonian H(t) or by additional work reservoirs. 
It is assumed that the dynamics can be described by a quantum master equation <cit.> (ħ = k_B = 1 throughout),

    dρ/dt = ℒ_tρ≡    -i[H(t), ρ] + ∑_n  𝒟[K_n] ρ
       +∑_α∈{h,c}, j(γ_α j^-𝒟[L_α j] + γ_α j^+𝒟[L_α j^†])ρ ,

where 𝒟[L]ρ = L ρ L^† - 1/2{L^† L, ρ}.
Here, {L_α j} are jump operators for the hot (α = h) and cold (α = c) baths, with 
L_α j denoting extractions and L_α j^† denoting injections, each occurring at rates γ_α j^∓, respectively.
Finally, K_n are jump operators of work reservoirs, which are often used in describing absorption refrigerators <cit.>.

We assume one can only monitor whether energy is injected (extracted) to (from) the hot or cold baths without identifying the specific jump operator (indexed by j) responsible. Therefore, the four corresponding jump superoperators are

    𝒥_E_αρ = ∑_j γ_α j^- L_α j^ρ L_α j^†,
          𝒥_I_αρ = ∑_j γ_α j^+ L_α j^†ρ L_α j^.


As our first result, we prove in the Supplemental Material <cit.> that the condition for the quantum trajectory to have alternating injections and extractions (i.e. at most a single excitation)  is achieved, if and only if, there exist two subspaces  and  spanning the system Hilbert space ℋ, such that: 



    L_α j = 𝒫_E L_α j𝒫_I   ∀ α,j; 
       𝒫_E H(t) 𝒫_I = 𝒫_I H(t) 𝒫_E =0; 
       𝒫_E K_n 𝒫_I = 𝒫_I K_n 𝒫_E=0,


where 𝒫_E/I are projection operators onto ℋ_E/I, satisfying 𝒫_E + 𝒫_I = 1.
Thus, the jump operators of the baths must be block upper-triangular, while those of the work reservoir, and Hamiltonian must be block diagonal in the joint basis of ℋ_E and ℋ_I.

Consequently, L_α j takes the system to  by extracting energy, while L_α j^† directs it to  by injecting.
We refer to  and  as post-extraction and post-injection subspaces. 
While the unitary dynamics and work reservoirs can inject (extract) work into (out of) the system, this result implies that such processes must occur within each subspace.
Transitions between these subspaces are only feasible through interactions with the baths.
An example is the three-level maser [see Fig. <ref> (a)]; other examples are hinted in <cit.>.

We henceforth assume, as is often the case, that there exists a rotating frame where H(t) is time-independent, and that the steady-state (ℒ=0) in this frame is unique.
The single-excitation hypothesis implies a conservation law for the average excitation current exchanged with the baths,

    := ( - )  = - ( - ) ,

which is deduced by noting that ddt𝒫_E ρ(t) 𝒫_E → 0 as the system approaches the steady-state. 
Equation (<ref>) does not imply that the heat currents to both baths are equal, as jumps to each bath generally involve different energies.
Indeed, their mismatch accounts for the work exchanged.

Statistics of cycles.—
Under the single-excitation assumption, the trajectory (<ref>) can be characterized in terms of the statistics of four possible pairs: I_h E_c, I_c E_h, I_h E_h, and I_c E_c.
We refer to each pair I_∙ E_∙ as a “cycle” and label them as X=1,2,3,4, respectively (see Fig. <ref>).
X=1 is a work extraction cycle [Alternatively, it could also represent an accelerator, wherein the direction of heat flow remains the same, but work is injected instead of extracted. For simplicity, we will continue to refer to this as a work cycle. This scenario always holds when the energy exchanged with the cold bath is smaller than that with the hot bath, an assumption we make implicitly.] and X=2 a refrigeration cycle, while X=3, 4 are idle cycles.

We are interested in the long-time steady-state behavior of strings of the form X_1X_2… = I_∙ E_∙ I_∙ E_∙…, adopting the convention that strings always begin with an injection. 
Then, as explained in <cit.>, the probability of observing a specific sequence X_1,…,X_n, with durations τ_1,…,τ_n, is given by

    p_X_1,…,X_n^(τ_1,…,τ_n) = 𝒪_X_n,τ_n…𝒪_X_1,τ_1π_E ,

where

    𝒪_X,τ≡∫_0^τ dt  e^ℒ_0(τ-t) e^ℒ_0t,

and ℒ_0 =  - ∑_α (𝒥_E_α + 𝒥_I_α) is the no-jump superoperator. 
In Eq. (<ref>), we have introduced the jump steady-state <cit.> 

    π_E = ( + )/( + )∈,

to ensure the jump sequence is stationary.

Marginalizing Eq. (<ref>) over all (X_i,τ_i) except one yields the probability that a single cycle is of type X and duration τ,

    p_X^(τ) = 𝒪_X,τ^π_E .

Integrating over τ yields the  probability that the cycle is of type X:

    p_X^ = ∫_0^∞ dτ p_X^(τ) = 𝒪_X π_E,

where 𝒪_X = ∫_0^∞ dτ 𝒪_X,τ =.

The average cycle time given it is of type X reads

    E(τ| X) = 1/p_X^∫_0^∞ dτ τ p_X^(τ).

In <cit.>, we show

    E(τ) = ∑_X=1^4 E(τ| X) p_X^ = 2/,

where  is the dynamical activity of the baths representing the average number of jumps per unit time in the steady-state.

The p_X^ in Eq. (<ref>) represent the relative occurrence of each cycle type over many trajectories, regardless of their duration.
The proportion of p_1/2 over p_3/4 yields information on the intermittency of the machine, revealing how often useful cycles occur over idle ones overall.
In <cit.>, we prove that p_1/2 and the excitation current from Eq. (<ref>) are related by 

    = p_1 - p_2/E(τ),

which provides a fundamental connection between usual steady-state currents and our results: 
the system functions as an engine when p_1 > p_2, and as a refrigerator when p_1 < p_2.

Statistics of useful cycles.— 
Drawing on the concept of intermittency, we can further address the distribution of idle cycles and the intervals between useful ones.
Let 𝒪_ u = 𝒪_1 + 𝒪_2 and 
𝒪_ id = 𝒪_3 + 𝒪_4 denote the superoperators associated with useful and idle cycles, respectively (and similarly for 𝒪_ u,τ and 𝒪_ id,τ).
We find the probability of n idles between two useful cycles to be <cit.>

    ℙ_u(n) = 𝒪_ u𝒪_ id^n𝒪_ uπ_E /𝒪_ uπ_E ,
             n = 0,1,2,…

and, similarly, the time interval t between them to be <cit.>,

    ℙ_u(t) = ∫_-∞^∞dz/2π𝒪_ u,z (1-𝒪_ id,z)^-1𝒪_ uπ_E /𝒪_ uπ_E e^ z t,

where 𝒪_X,z = ∫_0^∞ dτ e^- z τ𝒪_X,τ.
These probabilities provide a refined overview on the performance of the machine.
Remarkably, both these equations fully account for the statistical dependence between cycles.
This discussion concludes our characterization of the questions posed in the beginning, taking into account not only the behavior of individual cycles but also their correlations in sequences of arbitrary length.



Example: Three-level system.—
We apply our results to a three-level maser <cit.> whose schematic is depicted in Fig. <ref>. It is coupled to hot and cold baths at energy ω_α and temperature T_α with their populations following a Bose-Einstein distribution given by n̅_α=[exp(ω_α/T_α)-1]^-1. The maser is driven by a Hamiltonian H(t) = (ω_h-ω_c)σ_11 + ω_h σ_22 + ϵ(e^iω_d tσ_01 + e^-iω_d tσ_10) with a Rabi drive of strength ϵ and frequency ω_d. 
The jump operators are 
L_h = σ_02, L_c = σ_12 (and K_n = 0) with rates 
γ_α^- = γ_α(n̅_α + 1) and γ_α^+ = γ_αn̅_α.
Here, σ_ij = |i⟩⟨ j| are the transition operators.

The post-extraction subspace is spanned by {|0⟩,|1⟩}, and the post-injection by {|2⟩}. As anticipated, the Hamiltonian is block diagonal in the joint basis of these subspaces.
In this model, we find that the excitation current is directly related to the steady-state heat and work currents as
Q̇_h = ω_h, Q̇_c = - ω_c, and Ẇ = ω_d <cit.>.



Explicit formulas for the the cycle probabilities of this model are provided in <cit.>.
Fig. <ref> (b) illustrates p_X(τ) [Eq. (<ref>)] as a function of time.
The wiggles are reminiscent of coherent Rabi oscillations between |0⟩ and |1⟩. For large τ, all probabilities scale as p_X^(τ) ∼ e^-γτcos(ϵτ) where γ = (γ_h+γ_c)/2.

The marginals p_X [Eq. (<ref>)] are shown in Fig. <ref> (c) as a function of ratio of bath temperatures.
The plot highlights the different regimes of operation, which changes from refrigeration to engine at T_h/T_c = ω_h/ω_c.
It is noteworthy that

    p_1 - p_2 ∝n̅_h - n̅_c,     p_3/p_1 = p_2/p_4 = (n̅_h + 1) γ_h/(n̅_c + 1) γ_c,

implying that across all the parameter ranges, the probabilities of idle cycles bound those of useful ones <cit.>.
As a result, it is always more likely to observe the machine undergoing a cycle with no net heat transfer.

For small ϵ, the probability of a useful cycle p_u := p_1+p_2 scales as
ϵ^2/(Γ^2 + Δ^2), where Γ = (n̅_h γ_h + n̅_c γ_c)/2 
is the net decoherence rate
and
Δ = (ω_h-ω_c) - ω_d is the detuning.
This provides insights on maximizing the occurrence of useful cycles: stronger pumps, more resonant drives and lower damping. 
For this model Eq. (<ref>) simplifies to the geometric distribution, 
ℙ_ u(n) = p_u(1-p_u)^n. 
The distribution of time between two useful cycles [Eq. (<ref>)] is analyzed in <cit.>.

The average duration of cycles [Eqs. (<ref>), (<ref>)] are plotted in Fig. <ref> (d); for this model, it turns out that E(τ| 1) = E(τ| 3) and E(τ| 2)=E(τ| 4). 
Noticeably, the cycles tend to take much longer in the refrigeration regime.  
Moreover, at resonance (Δ = 0), all conditional averages tend to become very close (although not strictly equal), as shown in the inset of Fig. <ref> (d).

Conclusions.— We showed how to unravel the time-dependent statistics of quantum thermal machines, which allowed us to classify the stochastic dynamics in terms of distinct types of cycles, describe how often each cycle occurs, and how long they take.
Our results encompass all statistical correlations between cycles, and also connect with known results in FCS for the average excitation current and dynamical activity. 
This framework opens up a new avenue for characterizing quantum thermal machines, based entirely on experimentally observable data.
In particular, our formalism could be readily employed to analyse, e.g., mesoscopic transport in quantum dot experiments, shedding light on the underlying thermodynamics.

Of particular importance to this analysis is the notion of intermittency — i.e., how often a machine is actually performing a thermodynamically useful task. 
Since our approach enables the identification of both useful and idle cycles, we now have the tools to optimize the intermittency for fixed efficiency and output power.
Our results [cf. Eq. (<ref>)] also allow us to examine cycle “bunching”, specifically how the occurrence of one useful cycle influences the probability of observing another immediately after.
These insights have the potential to significantly deepen our understanding and interpretation of quantum stochastic processes.


A.S.H. expresses thanks to Joey Smiga, Felix Binder, and Guilherme Fiusa for fruitful discussions.







Supplemental Material for “Time-Resolved Stochastic Dynamics of Quantum Thermal Machines"
















Sections S1 to S5 prove various results mentioned in the main text, and the last section S6 pertains to the technical details of three-level maser example.



§ S1. PROOF OF THE SINGLE-EXCITATION CONSTRAINTS ON A MASTER EQUATION

In this section, we prove the necessary and sufficient conditions given in Eq. (<ref>) on the master equation from  Eq. (<ref>) to achieve quantum trajectories that have alternating injections and extractions (I_∙E_∙I_∙E_∙…) of excitations in a system.
Specifically, we comment on the structure of Hamiltonian, jump operators, and work reservoirs.
In what follows, without loss of generality, we shall drop the index referring to the bath α for jump operators L_α j as it is not relevant for this proof in particular. We first assume that there are no work reservoirs. We also assume that the Hamiltonian is time independent, even though the proof is unaltered otherwise.

We begin by imposing that any two successive extractions out of the system cannot occur.
For any k, q and for all times t_1, t_2, we demand

    𝒥_E_k e^ℒ_0 t_2𝒥_E_q e^ℒ_0 t_1ρ = 0,

and likewise for two consecutive injections. 
Then, the action of a no-jump operator ℒ_0 is equivalent to an effective non-Hermitian operator, ℒ_0 ρ = -i [H_e ρ - ρ H_e^†], where H_e = H - i∑_jL_j^†L_j^†/2.
Thus, we can express e^ℒ_0tρ = e^-iH_e tρ e^iH_e^† t.
Expanding the extraction superoperators, we obtain

    γ_kγ_q L_k e^-i H_e t_2 L_q e^-i H_e t_1ρ e^i H_e^† t_1 L_q^† e^i H_e^† t_2 L_k^† = 0,

for γ_k, γ_q > 0. 
Since we want this to be true for all times t_1 and t_2, Taylor expanding e^-i H_e t = ∑_n=0^∞(-i H_e t)^n/n! then allows us to conclude that this will be possible if and only if all powers of H_e satisfy the condition,

    L_k H_e^n L_q = 0,      n = 1,2,…

as well as 

    L_k L_q = 0   ∀  k, q.




 §.§ Constraints on each jump operator

Setting k = q in Eq. (<ref>) renders L_k^2 = 0.
Let us temporarily ignore the index for the jump operator.
The condition L^2=0 means the jump operators must be nilpotent matrices of order two.
From Ref. <cit.> (p.no. 157) one can show that for any such matrix, it is always possible to find a unitary matrix V such that L = V Λ V^†, where

    Λ = σ_1 [ 0 1; 0 0 ]⊕⋯⊕σ_r [ 0 1; 0 0 ]⊕ 0_d - 2r,

with σ_1 ≥⋯≥σ_r > 0 being the positive singular values of L and ⊕ denoting direct sum. 
Here, d= dim(L) and r =  rank(L).
It follows from the rank-nullity theorem <cit.> that r ≤⌊d2⌋.
Thus, we can choose d-r orthonormal vectors {|μ_i⟩}_i=1^d-r 
satisfying Λ |μ_i⟩ = 0, as well as r vectors 
{|ν_j⟩}_j=1^r satisfying Λ |ν_j⟩ = σ_j |μ_i⟩. 

We now introduce a rotated basis defined by |ϵ_i⟩ = V |μ_i⟩ and |ι_j⟩ = V |ν_j⟩, so that the jump operator L acts in the following way:

    L|ϵ_i⟩   = 0, 
    
    L|ι_j⟩   = ∑_i=1^r c_i |ϵ_i⟩,    c_i ∈ℝ.

From this, we can form orthogonal projectors, 𝒫_E =  span{|ϵ⟩}
and 𝒫_I =  span{|ι⟩}
that obey 𝒫_E + 𝒫_I = 1.
This reformulation allows us to express the operators as L = 𝒫_E L 𝒫_I.
Thus, we conclude that there is a basis of the Hilbert space in which each jump operator is written as a strictly upper-triangular matrix.

To further clarify the procedure, 
let us consider an explicit example for d = 5 with r = 2 case. According to our proof, such a jump operator L must be unitarily similar to

    Λ = 
    [   0 σ_1   0   0   0;   0   0   0   0   0;   0   0   0 σ_2   0;   0   0   0   0   0;   0   0   0   0   0 ],

where σ_1 ≥σ_2 > 0 are positive singular values of L. As one may verify, we can find the orthonormal set of vectors and the action of Λ on them as,

    .
    |μ_1⟩   = (1, 0, 0, 0, 0)^T
     
    |μ_2⟩   = (0, 0, 1, 0, 0)^T
    
    |μ_3⟩   = (0, 0, 0, 0, 1)^T}Λ |μ_i ⟩   = 0  for i = 1, 2, 3,
    .
    |ν_1⟩   = (0, 1, 0, 0, 0)^T
     
    |ν_2⟩   = (0, 0, 0, 1, 0)^T}Λ |ν_i ⟩   = σ_i |μ_i⟩ for i = 1, 2.

Now, the exact forms of unitary matrices V are dependent on L.



 §.§ Joint injection/extraction subspaces for the entire set of jump operators


We now return to the full set of jump operators L_k.
As just shown, to each one we can associate two orthonormal projectors 𝒫_E_k and 𝒫_I_k such that L_k = 𝒫_E_k L_k 𝒫_I_k, where each pair satisfies 𝒫_E_k + 𝒫_I_k = 1.
Next, we show that all 𝒫_E_k must coincide. 
This follows from the additional condition supplied by Eq. (<ref>).

Since for any pair L_k L_q = (𝒫_E_k L_k 𝒫_I_k)(𝒫_E_q L_q 𝒫_I_q) = 0, this must imply that 𝒫_I_k𝒫_E_q = (1 - 𝒫_E_k)𝒫_E_q = 0.
Thus 𝒫_E_q = 𝒫_E_k𝒫_E_q, allowing us to conclude that 𝒫_E_q = 𝒫_E_k. 
This also implies that 𝒫_I_q = 𝒫_I_k leading to only two unique projectors  𝒫_E and 𝒫_I which must be the same for all jump operators associated to the hot and cold baths.



 §.§ Hamiltonian and work reservoir operators


From Eq. (<ref>), setting n = 1  we have that L_k H(t) L_q = 0. This implies (𝒫_E L_k 𝒫_I) H(t) (𝒫_E L_q 𝒫_I) = 0 which reduces to 𝒫_I H(t) 𝒫_E = 0 = 𝒫_E H(t) 𝒫_I. Since the work reservoir jump operators also do not induce any transitions between the subspaces  and , it follows that 𝒫_I K_n 𝒫_E = 0 = 𝒫_E K_n 𝒫_I.

The converse holds true trivially; i.e., given the above forms for the jump operators and the Hamiltonian, it is straightforward that conditions listed in Eq. (<ref>) are satisfied.



 §.§ Other schematic examples of thermal machines satisfying our condition

A key insight to take away is that a Rabi drive cannot connect post-injection and post-extraction subspaces for the single-excitation assumption to hold.
To clarify the restrictions imposed by the assumption from a physical point of view, we show in Fig. <ref> a schematic of a four-level system for which the restriction is applicable and another example for which it is not. 






§ S2. CYCLE STATISTICS IN THE LONG-TIME LIMIT

For a given initial state ρ, the probability of observing a trajectory of the sort given in Eq. (<ref>) takes the form <cit.>,

    𝒫(τ_1, k_1, …, τ_n, k_n) = 𝒥_k_ne^ℒ_0τ_n⋯𝒥_k_1e^ℒ_0τ_1ρ,

for all k_i ∈{I_h, E_h, I_c, E_c} representing the monitored channels and τ_i ≥ 0 referring to the time interval between the jumps.

For the case of single-excitations, we shall replace k_i's with alternating injections and extractions.
As in the main text, we adopt the convention that the cycle always begins with an injection. 
For each pair (τ_1,τ_2), (τ_3,τ_4), … we change variables to e.g. (τ_1,τ_2)→ (t_1,s_1-t_1), and then integrate over s_i. 
In this way, we bundle each pair I_∙ E_∙ together, thus making a cycle. 
Carrying out this operation in Eq. (<ref>) yields

    p_X_1,…,X_n^(τ_1,…,τ_n) = 𝒪_X_n,τ_n…𝒪_X_1,τ_1ρ,

with the cycle superoperators 𝒪_X,τ as defined in Eq. (<ref>).
This result differs from Eq. (<ref>) of the main text only in the choice of initial state, which is left arbitrary here. 
Next, we turn to the justification for beginning in a specific initial state π_E.

We are interested in the distribution (<ref>) in the long-time limit of operations of the machine. 
Specifically, we consider the machine to be in operation for an extended period before we begin counting cycles.
We therefore need to choose the initial state ρ which makes Eq. (<ref>) stationary.
Interestingly, it is not the steady-state of the master equation that possesses this property, but rather the so-called jump steady-state (JSS) <cit.>.
The steady-state captures how long the system stays in a given corner of Hilbert space, while the JSS captures how often it visits that region, an intuition borrowed from classical stochastic processes <cit.>.

Given the unique steady-state = 0, we define the JSS via 

    π = /,

where 

    = ∑_α∈{h, c} (𝒥_E_α + 𝒥_I_α),

is the sum of all jump operators of hot and cold baths, and 

    = ,

is the dynamical activity of the hot and cold baths together – i.e., the average number of jumps per unit time in the steady-state.

Because of the single-excitation hypothesis, the JSS in our case has to be block diagonal in the post-injection and post-extraction subspaces 

    π = p_E^π_E^  + p_I^π_I^.

To see this, we can expand Eq. (<ref>) to

    π   = 1/∑_α∈{h, c} (𝒥_E_α + 𝒥_I_α) 
       = 1/( ∑_α, jγ_α j^- L_α j^ L_α j^†)
    +  1/( ∑_α, jγ_α j^+ L_α j^† L_α j^).

Now, we define the (properly normalized) post-extracted and post-injected parts of the JSS, along with their weights as follows:

    π_E    = ∑_α, jγ_α j^- L_α j^ L_α j^†/∑_α, jγ_α j^-L_α j^† L_α j^,
         p_E^ = 1/(∑_α, jγ_α j^-L_α j^† L_α j^) ≡𝒦_E/, 
    π_I    = ∑_α, jγ_α j^+ L_α j^† L_α j^/∑_α, jγ_α j^+L_α j^ L_α j^†,
         p_I^ = 1/(∑_α, jγ_α j^+L_α j^ L_α j^†) ≡𝒦_I/,

leading to the desired form.

We can think of 𝒦_E and 𝒦_I as the dynamical activities of the  and  subspaces. 
As proved in Sec. S1, in case of single-excitation machines, the states are partitioned into mutually exclusive subspaces. 
Consequently, the state has an equal probability of visiting either subspace, leading to,

    p_E^ = p_I^ = 1/2,    𝒦_E = 𝒦_I = /2,    π = π_E + π_I/2.


We shall now prove that the distribution (<ref>) is stationary when the initial state ρ is the post-extraction JSS π_E. 
The key point is that since injections and extractions alternate, the state just before an injection must reside in the post-extraction subspace.



 §.§ Proof of stationarity


To prove stationarity it suffices to consider a sequence of two cycles of types X_1 and X_2, each with durations τ_1 and τ_2, respectively. The corresponding trajectory probability (<ref>) for the initial state as π_E is 

    p_X_1,X_2^(τ_1, τ_2) = 
    ∫_0^τ_2 dt_2
    ∫_0^τ_1 dt_1
    𝒥_E_X_2 e^ℒ_0(τ_2-t_2)𝒥_I_X_2 e^ℒ_0 t_2𝒥_E_X_1 e^ℒ_0(τ_1-t_1)𝒥_I_X_1 e^ℒ_0 t_1π_E .

Marginalizing over all quantities of the first cycle amounts to the following substitution:

    ∑_X_1∫_0^∞ dτ∫_0^τ dt 𝒥_E_X_1 e^ℒ_0(τ_1-t_1)𝒥_I_X_1 e^ℒ_0 t_1 = ℳ_E ℳ_I,

where ℳ_k = -∑_α∈{h, c}𝒥_k_αℒ_0^-1 for k ∈{I, E} such that ℳ_I and ℳ_E lie in  and , respectively. Combining these two operators gives

    ℳ = ℳ_E + ℳ_I = - ,

where the final equality follows from Eq. (<ref>). Notably, this operator has a special property that only the jump steady-state satisfies <cit.>: 

    ℳπ = -𝒥ℒ_0^-1𝒥/𝒥 = -𝒥ℒ_0^-1(ℒ-ℒ_0)/𝒥 = π.

Since ℳ_I π_I = 0 = ℳ_E π_E, it follows that


    ℳ_I π_E    = π_I,
    ℳ_E π_I    = π_E.


Thus, marginalizing Eq. (<ref>) over the first cycle leaves us with 

    ℳ_E ℳ_I π_E = ℳ_E π_I = π_E,
 
i.e., cycling back to where we started.

As a result, we obtain the following identities which in turn prove stationarity of Eq. (<ref>):

    ∑_X ∫_0^∞ dτ 𝒪_X,τπ_E    = π_E
        
    ∑_X ∫_0^∞ dτ 𝒪_X,τ∙   = ∙.

Thus, marginalizing Eq. (<ref>) from either side yields the same distribution, proving stationarity.
The above properties also ensure the proper normalization in Eq. (<ref>),

    ∑_X ∫_0^∞ dτ p_X(τ) = ∑_X p_X = 1.




§ S3. MOMENTS OF WAITING TIME DURATION OF A CYCLE

The n-th moment of the waiting time, given a cycle of type X is a generalization of  Eq. (<ref>), and reads

    𝔼(τ^n | X) = 1/p_X^∫_0^∞ dτ τ^n p_X^(τ).

Substituting for the probability from Eq. (<ref>) and changing the integration variable to u = τ - t results in the following:

    𝔼(τ^n | X)    = 1/p_X^∫_0^∞ dτ τ^n ∫_0^τ dt  e^ℒ_0(τ - t) e^ℒ_0 tπ_E 
       = 1/p_X^∫_0^∞ dτ τ^n ∫_0^∞ du  e^ℒ_0u e^ℒ_0(τ - u)π_E Θ(τ - u) 
       = 1/p_X^∫_0^∞ du  e^ℒ_0u(∫_0^∞ dτ τ^n e^ℒ_0τΘ(τ - u)) e^-ℒ_0uπ_E ,

where we introduced Heaviside step function Θ(τ - u) in the second equality.
Let us first solve the integral in the parentheses,

    I_n    ≡∫_0^∞ dτ τ^n e^ℒ_0τΘ(τ - u)
       = τ^n (-e^ℒ_0τ)Θ(τ - u)|_τ = 0^∞ + [n ∫_0^∞dτ τ^n-1e^ℒ_0τΘ(τ-u) + ∫_0^∞dτ τ^n e^ℒ_0 τδ(τ - u)]
       = n  I_n - 1 + u^n  e^ℒ_0 u

where the first term in the second equality vanishes at both the limits τ = 0 and τ = ∞ (the latter is because of the assumption that there are no dark subspaces). Solving this recursion relation yields,

    I_n = [∑_i=0^n-1n!/(n-i)!u^n-iℒ_0^-(i+1) - n! ℒ_0^-(n+1)]e^ℒ_0u,

Plugging this back into Eq. (<ref>), we find

    E(τ^n | X) = 1/p_X^(∑_i=0^n-1n!/(n-i)!ℒ_0^-(n-i+1)ℒ_0^-(i+1)π_E - n! ℒ_0^-1ℒ_0^-(n+1)π_E).

For example, if n = 1, we obtain the expectation value for the duration of a given cycle X,

    E(τ| X) = - 1/p_X^(ℒ_0^-2π_E  + ℒ_0^-2π_E ).

On the other hand, the unconditional waiting time is 

    E(τ) = ∑_X=1^4 p_X E(τ| X) = - ∑_X = 1^4 (ℒ_0^-2π_E  + ℒ_0^-2π_E).


We can relate E(τ) in Eq. (<ref>) to the dynamical activity defined in Eq. (<ref>) through the notion of residence times.
We refer P_ no(τ|ρ) = e^ℒ_0 τρ to denote the survival probability that the system remains in a given state ρ for a time t.
Since -dP_ no/dτ is then the waiting-time distribution, we can determine the average time the system will remain in state ρ before jumping as 

    E(τ|ρ) = -∫_0^∞ dτ τ dP_ no(τ|ρ)/dτ = -ℒ_0^-1ρ.

A consequence of assuming at most a single-excitation in the system is that the no-jump superoperator ℒ_0 cannot produce transitions between subspaces. Accordingly,


    -ℒ_0^-1π_I    = E(τ|π_I) π_I,
    
    
    -ℒ_0^-1π_E    = E(τ|π_E) π_E.


Specifically, starting in the jump steady-state yields the well-known result that the dynamical activity is the inverse of the average waiting time in the jump steady-state,

    E(τ|π) = - / = -  ( - ℒ_0) / = 1/,

where we have used =0 and 
(·) = 0.
This relation can be extended to the unconditional expectations as well:

    E(τ)    = ∑_X ∫_0^∞ dτ τ∫_0^∞ dt  e^ℒ_0 (τ-t) e^ℒ_0 tπ_E 
       = ∑_X ∫_0^∞ du ∫_0^∞ dt (u+t)  e^ℒ_0u e^ℒ_0 tπ_E 
       = -ℳ_E ℒ_0^-1ℳ_I π_E + ℳ_E ℳ_I ℒ_0^-1π_E 
       = E(τ|π_I)π_I  + E(τ|π_E)π_E 
       = 2E(τ|π)  
       = 2/,

where the second equality comes from a substitution u = τ - t and the last equality follows from Eq. (<ref>).



§ S4. CONNECTION WITH THE AVERAGE EXCITATION CURRENT


Eq. (<ref>) of the main text provides a fundamental connection between the statistics of cycle and the average excitation current, a much more standard measure to characterize the operation of an engine. 
In order to prove that relation  we shall first recast the probabilities from Eq. (<ref>) as,

    p_X^   = π_E
       = 2 π
       = 2/,

where we have used Eq. (<ref>) in going to the second equality, and Eq. (<ref>) in the third.
Since E(τ) = 2/ from Eq. (<ref>), we have 

    p_X^/E(τ)   = 
       =  ( - ℒ_0) 
       = -,

where made use of (·) = 0 in the last step. Thus, we can express

    p_1 - p_2/E(τ)   =  - 
       = ( - ℒ_0 -  -  - )  -  ( - ℒ_0 -  -  - ) 
       =  - 
       ≡,

which proves Eq. (<ref>). In simplifying the above, we repeatedly made use of (·) = 0 and ℒ = ℒ_0 +  +  +  +.




§ S5. PROBABILITIES FOR OBSERVING INTERMEDIATE CYCLES BETWEEN TWO OF THE SAME CYCLES


Eqs. (<ref>) to (<ref>) in the main text refer to the statistics of individual cycles. 
However, armed with the full trajectory distribution (<ref>), we can take this a step further and ask questions pertaining to the occurrence of multiple cycles in specific orders, and how correlations between  outcomes might affect that. 
In particular, here we prove the following two results from the main text: 

    
  * Eq. (<ref>): the probability that there are n idle cycles between two useful ones;
    
  * Eq. (<ref>): the probability that it takes a time t between the conclusion of two useful cycles. 

Recall that the useful cycles are X=1,2 and the idle cycles are X=3,4. 
Starting with the cycle superoperators in Eq. (<ref>), we define

    𝒪_ u = 𝒪_1 + 𝒪_2, 
            𝒪_ id = 𝒪_3 + 𝒪_4,

and similarly for 𝒪_ u,τ and 𝒪_ id,τ.


The proof of Eq. (<ref>) is straightforward.
We are seeking the probability of observing n idle cycles until a next useful one given that we have already observed a useful cycle. Post the first useful cycle, the state is updated to 𝒪_ uπ_E/𝒪_ uπ_E. We then readily obtain,

    ℙ_u(n) = 𝒪_ u𝒪_ id^n𝒪_ uπ_E/𝒪_ uπ_E.


To prove Eq. (<ref>) we proceed as follows. 
Suppose we are interested in observing a specific cycle X = x by time t such that we implicitly assume only other cycles X x occur until then.
This cumulative probability ℙ_X = x, t is then an infinite sum where each term defaults to observing X in the last trial while assigning the probability for observing X until the last but one, 

    ℙ_X = x(t) =    p_X = x(t) 
       + ∑_X_1∫_0^t dt_1 p_X_1, X_2 = x(t_1, t-t_1) 
       + ∑_X_1, X_2∫_0^tdt_2∫_0^t_2dt_1 p_X_1, X_2, X_3 = x(t_1, t_2 - t_1, t - t_2) 
       + ⋯,

where we have made use of p_X(t) from Eq. (<ref>). 
Written in terms of the cycle operators, this becomes

    ℙ_X = x(t) =    𝒪_X = x, tπ_E 
       +∑_X_1∫_0^t dt_1 𝒪_X_2 = x, t - t_1𝒪_X_1, t_1π_E 
       +∑_X_1, X_2∫_0^t dt_2∫_0^t_2 dt_1 𝒪_X_3 = x, t-t_2𝒪_X_2, t_2 - t_1𝒪_X_1, t_1π_E 
       +⋯.

We can now recast it in terms of Eq. (<ref>) after replacing for the duration of cycles, τ_i = t_i - t_i-1, for i = 1, 2, … with τ_1 = t_1 and τ_n = t - t_n-1,

    ℙ_X = x(t)    = ∑_n=0^∞∑_X_1, ⋯, X_n-1∫_0^∞ dτ_n ⋯ dτ_1 𝒪_X_n = x, τ_n^𝒪_X_n-1, τ_n-1⋯𝒪_X_1, τ_1π_E  δ(t - ∑_n=1^∞τ_i).

Here we shall make use of the Fourier expansion of delta function, δ(t) = ∫_-∞^∞ dz e^izt/2π, and also introduce the Fourier transformed version of the cycle operator,

    𝒪_X,z = ∫_0^∞ dt 𝒪_X, te^-izt,

so that we can simplify the above to,

    ℙ_X = x(t)
       = ∑_n=0^∞ ∑_X_1, …, X_n-1 ∫_-∞^∞dz/2π𝒪_X_n = x, z^𝒪_X_n - 1, z⋯𝒪_X_1, zπ_E  e^izt.

Since all X_k for k = 1, ⋯, n-1 are identical so that X_k ≡X, we can further reduce the above equation to,

    ℙ_X = x(t)   = ∑_X∫_-∞^∞dz/2π𝒪_X_n = x, z(∑_n = 0^∞𝒪_X, z^n - 1)π_E e^ zt, 
       = ∑_X∫_-∞^∞dz/2π𝒪_X_n = x, z(1 - 𝒪_X, z)^-1π_E e^ zt,

where we have made use of the summation of infinite geometric series: ∑_n=0^∞ x^n = 1/(1-x) for x < 1.

Given that we begin with a useful cycle, we can update the initial state to π_E →𝒪_ uπ_E/𝒪_ uπ_E.
Summing over the idle cycles X = {3, 4} in Eq. (<ref>), we obtain the result from Eq. (<ref>),

    ℙ_u(t) = 1/𝒪_ uπ_E ∑_X = {3, 4} ∫_-∞^∞dz/2π 𝒪_X, z(1 - 𝒪_X, z)^-1𝒪_ uπ_E e^izt.




§ S6. EXPRESSIONS FOR STATISTICAL QUANTITIES IN THREE-LEVEL MASER


In this section, we provide the equations specific to three-level masers for the statistical measures introduced in the main text. We begin with evaluating the essential operators involved in obtaining those measures.
The computational basis is spanned by |0⟩, |1⟩, |2⟩.
We use the notation σ_ij = |i⟩⟨ j| throughout.



 §.§ Time-independent Hamiltonian

The maser Hamiltonian is

    H(t) = (ω_h-ω_c)σ_11 + ω_h σ_22 + ϵ(e^iω_d tσ_01 + e^-iω_d tσ_10).

Choosing an appropriate rotating frame of reference X = ω_d σ_11 + ω_h σ_22 and rotate according to H_ rot = U H U^† where U = e^ X t as in Ref. <cit.>, we obtain a time-independent Hamiltonian

    H = H_ rot - X = Δσ_11 + ϵ(σ_01 + σ_10),

where Δ = (ω_h - ω_c) - ω_d is the detuning parameter. We use this Hamiltonian for all further evaluations.



 §.§ Jump operators and steady-state

According to Eq. (<ref>) the jump operators for the three-level maser appear as,

    2(ρ)    = γ_h(n̅_h+1)σ_02ρσ_02^†,           (ρ) = γ_h n̅_h σ_20ρσ_20^†,
    (ρ)    = γ_c(n̅_c+1) σ_12ρσ_12^†,           (ρ) = γ_c n̅_cσ_21ρσ_21^†.

In this case, the jump steady-state along with its post-extracted and post-injected parts are,


    π_E    = ( γ_c(n̅_c + 1)γ_h(n̅_h + 1) + 1 )^-1 |0⟩⟨ 0| + ( γ_h(n̅_h + 1)γ_c(n̅_c + 1) + 1 )^-1 |1⟩⟨ 1|, 
    π_I    = |2⟩⟨ 2|,
    π   = π_E + π_I/2,


Because there are no work reservoirs, the action of the no-jump superoperator can be described solely in terms of the non-Hermitian Hamiltonian H_e = H - i/2 ∑_j L_j^† L_j^. 
For the maser, this reads

    H_e = 1/2[                      n̅_h γ_h                             ϵ                             0;                             ϵ                 n̅_c γ_c + 2Δ                             0;                             0                             0 γ_c (1+n̅_c) + γ_h (1 + n̅_h) ].




 §.§ Time-resolved probability of completing a cycle

Plugging the above operator forms into Eq. (<ref>) describing time-resolved cycle probabilities, we obtain

    p_1^(τ)    = ∫_0^τ dt 𝒥_E_c e^ℒ_0 (τ-t) 𝒥_I_h e^ℒ_0 tπ_E 
    
       = γ_h^2 n̅_h(n̅_h + 1) γ_c (n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|0⟩|^2 
           + γ_h n̅_h γ_c^2 (n̅_c + 1)^2/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|1⟩|^2.


Likewise, we arrive at similar expressions for the other cycles:

    p_2^(τ)     = γ_h^2 (n̅_h + 1)^2 γ_c n̅_c/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|0⟩|^2 
           + γ_h (n̅_h + 1) γ_c^2 n̅_c(n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|1⟩|^2, 
    
     p_3^(τ)    = γ_h^2  γ_c n̅_h(n̅_h + 1)^2/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|0⟩|^2 
           + γ_h n̅_h (n̅_h + 1) γ_c^2 (n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|1⟩|^2,
    
     p_4^(τ)     = γ_h^2 (n̅_h + 1) γ_c n̅_c (n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|0⟩|^2 
           + γ_h γ_c^2 n̅_c(n̅_c + 1)^2/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|1⟩|^2.

These are plotted in Fig. <ref> (a) of the main text.



 §.§ Probability of observing a cycle of each kind irrespective of waiting times


Marginalizing the previous results over τ yield the probabilities p_X of observing a cycle of type X [Eq. (<ref>)]. 
Let us introduce the simplifying variables:


    Γ   = 12(n̅_h γ_h + n̅_c γ_c),
        
    γ   = 12(γ_h + γ_c),
        
    χ   = n̅_h n̅_c γ_h γ_c,
        
    n̅_ ave   = Γγ = n̅_h γ_h + n̅_c γ_cγ_h + γ_c,
        
    κ   = χ2Γ = n̅_h n̅_c γ_h γ_cn̅_h γ_h + n̅_c γ_c,
        
    γ_ cl   = 2ϵ^2 ΓΔ^2 + Γ^2.


The last quantity is the classical transition rate between |0⟩ and |1⟩ derived by assuming Lorentzian broadening in Fermi's golden rule, as explained in Ref. <cit.>.
With these definitions, we find

    p_1^   = 𝒥_E_cℒ_0^-1𝒥_I_hℒ_0^-1π_E , 
       =
    κ/2Γ(1 + 1/n̅_c)/(1+1/n̅_ ave)[(1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_h)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1],
    
    
    p_2^   = 𝒥_E_hℒ_0^-1𝒥_I_cℒ_0^-1π_E 
       = 
    κ/2Γ(1 + 1/n̅_h)/(1+1/n̅_ ave)[(1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_c)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1],
    
    
    p_3^   = 𝒥_E_hℒ_0^-1𝒥_I_hℒ_0^-1π_E 
       =
    κ/2Γn̅_h γ_h/n̅_c γ_c(1 + 1/n̅_h)/(1+1/n̅_ ave)[
    (1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_h)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1],
    
    
    p_4^   = 𝒥_E_cℒ_0^-1𝒥_I_cℒ_0^-1π_E 
       = 
    κ/2Γn̅_c γ_c/n̅_h γ_h(1 + 1/n̅_c)/(1+1/n̅_ ave)[
    (1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_c)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1].

It is straightforward to verify that p_1 + p_2 + p_3 + p_4 = 1.
Remarkably,

    p_1 - p_2    = 1/2Γ(1/κ + 1/γ_ cl)^-1(1 + 1/n̅_ ave)^-1(1/n̅_c - 1/n̅_h), 
    p_3/p_1 = p_2/p_4   = (n̅_h + 1) γ_h/(n̅_c + 1) γ_c.

As expected from Eq. (<ref>), p_1 - p_2 relates to the particle current. In this model, this is indeed evidenced by its dependence on the difference in Bose occupations, n̅_h - n̅_c.
These expressions are plotted in Fig. <ref> (b) of the main text, where we also set γ_h = γ_c leading to the following conclusions:

    
  * In the refrigeration regime (n̅_h < n̅_c), we have p_3 < p_1 < p_2 < p_4. 
    
  * In the engine regime (n̅_h > n̅_c), we have  p_4 < p_2 < p_1 < p_3.

Thus, the idle cycle probabilities bound those of useful cycles across all parameter ranges.



 §.§ Useful cycle probabilities

The probability of observing a useful cycle is  p_u := p_1 + p_2. 
Because the post-injection subspace in this case is one-dimensional, it turns out that the cycles are independent of each other. 
As a consequence, the probability of observing n idle cycles between two useful ones [Eq. (<ref>)] will be given simply by the geometric distribution 

    ℙ_ u(n) = p_u(1-p_u)^n.




Figure <ref> (a) shows the mean and variance: ⟨ n ⟩ and Var(n) of this distribution.
We find the optimal temperature ratio T_h/T_c = ω_h/ω_c minimizes the mean number of idle cycles, thereby enhancing resource efficiency.
However, a steeper temperature gradient between the baths does not necessarily result in efficient heat transport, as it also promotes the participation of idle cycles in system dynamics.
To quantify this, we turn to the time between two useful cycles given in Eq. (<ref>) and computed numerically in this case.
The results plotted in Fig. <ref> (b) show that a steeper temperature gradient facilitates faster heat transport.


 Paper Title: Perfect Wave Transfer in Continuous Quantum Systems.







per.moosavi@fysik.su.se

Department of Physics, Stockholm University, 10691 Stockholm, Sweden



christandl@math.ku.dk





gmgraf@phys.ethz.ch

Institute for Theoretical Physics, ETH Zurich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland



spyros.sotiriadis@physics.uoc.gr









§ ABSTRACT 
We study the perfect transfer of information in 1+1D continuous quantum systems. This includes effective descriptions of inhomogeneous spin chains, for which the notion of perfect state transfer in quantum information was introduced, and here phrased in terms of waves. We show that reflection symmetry is necessary for perfect wave transfer (PWT) in any inhomogeneous conformal field theory, and even sufficient when restricted to one-particle excitations. To determine if or when it is sufficient more generally, we first break conformal invariance and study a broad class of 1+1D bosonic theories. We show that the question can then be posed as an inverse Sturm-Liouville problem that determines when the bosonic theory exhibits PWT. We demonstrate how to uniquely solve this problem, which also shows that reflection symmetry is sufficient for the special case with conformal invariance. Using bosonization, our continuum results extend these notions to interacting quantum field theories.




Perfect Wave Transfer in Continuous Quantum Systems

    Spyros Sotiriadis
    
August 1, 2024

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







Introduction.—
phantomsection
tocsectionIntroduction


Non-photonic quantum communication enables the efficient transmission of information from one quantum register to another <cit.>.
This is desirable for quantum networks and computation <cit.>, not least since it
allows for the perfect transfer of states even without dynamical control <cit.>.
First shown theoretically for communication channels in the form of a spin chain with spatially varying hopping, such perfect state transfer (PST) was extended to spins on graphs and other networks <cit.> and demonstrated experimentally <cit.>.
However, the notion has remained squarely in the discrete realm.

In this paper, we extend PST to continuous systems in one space and one time dimension (1+1D).
In so doing, a rephrasing in terms of waves becomes natural, motivating the term perfect wave transfer (PWT).
As for states in PST, a system on the interval [-L/2, L/2] of length L with suitable boundary conditions (BCs) exhibits PWT if the propagation of any initial wave, after some time T, amounts to its reflection; see Fig. <ref>.
Concretely: The system enjoys PWT if

    ⟨(x, 0) ⟩ = ⟨(-x, T) ⟩,

for all time-evolved observables (x, t) and any expectation value ⟨·⟩. 
We focus on realizations that do not need dynamical control, but the same problem is also interesting for driven systems <cit.>.

Related recent advances in the continuum include non-equilibrium studies of inhomogeneous versions of conformal field theory (CFT).
These are conformally-invariant quantum field theories deformed so that the usual constant propagation velocity becomes a profile v(x) <cit.>.
Among their applications, such inhomogeneous CFTs effectively describe gapless quantum many-body systems with mesoscopic inhomogeneities, such as spin chains with spatially varying hopping, trapped ultracold atoms, and arrays of quantum circuits; cf. <cit.>.
Special cases include the square-root profile v(x) = v √(1 - (2x/L)^2) for some constant v > 0 <cit.>—the continuum counterpart of the inhomogeneous spin chain in <cit.> that first exhibited PST—and profiles produced by Möbius transformations <cit.>.
Note that for PWT, as for other transport properties of finite systems, the BCs <cit.> are crucial.
Furthermore, inhomogeneous CFT can be viewed as a theory in curved spacetime.
As such, besides applications to quantum communication in solid-state devices <cit.>, our work also relates to relativistic quantum information <cit.>.
Since the same theories are also natural to study on the half or full infinite line, our results may likewise be relevant for long-distance quantum communication <cit.>.






Engineering PST/PWT is at  heart a spectral problem, amounting to whether the propagator coincides with the parity operator at certain times.
We address this engineering question for quantum field theories describing a continuous quantum communication channel.
Remarkably, for bosonic theories <cit.>, we arrive at an inverse problem for Sturm-Liouville operators, which we solve and whose solution determines if the channel enjoys PWT.
Using bosonization <cit.>, our results extend the notion of PST/PWT to a large class of systems, including interacting fermionic quantum field theories.


Inhomogeneous CFT.—
phantomsection
tocsectionInhomogeneous CFT


To begin with, we study the continuum limit of the inhomogeneous spin chains in <cit.>.
In the gapless regime, the continuous counterpart is an inhomogeneous CFT with a velocity profile v(x) > 0, x ∈ [-L/2, L/2] <cit.>.
The simplest example is that of inhomogeneous free fermions, given by the Hamiltonian (up to an overall constant and setting ħ = 1)

    H_iFF≡∫_-L/2^L/2 x v(x)/2[ ψ^†_+ (-) ∂_x ψ_+
      + ψ^†_-∂_x ψ_-
      + ]

with fields ψ_±(x) of right(+)/left(-) fermions satisfying {ψ_±(x), ψ^†_±(x') } = δ(x-x') and {ψ_±(x), ψ^(†)_∓(x') } = 0.
Here, ⋯ denotes Wick (normal) ordering with respect to the ground state |Ω⟩ (which for bilinears in the fields amounts to an additive renormalization).
For simplicity, we focus on the theory (<ref>), while general CFTs are discussed in the Supplementary Material (SM) <cit.>.

We assume open (Neumann) BCs at x = ± L/2.
These can be incorporated by standard unfolding methods; cf. <cit.>:
Couple right and left movers by identifying ψ_-(x) = ψ_+(L-x), giving a chiral theory of only right-moving fermions on [-L/2, 3L/2] with periodic BCs and velocity v(x) for x ∈ [-L/2, L/2] and v(L-x) for x ∈ [L/2, 3L/2].
This implies zero particle flow at x = ± L/2 since the two fermion densities ρ_±(x) = ψ^†_±(x)ψ_±(x) are equal at each boundary and the current is ∝ρ_+ - ρ_-.

PST was probed in <cit.> using a time-dependent overlap between one-particle states that are reflections of each other.
Its continuum counterpart is to study states created by fields

    ψ(x)
    ≡^ xψ_+(x) + ^- xψ_-(x),

where  denotes the Fermi momentum.
Eq. (<ref>) corresponds to the low-energy linearization of the spin-chain dispersion relation around its Fermi points ±, followed by recentering the resulting fields so they describe right/left-moving excitations around zero momentum.
As a preparatory subset of possible states, consider

    |Ψ_j⟩≡∫_-L/2^L/2 x ξ_j(x) ψ^†(x) |Ω⟩     (j = 1,2)

for smooth enough functions ξ_j(x), interpreted as distributions or “wave packets” of fermionic excitations in each state, and study the time-dependent overlap

    F(t)
    ≡⟨Ψ_2| ^- H_iFF t |Ψ_1⟩.

The states (<ref>) form a restricted class as only one-particle excitations are possible and as right/left movers have the same particle distributions; cf. the more general treatment in the SM <cit.> where we lift the latter restriction.

To probe PWT for one-particle excitations as done in <cit.>, we restrict to specular waves, ξ_2(x) = ξ_1(-x), and compare F(t) in Eq. (<ref>) with ⟨Ψ_1|Ψ_1⟩.
We say that:
There is one-particle PWT if there exists T > 0 so that

    |F(T)|
    = |⟨Ψ_1|Ψ_1⟩|

for all initial waves ξ_1(x) in Eq. (<ref>) [If Eq. (<ref>) holds for any state |Ψ_1⟩ in the Fock space, then general PST/PWT (i.e., for any number of excitations) would be enjoyed.].

Eq. (<ref>) amounts to a comparison between two-point correlation functions at equal and non-equal times.
Such can be computed in any inhomogeneous CFT by mapping them to the corresponding homogeneous correlations <cit.>.
The details are given in the SM <cit.> and boil down to comparing spacetime points after conformal transformations to coordinates given by

    y = y(x) ≡∫_0^x s v_0/v(s),
        1/v_0≡1/L∫_-L/2^L/2 s/v(s).

The outcome of the comparison shows that reflection symmetry is essential:


An inhomogeneous CFT exhibits one-particle PWT 

if and only if v(x) is even.


The corresponding times are odd multiples, (2m+1)T for any integer m, of the time T = L/v_0 that excitations take to cross the entire system's length.
For later usage, we note that y ∈ [-L/2, L/2] if v(x) is even.

As an example, consider the square-root profile v(x) = v √(1 - (2x/L)^2), which gave the effective low-energy theory of the spin chain in <cit.>.
This continuum theory exhibits PWT, just like for any even profile v(x), in contrast to <cit.> where that profile was exceptional in this respect.
Notably, the square-root profile is positive everywhere except at x = ± L/2.
Thus, strictly speaking, v(x) should be regularized before using the tools in <cit.>, but since the integral defining 1/v_0 in Eq. (<ref>) converges, it has no consequence for PWT. Oppositely, the limiting case of inhomogeneous CFT given by Möbius transformations so that v(x) decays to zero at the boundaries cannot exhibit PWT since 1/v_0 diverges.


Bosonic theories.—
phantomsection
tocsectionBosonic theories


The above result and observations should not come as a surprise.
After all, if one tries to compare discrete and continuous theories, then an inhomogeneous CFT can only describe low-energy excitations of a given lattice model.
Moreover, one can show <cit.> that any Hamiltonian (with pure point spectrum) must commute with the parity operator for PWT to be possible, i.e., v(x) must be even.
The general structure of the spectrum due to conformal invariance in 1+1D <cit.> opens up the possibility that it might also be sufficient; e.g., the example (<ref>) has a linear spectrum and is one-particle reducible, suggesting that if PWT holds for one-particle excitations then this should extend to many particles.
The remainder of this paper concerns to what extent this is true more generally.
Namely:
What are necessary and sufficient conditions for PWT (<ref>) for any number of excitations?

To answer this question, we first break conformal invariance. 
Examples include 1+1D fermions with non-local interactions <cit.>, while here we focus on bosons, specifically inhomogeneous Tomonaga-Luttinger liquids (TLLs) <cit.>.
This theory is formulated using a compactified bosonic field φ(x) (modulo 2π) with conjugate Π(x) satisfying
[∂_x φ(x), Π(x')] = ∂_x δ(x-x').
The Hamiltonian is (up to an overall constant)

    H_iTLL≡∫_-L/2^L/2 x/2π v(x)
      (
          [πΠ(x)]^2/K(x)
          + K(x)[∂_x φ(x)]^2
        )
       ,

where, besides v(x) > 0, the so-called Luttinger parameter K(x) > 0 also depends on x.
For simplicity, we assume that both functions are smooth.
When K(x) = K is constant, Eq. (<ref>) is an inhomogeneous CFT, and when K = 1, it is the bosonized version of the free fermions (<ref>) <cit.>.
However, K'(x) ≠ 0 locally couples right and left movers non-trivially, thus they do not commute and the theory is not conformal <cit.>.
Again, we impose open BCs, namely zero current at the boundaries:

    (x) |_x = ± L/2 = 0,
      where  (x)
    ≡
    - v(x)K(x)/π∂_x φ(x)

is the total particle current.
Unless stated otherwise, we also assume that K'(x)|_x = ± L/2 = 0 so that right and left movers are only coupled via the BCs (<ref>) at those points.


Sturm-Liouville (SL) theory.—
phantomsection
tocsectionSturm-Liouville (SL) theory


Following <cit.>, we use SL theory to find eigenfunctions, using which the fields can be expanded and the Hamiltonian (<ref>) diagonalized.
(This generalizes <cit.> as v(x), K(x) are unrelated and differs from the approach in <cit.>.)
Indeed, using integration by parts and Eq. (<ref>),

    H_iTLL
    =
    ∫_-L/2^L/2 x/2π w(x)
      (
          [ πΠ(x)/w(x)]^2
          + φ(x) φ(x)
        )

with the SL operator

    ≡
    - 1/w(x)[ ∂_x p(x) ∂_x + q(x) ],
 


    w(x)
    ≡K(x)/v(x),
      
    p(x)
    ≡
    v(x)K(x),
      
    q(x)
    ≡
    0,

where  acts on the space  of smooth [More precisely, twice continuously differentiable.] functions u(x) on [-L/2, L/2] that satisfy v(x)K(x) ∂_x u(x)|_x = ± L/2 = 0, i.e., Neumann BCs when v(x), K(x) > 0.

We want to solve the SL problem

    u = λ u,
      
    u ∈.

A distinction is warranted: Eq. (<ref>) is regular if certain technical conditions are satisfied [Here, these amount to v(x), K(x) being strictly positive and v(x), K(x), ∂_x p(x) [and q(x) if non-zero] continuous on the whole interval [-L/2, L/2].]; otherwise it is irregular.
Its importance stems from properties that hold for regular problems <cit.>:



  * 
The spectrum consists of eigenvalues λ_n that are real and ordered: 0 ≤λ_0 < λ_1 < …. 


  * 
Each λ_n has a unique square-integrable eigenfunction u_n(x) with n isolated zeros in [-L/2, L/2].


  * 
The u_n(x) form a complete orthonormal basis for square-integrable functions on [-L/2, L/2] with weight w(x):
∫_-L/2^L/2 x  w(x) u_n(x) u_n'(x)
= δ_n,n'.


Furthermore, assuming that the integral defining 1/v_0 in Eq. (<ref>) converges:




  * 
The λ_n can at most grow as n^2 for large n, i.e., λ_n = O(n^2).


The last follows by Weyl's law; see the SM <cit.> and, e.g., <cit.>.
Simply put, it gives a semi-classical estimate of the number of eigenvalues below a given large λ > 0 for Laplace-type operators.

Unless otherwise stated, our SL problem will be regular.
If irregular, the above properties are not guaranteed, while in certain cases they survive.
Note also that, since q(x) = 0 here, u_0(x) = const solves Eq. (<ref>) with λ_0 = 0.

The stated properties allow us to expand

    φ(x)
    = ∑_n = 0^∞φ_n u_n(x),
        Π(x)
    = ∑_n = 0^∞Π_n w(x) u_n(x),

whereby we obtain [ φ_n, Π_n'] = δ_n,n' and can diagonalize H_iTLL in Eq. (<ref>).
Indeed, the latter becomes

    H_iTLL
    = 1/2π[ πΠ_0]^2
      + ∑_n = 1^∞
        E_n a^†_n a_n

with eigenenergies

    E_n = √(λ_n),

where we expressed φ_n = √(π/2E_n)( a_n + a^†_n) and Π_n = -√(E_n/2π)( a_n - a^†_n) for n ≥ 1 in bosonic operators a^†_n, a_n with [ a_n, a^†_n'] = δ_n,n'.
Note that φ_0 (and zero modes in general) should be treated separately, but this will have no consequence here.
In the SM <cit.>, we also discuss SL operators (<ref>) with q(x) ≠ 0, corresponding to an inhomogeneous (possibly imaginary) mass M(x) by adding v_0^2 M(x)^2 φ(x)^2 inside the big parentheses in Eq. (<ref>).

The time evolution of any observable, such as φ(x), will contain terms like a^†_n^ E_n t u_n(x) and a_n^- E_n t u_n(x).
Thus, as shown in the SM <cit.>, to exhibit PWT as defined by Eq. (<ref>), all u_n(x) must have definite parity, i.e., be invariant under reflections x ↦ -x up to a sign, which must equal the value of each phase factor
^± E_n T for a common T > 0.
Moreover, the u_n(x) have definite parity if and only if v(x), K(x) [and q(x) if non-zero] are even, i.e., H_iTLL is reflection symmetric.
In conclusion, PWT boils down to requirements on the spectrum:


An inhomogeneous TLL exhibits PWT if and only if v(x) and K(x) are even and all eigenvalues are of the form E_n = π m_n/T for certain integers m_n.


From Properties <ref> and <ref> and since λ_0 = 0 here, these integers m_n satisfy 0 = m_0 < m_1 < … with m_n even (odd) if n is even (odd).
Moreover, from Property <ref>, they scale as m_n ∼ n for large n, i.e., m_n = n + o(n).

Lastly, to probe PWT based on Eq. (<ref>), we note that the results above allow one to compute correlation functions for inhomogeneous TLLs; see the SM <cit.>.
In particular, the ground-state φφ correlation function is

    ⟨Ω| φ(x,t)φ(x',t') |Ω⟩
    = ∑_n = 1^∞π/2 E_n
      u_n(x) u_n(x') ^- E_n(t-t').

Thus, for a spectrum satisfying the PWT requirements,
⟨Ω| φ(x,T)φ(x',0) |Ω⟩ = ⟨Ω| φ(-x,0)φ(x',0) |Ω⟩,
meaning that, for fixed x', the correlations at t = T are reflections of those at t = 0.
In Fig. <ref>, we plot Eq. (<ref>) for K(x) = 1, which, by bosonization, is related to log F(t) in Eq. (<ref>), in that it encodes the same information; see the SM <cit.>.


Inverse Sturm-Liouville problem.—
phantomsection
tocsectionInverse Sturm-Liouville problem


Reversing the logic, the PWT requirements on the spectrum yield an inverse spectral problem for SL operators with Neumann BCs
[and q(x) = 0]:


Given a spectrum E_n = π m_n/T for integers m_n as above, determine the even functions v(x) and K(x).


A related prominent problem is the inverse scattering occurring in the theory of integrable PDEs, whose solution is given by the Gelfand-Levitan-Marchenko equation.
However, that case is formulated on an infinite line, while here we are interested in a finite interval.

Before a general discussion, we consider two examples:

The inverse SL problem has a known solution when the spectrum is exactly linear:
For K(x) = K > 0 constant but v(x) > 0 general, the SL operator is = - v(x) ∂_x v(x) ∂_x = -v_0^2 ∂_y^2 using Eq. (<ref>) with eigenenergies E_n = π n v_0/L and eigenfunctions u_n > 0(x) = √(2 v_0/L)cos( nπ [L + 2y]/2L ) and u_0(x) = √(v_0/L).
This is precisely the conformal case of non-interacting right/left movers, enjoying PWT for any even v(x) > 0.

In the previous example, the SL operator was regular.
Relaxing this condition by taking v(x) = v √(1 - (2x/L)^2) and K(x) = K [ 1 - (2x/L)^2 ]^α for α > -1/2 and constants v, K > 0, we get an irregular problem for which Properties <ref>–<ref> remain valid; the BCs being that a solution is sufficiently well-behaved for Eq. (<ref>) to hold.
The eigenfunctions are given by Gegenbauer polynomials C^(α)_n(2x/L) <cit.> up to normalizations, which have definite parity and satisfy Eq. (<ref>) with λ_n = (π v_0/L)^2 n(n + 2α) for v_0 = 2v/π.
Clearly, only the case α = 0 (Chebyshev polynomials of the first kind) exhibits PWT, while all α≠ 0 do not, including Legendre polynomials (α = 1/2) and Chebyshev polynomials of the second kind (α = 1) <cit.>.
The respective φφ correlations are plotted in Fig. <ref>.

To solve the inverse SL problem in general, we write the operator (<ref>) in Liouville normal form, as is always possible for regular problems <cit.>: Let (y) ≡ u(x) √(K(x)) for y = y(x) given by Eq. (<ref>) and define 











    q̂(y)
    ≡
    - q(x)/w(x) + 1/√(K(x)) v_0^2 
     ∂_y^2 √(K(x)).

The eigenvalue problem (<ref>) then becomes

    v_0^2 ∂_y^2 (y) + [λ - q̂(y)] (y) = 0

with unchanged BCs
[For even v(x) > 0, they are v_0∂_y(y) - v(x) Λ(x) (y) = 0 at y = ± L/2 with Λ(x) ≡∂_xln√(K(x)) <cit.>, which imply Neumann BCs if K'(x)|_x = ± L/2 = 0.] and if reflection symmetric in x, then so in y.
Clearly, _n(y) have the same eigenvalues as the respective u_n(x) and similarly form a complete orthonormal basis but with identity weight.

Results in the literature often require two sets of eigenvalues for different BCs to get a unique solution to the inverse problem.
However, one set suffices for reflection-symmetric SL operators and certain BCs, including ours.
Indeed, <cit.> and <cit.> say that an even q̂(y) is uniquely determined by the spectrum for Neumann BCs.
Thus, viewing q̂(y) as given and since q(x) = 0, Eq. (<ref>) becomes a second-order differential equation for √(K(x(y))).
By general uniqueness results [assuming q̂(y) is continuous] and since K(x) must be even, this determines K(x) up to changing coordinates (<ref>) and an overall rescaling (discarding solutions that are not positive).

The results <cit.> do not reconstruct q̂(y).
However, since we know that K(x) = K gives an SL problem with m_n = n and q̂(y) = 0, a constant Luttinger parameter is the unique solution for the inverse problem given an exactly linear spectrum.
Beyond this special case, nothing a priori prevents the spectrum from deviating from the leading n behavior as long as the requirements on the m_n remain satisfied.
This would then give rise to a class of possible bosonic theories that exhibit PWT, where, by uniqueness, q̂(y) is non-zero, thus K(x) non-constant.
While not usually analytically tractable, there are constructive approaches to find q̂(y) <cit.>.


Discussion.—
phantomsection
tocsectionDiscussion


We introduced the notion of PWT and showed that it is exhibited by one-particle excitations in inhomogeneous CFT whenever the velocity profile v(x) > 0 is even.
Breaking conformal invariance, by considering inhomogeneous TLLs with a Luttinger-parameter profile K(x) > 0, we showed that PWT for general excitations amounts to an inverse spectral problem, which we solved.
This continuum result for bosons is remarkably similar to earlier results: 
For spin chains with nearest neighbor hopping, the corresponding class of operators can be represented as Jacobi matrices, and the resulting inverse eigenvalue problem was studied <cit.>.
As in the continuum, two sets of eigenvalues are generally needed for uniqueness <cit.>, while one set suffices for matrices symmetric with respect to the anti-diagonal (corresponding to reflection symmetry) <cit.>.

By bosonization, our results for inhomogeneous TLLs cover fermionic theories with position-dependent interactions <cit.>.
Indeed, even when K(x) = K is constant, as long as K ≠ 1, the theory describes an interacting quantum many-body system, albeit at low-energies.

Lastly, we showed that an inhomogeneous TLL with constant K(x) = K enjoys PWT for general excitations whenever v(x) > 0 is even.
Such a theory is an example of an inhomogeneous CFT.
It would be interesting to study PWT for any number of excitations in general CFTs, and also extend to systems on the half or full infinite line.




  
*Acknowledgments.



We are thankful to Pawel Caputa and Shinsei Ryu for motivating discussions on related topics.
P.M. gratefully acknowledges financial support from the Wenner-Gren Foundations (Grant No. FT2022-0002).
M.C. acknowledges financial support from the European Research Council (ERC Grant Agreement No. 818761), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the Novo Nordisk Foundation (Grant No. NNF20OC0059939 ‘Quantum for Life’), and thanks the NCCR SwissMAP of the Swiss National Science Foundation and the Section of Mathematics at the University of Geneva for their hospitality.












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[Note1()]Note1
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[Note2()]Note2
  note More precisely, twice continuously
  differentiable.Stop
[Note3()]Note3
  note Here, these amount to v(x), K(x) being strictly positive
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  the whole interval [-L/2, L/2].Stop
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Supplemental Material


phantomsection
tocpartSupplemental Material



This supplemental material has three parts.
Part A gives details for two-point correlations in inhomogeneous 1+1D CFT with open (Neumann) boundary conditions (BCs).
Part B describes how to diagonalize general inhomogeneous Tomonaga-Luttinger liquids (TLLs) using Sturm-Liouville (SL) theory, leading to a formulation of perfect wave transfer (PWT) as an inverse SL problem, including extension to massive TLLs.
Part C contains details for correlation functions in inhomogeneous TLLs.






§ PART A: CORRELATIONS IN INHOMOGENEOUS 1+1D CFT WITH OPEN BOUNDARIES





The general inhomogeneous CFT Hamiltonian is (setting ħ = 1)

    H_iCFT≡∫_-L/2^L/2 x  v(x) [ T_+(x) + T_-(x) ],

where T_±(x) are the right/left-moving components of the stress-energy tensor in light-cone coordinates, whose Fourier transforms yield two commuting copies of the Virasoro algebra.
In position space, they satisfy the equal-time commutation relations

    [ T_±(x), T_±(x') ]
    = ∓ 2∂_xδ(x-x') T_±(x')
      ±δ(x-x') ∂_x'T_±(x')
      ±c/24π∂_x^3δ(x-x')

and [ T_±(x), T_∓(x') ] = 0.
Note that, at non-equal time, T_± should be understood to only depend on the “right/left-moving coordinate” x^∓; for now, we only view these as labels, distinguishing dependencies on right/left components.

Besides the T_±(x) fields, an important class of operators are Virasoro primary fields Φ(x^-, x^+), defined by how they behave under conformal transformations.
The latter are given by orientation-preserving circle diffeomorphisms f_+, f_- and primary fields transform as

    Φ(x^-, x^+)
    →
    f'_+(x^-)^Δ_Φ^+ f'_-(x^+)^Δ_Φ^-Φ(f_+(x^-), f_-(x^+)),

where (Δ_Φ^+, Δ_Φ^-) denote the field's conformal weights.
To impose open BCs at x = ± L/2, as in the main text, we will assume that primary fields come in pairs Φ_+, Φ_- with conformal weights satisfying Δ^+_Φ_+ = Δ^-_Φ_-, Δ^-_Φ_+ = Δ^+_Φ_-, since this allows one to couple excitations so that right movers are reflected to left movers when reaching x = L/2, and vice-versa at x = -L/2.
To ease notation, we will use Δ^+_Φ≡Δ^+_Φ_+, Δ^-_Φ≡Δ^-_Φ_+ for each such pair.

Before continuing, we mention two examples:



  * 
For the inhomogeneous free fermions in the main text, the components of the stress-energy tensor are
T_±(x)
= (1/2) [
    ψ^†_±(x)
    (∓∂_x ) ψ_±(x)  + ]
  - π/ 12L^2,
and the primary fields of the theory are the fermionic fields ψ_±(x) themselves, whose conformal weights are
Δ^+_ψ_+ = Δ^-_ψ_- = 1/2, Δ^-_ψ_+ = Δ^+_ψ_- = 0.


  * 
For an inhomogeneous TLL with constant Luttinger parameter K(x) = K, one has
T_±(x)
= (1/4π K)
  [ πΠ(x) ∓ K ∂_x φ(x) ]^2
  - π/ 12L^2,
and the primary fields include vertex operators constructed as exponentials of the bosonic fields.
Applied to the Luttinger model <cit.>, these vertex operators are (renormalized) fermionic fields ψ_± with conformal weights Δ^+_ψ_+ = Δ^-_ψ_- = (1+K)^2/8K, Δ^-_ψ_+ = Δ^+_ψ_- = (1-K)^2/8K.
(Clearly, free fermions are recovered for K = 1.)



As in the main text, open BCs can be incorporated by standard unfolding methods; cf. <cit.>:
Couple right- and left-moving excitations by identifying

    T_-(x) = T_+(L-x),
        Φ_-(x^-, x^+) = Φ_+(L-x^+, L-x^-)

for all pairs Φ_+, Φ_- of Virasoro primary fields.
It follows that we can equivalently consider a chiral inhomogeneous CFT of only right movers on the interval [-L/2, 3L/2] of length 2L with velocity

    (x)
    ≡
      v(x)    for  x ∈ [-L/2, L/2], 
    
      v(L-x)    for  x ∈ [L/2, 3L/2]

and with every quantity satisfying periodic BCs.
Note that the Hamiltonian (<ref>) can then be written as

    H_iCFT≡∫_-L/2^3L/2 x (x) T_+(x).


For later reference, we define

    (x) 
    ≡∫_0^x s _0/(s),
        1/_0≡1/2L∫_-L/2^3L/2 s/(s),
        _t^±(x)
    ≡^-1( (x) ±_0t ).

Naturally,  lies in the universal cover _+(S^1_2L) of the group of orientation-preserving diffeomorphisms of the circle S^1_2L with circumference 2L. 
Given any v(x), one can show the following properties:

    (L-x) = (x),
        (L-x) = (L) - (x),
        ^-1( (L)-y ) = L - ^-1(y),
        _t^±(L - x) = L - _t^∓(x).

Lastly, we introduce the undeformed Hamiltonian

    H_0 = ∫_-L/2^3L/2 x _0 T_+(x),

whose ground state we denote by |Ω_0⟩.
Similarly, we let |Ω⟩ denote the ground state of H_iCFT.
Following <cit.>, the latter are related to their undeformed counterparts by

    U_+() H_iCFT U_+()^-1
    = H_0 + const,
        
    |Ω⟩
    = U_+()^-1 |Ω_0⟩,

where U_+() is a projective unitary representation of  on the Hilbert space of the CFT.

Similar to the main text, we define

    |Φ_j⟩≡∫_-L/2^L/2 x ( ^- k xξ^+_j(x) Φ_+^†(x,x) + ^ k xξ^-_j(x) 
     Φ_-^†(x,x) ) |Ω⟩     (j = 1,2)

for smooth enough functions ξ^±_j(x), interpreted as distributions or “wave packets” of right/left-moving excitations in each state, and some k ∈ℝ in order to accommodate for  in Eq. (<ref>).
(The latter will have no effect on PWT, and one might as well set k = 0.)
As in the main text, this is a restricted class of states which only allow for one-particle excitations, while we eased other restrictions in the sense that the excitations are not necessarily fermionic and the distributions of right/left movers can be different.

To probe PWT for one-particle excitations, we seek to compute

    ⟨Φ_j| Φ_j⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_j(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_j(x_2)Φ_-(x_2,x_2) ) 
    ×( ^- k x_1ξ^+_j(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_j(x_1) Φ_-^†(x_1,x_1) )
    |Ω⟩

and

    F(t)
    ≡⟨Φ_2| ^- H_iCFTt |Φ_1⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_2(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_2(x_2)Φ_-(x_2,x_2) )
      ^- H_iCFTt
    ×( ^- k x_1ξ^+_1(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_1(x_1) Φ_-^†(x_1,x_1) )
    |Ω⟩,

in order to eventually compare |F(t)| with |⟨Φ_1| Φ_1⟩| for specular waves, ξ^±_2(x) = ξ^∓_1(-x).
Using the unfolding relations (<ref>), we obtain

    ⟨Φ_j| Φ_j⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_j(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_j(x_2)Φ_+(L-x_2,L-x_2) ) 
    ×( ^- k x_1ξ^+_j(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_j(x_1) Φ_+^†(L-x_1,L-x_1) )
    |Ω⟩

and

    F(t)
    = ^- Ct∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ^ H_iCFTt( ^ k x_2ξ^+_2(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_2(x_2)Φ_+(L-x_2,L-x_2) )
      ^- H_iCFTt
    ×( ^- k x_1ξ^+_1(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_1(x_1) Φ_+^†(L-x_1,L-x_1) )
    |Ω⟩,

where we used that H_iCFT |Ω⟩ = C |Ω⟩ for some real constant C that encodes the Casimir energy.
(This only gives rise to an overall phase that one could safely ignore.)
Clearly, as stated in the main text, the above amounts to evaluating two-point correlations for primary fields, specifically those of right movers in a chiral CFT on [-L/2, 3L/2].

First, as a preliminary step, for primary fields with periodic boundary conditions on the system with length 2L, conformal invariance implies that the undeformed two-point correlations are

    ⟨Ω_0| Φ_+(x_2^-, x_2^+) Φ^†_+(x_1^-, x_1^+) |Ω_0⟩
    = 1/2π
      G^+_ΦΦ(x_2^- - x_1^-)^2Δ^+_Φ
      G^-_ΦΦ(x_2^+ - x_1^+)^2Δ^-_Φ,

where G^±_ΦΦ(x) are 2L-periodic functions.
For example, as can be seen from, e.g., <cit.>, the correlations for right-moving free fermions (cf. Example <ref> above) on the system with length 2L and periodic BCs are

    ⟨Ω_0| ψ_+(x_2) ψ^†_+(x_1) |Ω_0⟩
    = 1/2π G^+_FF(x_2 - x_1)

with

    G^+_FF(x)
    ≡πexp( -π/2L x )
    /
      2L sin( π/2L [x + 0^+] )
    ,

where we used that Δ^+_ψ = 1/2, Δ^-_ψ = 0.
Similarly, for the Luttinger model (cf. Example <ref> above), the correlations for the (renormalized) right-moving fermionic fields are

    ⟨Ω_0| ψ_+(x_2^-, x_2^+) ψ^†_+(x_1^-, x_1^+) |Ω_0⟩
    = 1/2π
      G^+_FF(x_2^- - x_1^-)^(1+K)^2/4K
      G^+_FF(x_1^+ - x_2^+)^(1-K)^2/4K,

where we used that

    G^-_FF(x)
    ≡
      -πexp( +π/2L x )
    /
      2L sin( π/2L [x - 0^+] )
    
    = πexp( - π/2L [-x] )
      /
        2L
        sin( π/2L [- x + 0^+] )
      
    = G^+_FF(-x).


Second, by results spelled out in Proposition 3.1 in <cit.> for inhomogeneous CFT, the time dependence in F(t) means that those correlations are evaluated at spacetime points given by ^±_t(x) in Eq. (<ref>).
In addition, the action of U_+() from the expression (<ref>) for |Ω⟩ also leads to a conformal transformation of the fields.
It follows from this and the properties (<ref>) that

    ⟨Φ_j| Φ_j⟩
    = 1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_j(x_2)ξ^+_j(x_1)
    G^+_ΦΦ((x_2) - (x_1))^2Δ^+_Φ
    G^-_ΦΦ((x_2) - (x_1))^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_j(x_2)ξ^-_j(x_1)
    G^+_ΦΦ((x_2) + (x_1) - (L))^2Δ^+_Φ
    G^-_ΦΦ((x_2) + (x_1) - (L))^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_j(x_2)ξ^+_j(x_1)
    G^+_ΦΦ( - (x_2) - (x_1) + (L))^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) - (x_1) + (L))^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_j(x_2)ξ^-_j(x_1)
    G^+_ΦΦ( - (x_2) + (x_1))^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) + (x_1))^2Δ^-_Φ)

and

    F(t)
    = ^- Ct1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_2(x_2)ξ^+_1(x_1) 
    G^+_ΦΦ((x_2) - (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((x_2) - (x_1) + _0t)^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_2(x_2)ξ^-_1(x_1) 
    G^+_ΦΦ((x_2) + (x_1) - (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((x_2) + (x_1) - (L) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_2(x_2)ξ^+_1(x_1) 
    G^+_ΦΦ( - (x_2) - (x_1) + (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) - (x_1) + (L) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_2(x_2)ξ^-_1(x_1) 
    G^+_ΦΦ( - (x_2) + (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) + (x_1) + _0t)^2Δ^-_Φ),

where we used that
(_t^±(x) )
= (x) ±_0t
and
(_t^±(L - x) )
= (L) - (_t^∓(x))
= (L) - (x) ±_0t.

Finally, for specular waves, ξ^±_2(x) = ξ^∓_1(-x), we obtain

    F(t)
    = ^- Ct1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(-x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(-x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_1(x_2)ξ^+_1(x_1)
    G^+_ΦΦ( - (-x_2) - (x_1) + (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (-x_2) - (x_1) + (L) + _0t)^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_1(x_2)ξ^-_1(x_1)
    G^+_ΦΦ( - (-x_2) + (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (-x_2) + (x_1) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_1(x_2)ξ^+_1(x_1)
    G^+_ΦΦ((-x_2) - (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((-x_2) - (x_1) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_1(x_2)ξ^-_1(x_1)
    G^+_ΦΦ((-x_2) + (x_1) - (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((-x_2) + (x_1) - (L) + _0t)^2Δ^-_Φ),

after a change of variable and rearranging.

By comparing the terms appearing in Eqs. (<ref>) and (<ref>), it follows that |F(t)| for certain t equals |⟨Φ_1| Φ_1⟩| for all ξ^±_1(x) if and only if (x) is even, meaning (x) must be odd.
This in turn implies (L) = L and _0 = v_0 given by Eq. (<ref>), which means that equality holds for all t = (2m+1)L/v_0 due the 2L periodicity of G^±_ΦΦ(x).
[Moreover, for ξ^±_2(x) = ξ^∓_1(-x), one has |⟨Φ_1| Φ_1⟩| = |⟨Φ_2| Φ_2⟩| when v(x) is even.]
Since (x) = v(x) for all x ∈ [-L/2, L/2], this proves the claim that one-particle PWT is exhibited for all even inhomogeneous CFTs.

As a last remark, one can unfold in a different way to take the spin/statistics of the primary fields into account, so that, e.g., one gets anti-periodic BCs for fermions
on the system of size 2L, and twisted BCs for general anyonic fields.
This can be used to obtain simple explicit formulas for the two-point correlations of any primary fields.
However, since we are interested in recombinations of waves traveling in different directions, it is necessary to identify so as to always have periodic BCs.
In any case, for our purposes, all we needed was the spacetime dependence of the correlations, which are fixed on general grounds in CFT, circumventing any need for explicit formulas.






§ PART B: SPECTRAL PROPERTIES OF INHOMOGENEOUS TLLS





In this part, we first show that reflection symmetry is a necessary condition for PWT for any theory; here under the assumption that the Hamiltonian has pure point spectrum.
Second, we give the details for how to use SL theory to diagonalize the inhomogeneous TLL Hamiltonian (<ref>) and to formulate the PWT requirements on the spectrum.
Third, we review some details on Weyl's law applied to SL operators and, fourth, extend the previous discussion to TLL Hamiltonians with a mass term.
Fifth, we comment on the Liouville normal form used to state the results for the inverse SL problem in the main text.






 §.§ Necessity of reflection symmetry





Before considering TLL theory, we note that all models studied in the main text have pure point spectrum [as long as the integral defining 1/v_0 in Eq. (<ref>) converges].
Thus, we first consider a general Hamiltonian H with pure point spectrum and show that for such a system to exhibit PST/PWT, it is necessary that H commutes with the parity operator , which maps x ↦ -x.
In other words, we will show that reflection symmetry is a necessary condition.

To prove the statement, let E_n denote the eigenenergies of H with the corresponding energy eigenstates |n⟩.
On general grounds, the latter form a complete orthonormal basis for the Hilbert space  of the theory.
Let |Φ_1⟩ be an arbitrary state in  and let |Φ_2⟩≡ |Φ_1⟩.
(We recall that  is a unitary operator satisfying ^2 = 1, meaning that ^-1 = ^† =.)
As usual, define F(t) ≡⟨Φ_2| ^- H t |Φ_1⟩ =  ⟨Φ_1| ^- H t |Φ_1⟩.
For general PST/PWT, we require that there exists T > 0 such that |F(T)| = |⟨Φ_1|Φ_1⟩| for any state |Φ_1⟩.
In particular, this must hold for any energy eigenstate |n⟩, implying |⟨ n||n⟩| = 1.
If |n⟩| is not an eigenstate of , it is straightforward to show that this equality cannot hold.
Consequently, all energy eigenstates must also be eigenstates of , which implies that [, H] = 0.






 §.§ Inhomogeneous TLLs and SL theory





We now turn to inhomogeneous TLLs studied in the main text.
Inserting the expansions (<ref>) into Eq. (<ref>) yields

    H_iTLL
    = 1/2π [πΠ_0]^2
      + 1/2π∑_n = 1^∞( [πΠ_n]^2 + λ_n φ_n^2 )^2 ,

where φ_0 is absent (when E_0 = 0), no different from the standard case with constant K(x) [and v(x)].
For n ≥ 1, let

    φ_n
    = √(π/2)1/λ_n^1/4( a_n + a^†_n),
        Π_n
    = -λ_n^1/4/√(2π)( a_n - a^†_n)

in terms of bosonic creation and annihilation operators a^†_n and a_n, satisfying [ a_n, a^†_n'] = δ_n,n', as in the main text.
Inserting Eq. (<ref>) into Eq. (<ref>) yields Eqs. (<ref>)–(<ref>).

Using the above, one can compute the time evolution of any observable in the algebra of operators generated by the fields φ(x) and Π(x).
For instance, for φ(x, t) ≡^ H_iTLL tφ(x) ^- H_iTLL t and Π(x, t) defined similarly, we obtain

    φ(x, t)
    = ∑_n = 1^∞√(π/2E_n)( a_n^- E_nt + a^†_n^ E_nt)
      u_n(x),
        Π(x, t)
    = ∑_n = 1^∞ (-) √(E_n/2π)( a_n^- E_nt - a^†_n^ E_nt)
      w(x) u_n(x),

where we omitted the zero modes (as they have to be treated separately when E_0 = 0) and used ^ H_iTLL t a_n^- H_iTLL t = a_n^- E_nt and ^ H_iTLL t a^†_n^- H_iTLL t = a^†_n^ E_nt.
Generally, any observable (x, t) in this algebra will contain contributions of the form a^†_n^ E_n t u_n(x) and a_n^- E_n t u_n(x).
Now, in order for PWT to be exhibited, Eq. (<ref>) must hold for any state, which means that it holds true without expectation values, i.e., (x, 0) must equal (-x, T) at some later time T > 0 as an operator identity.
Taking = φ and using the linear independence of the generators a_n and a^†_n, Eq. (<ref>) yields

    u_n(x) = u_n(-x) ^± E_n T  ∀ n = 0, 1, …

for a common T > 0, where we included the trivial case n = 0 [recall that u_0(x) = const and E_0 = 0 when q(x) = 0 in Eq. (<ref>)].
Since u_n(x) = u_n(-x), Eq. (<ref>) implies that each u_n(x) must be an eigenfunction of the parity operator , whose eigenvalues are +1 or -1 (since ^2 = 1).
It follows that all eigenfunctions must have definite parity, and the only possibility for PWT to hold is that the corresponding eigenenergies satisfy E_n T = π m_n for integers m_n for each n, with m_n even (odd) if u_n(x) has even (odd) parity.

Finally, it is straightforward to show the following statement:
The eigenfunctions u_n(x) of a regular SL operator  have definite parity if and only if [, ] = 0.
For our operator in Eqs. (<ref>)–(<ref>), this means that both v(x) and K(x) must be even functions (using that they must be positive to exclude the possibility that both are instead odd).

Given the above, together with Properties <ref>–<ref> for regular SL operators stated in the main text, we have derived our main result on the spectral requirements for an inhomogeneous TLL to exhibit PWT.

Moreover, as stated in the main text, reversing the logic, namely, given a spectrum that satisfy the spectral properties for PWT to be exhibited, our formulation of PWT for inhomogeneous TLLs as an inverse SL problem follows.






 §.§ Weyl's law for SL operators





Weyl's law applied to SL operators, see, e.g., <cit.>, states that the number N(λ) of eigenvalues below a given λ > 0 asymptotically behaves as

    N(λ)/√(λ)∼1/π∫_-L/2^L/2 x √(w(x)/p(x))

for large λ.
This holds under certain technical conditions on the coefficients w(x), p(x) in Eq. (<ref>) [which are satisfied assuming our v(x), K(x) are positive and smooth functions].
Using Eqs. (<ref>) and (<ref>) as well as Property <ref> in the main text, it follows that

    n/E_n∼1/π∫_-L/2^L/2 x/v(x)
    = L/π v_0

for large n, where in the last step we used Eq. (<ref>).
Thus, assuming that the integral defining 1/v_0 converges, then this gives the stated asymptotic behavior, E_n = π v_0 n/L + o(n), in the main text.
(Another possibility would be if the integral diverges, in which case E_n∼const× n^γ for γ < 1, but this is inconsistent with that the eigenenergies must be proportional to integers m_n with a common proportionality constant in order for the theory to exhibit PWT.)






 §.§ “Massive” inhomogeneous TLLs





For massless TLL bosons, as discussed above and in the main text, given a spectrum satisfying the PWT requirements, we saw that K(x) can be found uniquely up to rescaling and changing coordinates.

We now extend to SL operators (<ref>) with q(x) ≠ 0, corresponding to an inhomogeneous (possibly imaginary) mass M(x) = √(-q(x)/v_0^2 v(x)) by adding v_0^2 M(x)^2 φ(x)^2 inside the big parentheses in Eq. (<ref>):

    H_iTLL≡∫_-L/2^L/2 x/2π v(x)
      (
          [πΠ(x)]^2/K(x)
          + K(x)[∂_x φ(x)]^2
          + v_0^2 M(x)^2 φ(x)^2
        )
       ,

Indeed, it is straightforward to extend the previous discussion to such models.
For instance, the PWT requirements on the spectrum are the same, except the spectrum does not start at zero:
For q(x) ≠ 0, a constant u_0(x) no longer solves Eq. (<ref>), thus E_0 > 0.
Instead, we have a “massive” inhomogeneous TLL, for which one can show:


A “massive” inhomogeneous TLL exhibits PWT if and only if v(x), K(x), q(x) are even
 and all eigenvalues are of the form E_n = πm̃_n/T for certain integers m̃_n.


Here, these integers m̃_n = m_n + in terms of the integers m_n for the massless case in the main text, where we allow for an overall shift by a finite even integer .
[The latter must be even so that m̃_n is even (odd) if n is even (odd), as required due to Property <ref> in the main text.]
Moreover, given such a spectrum, the results in <cit.> still imply that q̂(y) is uniquely determined by the spectrum, but this can no longer uniquely determine K(x) since the first term in Eq. (<ref>) is now non-zero in general.

As an example, we revisit the case with v(x) = v √(1 - (2x/L)^2) and K(x) = K [ 1 - (2x/L)^2 ]^α for α > -1/2 and constants v, K > 0 encountered in the main text, whose eigenfunctions were given by Gegenbauer polynomials C^(α)_n(2x/L), but now we include a fine-tuned potential q(x) = - (2vα/L)^2 w(x), where we recall that w(x) = K(x)/v(x).
This fine-tuned potential allows us to complete the square so that the eigenvalues in Eq. (<ref>) are λ_n = (π v_0/L) (n + α)^2  with v_0 = 2v/π, meaning that E_n = π (n + α) v_0/L.
Consequently, any c = α = 0, 2, 4, … works, yielding a discrete family of models with fine-tuned inhomogeneous mass that exhibit PWT.
We stress that the mass in this example satisfies M(x) ∝ 1/L and therefore vanishes in the thermodynamic limit.
Thus, one should not expect hallmarks of massive theories, such as exponential decay of two-point correlations with respect to the spatial separation.






 §.§ Liouville normal form and compactified free bosons





It is curious to note the following “shortcut” to writing our massless bosons (<ref>) in Liouville normal form:
Recall that an inhomogeneous TLL is a theory of compactified free bosons with a position-dependent compactification radius R(x) in curved spacetime given by the metric (h_μν) = (v(x)^2/v_0^2, -1) for coordinates (x^0, x^1) ≡ (v_0t, x); see, e.g., <cit.>.
Given a Luttinger parameter profile K(x), the compactification radius is R(x) = √(2α') K(x) (α' has dimension length squared and is commonly set equal to 2).
In terms of the bosonic field φ(x) (with values taken modulo 2π), the action corresponding to the Hamiltonian (<ref>) is

    S
    = 1/4πα'∫^2 x √(-h)
      R(x)^2 (∂_μφ) (∂^μφ),

where we recall that h ≡(h_μν).
This can be viewed as a non-exactly marginal deformation of a theory with constant compactification radius; cf. <cit.>.
The corresponding action for X(x) ≡ R(x) φ(x) becomes

    S
    = 1/4πα'∫^2 x √(-h)[ (∂_μ X) (∂^μ X) - q̂ X^2/v(x)^2 ]

with q̂ = q̂(y) in Eq. (<ref>) for y = y(x) given by Eq. (<ref>), which agrees with Eq. (<ref>) for (y) = X(x)/√(2α').
I.e., as a theory for X(x), even when q(x) = 0, the position-dependent K(x) corresponds a (possibly imaginary) inhomogeneous mass √(q̂(y))/v(x).
This also extends to the case

    S 
    = 1/2π∫^2 x √(-h)[ K(x) (∂_μφ) (∂^μφ) - v_0^2 M(x)^2 φ^2 ]

with a non-zero mass M(x) = √(-q(x)/v_0^2 v(x)) for the bosonic field φ(x), corresponding to the Hamiltonian (<ref>), the only difference being that the first term in Eq. (<ref>) is then no longer zero.






§ PART C: CORRELATION FUNCTIONS IN INHOMOGENEOUS TLLS





Consider the inhomogeneous TLL theory with Hamiltonian H_iTLL in Eq. (<ref>).
Using the eigenfunctions obtained from SL theory, we can compute any ground-state two-point correlation functions for the fields φ(x) and ϑ(x), where the latter is defined by

    ∂_x ϑ(x)
    ≡πΠ(x).

Indeed, using the expansions (<ref>), that a_n annihilates the ground state |Ω⟩ of H_iTLL, as well as the commutation relations [ a_n, a^†_n'] = δ_n,n', it is straightforward to show that

    ⟨Ω| φ(x,t) φ(x',t') |Ω⟩    = ∑_n = 1^∞π/2E_n u_n(x)u_n(x') ^- E_n (t-t'), 
    ⟨Ω| φ(x,t) ϑ(x',t') |Ω⟩    = ∑_n = 1^∞π/2 u_n(x)U_n(x') ^- E_n (t-t'), 
    ⟨Ω| ϑ(x,t) ϑ(x',t') |Ω⟩    = ∑_n = 1^∞π E_n/2 U_n(x)U_n(x') ^- E_n (t-t'),

where we introduced U_n(x) defined by

    ∂_x U_n(x)
    ≡
    w(x) u_n(x).

Similar correlation functions can be computed for other objects, including vertex operators (i.e., normal-ordered exponentials of the bosonic fields) <cit.>.
It is also straightforward to compute Green's functions; see, e.g., <cit.>, the only difference with the present case being that one must account for the non-trivial weight function w(x) that was simply a constant in <cit.>.

As an important example, consider a theory with K(x) = K > 0 constant but v(x) > 0 general [and q(x) = 0].
We recall that the SL operator in this case is = - v(x) ∂_x v(x) ∂_x = -v_0^2 ∂_y^2 using Eq. (<ref>) with eigenenergies E_n = π n v_0/L and eigenfunctions u_n > 0(x) = √(2 v_0/L)cos( nπ [L + 2y]/2L ) and u_0(x) = √(v_0/L).
Then, one can show that

    ⟨Ω| φ(x,t) φ(x',t') |Ω⟩
    
    = 1/4   (
      logexp( -π/2L [f(x)-f(x')-v_0(t-t')] )
      /
        2 sin( π/2L [f(x)-f(x')-v_0(t-t')+0^+] )
      
    + logexp( -π/2L [f(x)+f(x')+L-v_0(t-t')] )
      /
        2 sin( π/2L [f(x)+f(x')+L-v_0(t-t')+0^+] )
      . 
          + .
      log
        -exp( π/2L [f(x)+f(x')+L+v_0(t-t')] )
      /
        2 sin( π/2L [f(x)+f(x')+L+v_0(t-t')-0^+] )
      
    + log
        -exp( π/2L [f(x)-f(x')+v_0(t-t')] )
      /
        2 sin( π/2L [f(x)-f(x')+v_0(t-t')-0^+] )
      ),

where we included the necessary regularizations in order to interpret coincident spacetime points and used the identity ∑_n = 1^∞1/n^-ξ n = - log( 1 - ^-ξ) for any ξ such that ξ > 0.
Eq. (<ref>) is the analytical expression for what was plotted in Fig. <ref> (assuming K = 1).
Moreover, for delta-like waves, the terms appearing in log F(t) given by Eq. (<ref>) for the free-fermion case (<ref>) closely resemble those in Eq. (<ref>).
Indeed, the former agrees with 4 ⟨Ω| φ(x,t) φ(x',t') |Ω⟩ to leading order, and one can furthermore show that the agreement between the two ways of computing (CFT methods versus bosonization) is exact (as it should be) by evaluating and summing all combinations of two-point correlation functions of the vertex operators describing the right/left-moving free fermionic fields.





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 Paper Title: Minimum time connection between non-equilibrium steady states: the Brownian gyrator.










apatron@us.es
Física Teórica, Universidad de Sevilla, Apartado de
  Correos 1065, E-41080 Sevilla, Spain

cplata1@us.es
Física Teórica, Universidad de Sevilla, Apartado de
  Correos 1065, E-41080 Sevilla, Spain

prados@us.es
Física Teórica, Universidad de Sevilla, Apartado de
  Correos 1065, E-41080 Sevilla, Spain


§ ABSTRACT 
We study the problem of minimising the connection time between non-equilibrium steady states of the Brownian Gyrator. This is a paradigmatic model in non-equilibrium statistical mechanics, an overdamped Brownian particle trapped in a two-dimensional elliptical potential, with the two degrees of freedom (x,y) coupled to two, in principle different, thermal baths with temperatures T_x and T_y, respectively. Application of Pontryagin's Maximum Principle reveals that shortest protocols belong to the boundaries of the control set defined by the limiting values of the parameters (k,u) characterising the elliptical potential. We identify two classes of optimal minimum time protocols, i.e. brachistochrones: (i) regular bang-bang protocols, for which (k,u) alternatively take their minimum and maximum values allowed, and (ii) infinitely degenerate singular protocols. We thoroughly investigate the minimum connection time over the brachistochrones in the limit of having infinite capacity for compression. A plethora of striking phenomena emerge: sets of states attained at null connection times, discontinuities in the connection time along adjacent target states, and the fact that, starting from a state in which the oscillators are coupled, uncoupled states are impossible to reach in a finite time.


Valid PACS appear here
                             

                              
Minimum time connection between non-equilibrium steady states: the Brownian gyrator
    A. Prados
    August 5, 2024
===================================================================================






§  INTRODUCTION


In the pursuit of enhancing the efficiency of physical processes, researchers have recently turned their attention towards the field of shortcuts to adiabaticity <cit.> (STA). Originally born within the framework of quantum mechanics <cit.>, STA were developed with the aim of driving quantum systems towards a desired target state, while beating the natural relaxation timescale characterising the dynamics of the system under scrutiny. It was not long after when these ideas were extended to alternative contexts, such as those of classical mechanics <cit.> and non-equilibrium statistical mechanics <cit.>. For the latter, the term swift state-to-state transformations (SST) have been recently proposed <cit.> to include all the catalogue of different shortcut strategies, since other terms like engineered swift equilibration <cit.> or shortcuts to isothermality <cit.> are rather restrictive: in general, the shortcuts are neither an equilibration process nor an isothermal process.   

The field of STA and SST is deeply connected with control theory: for the physical system of interest, one aims at connecting given initial and final states by engineering a time-dependent driving—the control function—that gets the desired connection done.  Within the context of the control of Brownian particles,  one assumes most commonly that the controllable parameters characterise the external force or potential acting on the system—for instance, the amplitude and direction of an electric or magnetic field, or the stiffness or centre of an optical trap <cit.>. Moreover, in some situations, the whole potential may be regarded as the control function <cit.>. Also, instead of controlling the potential, one may engineer the thermal environment by effectively controlling the time dependence of the bath temperature <cit.>.

 Once the connection is shown to be feasible, there appears the problem of optimising it in a certain sense <cit.>. Therefore, one enters the realm of optimal control theory <cit.>, in which the minimisation of a certain functional of the system's trajectory—entropy production <cit.>, connection time <cit.>, or other figures of merit <cit.>—is sought after. The optimisation problem is tackled by using the mathematical tools of optimal control theory, specifically Pontryagin's Maximum Principle when non-holonomic constraints are present <cit.>. With a few exceptions <cit.>, optimal SST protocols in non-equilibium statistical mechanics have only been analysed for connections between equilibrium states.

 The Brownian Gyrator (BG) is a paradigmatic example of a system with a non-equilibrium steady state (NESS) <cit.>. An overdamped Brownian particle is confined in two dimensions  by an elliptic potential and, in addition, its two coordinates (x,y) are submitted to the action of two white noise forces stemming from two thermal baths at different temperatures T_x and T_y, respectively.  The two different temperatures make the system reach a NESS in the long-time limit, and the term “gyrator” refers precisely to this system presenting a non-vanishing average torque proportional to the temperature difference |T_x-T_y| at the NESS. 


From a theoretical perspective, the quadratic dependence of the potential on the system variables (x,y) makes it possible to attack the problem analytically, since the evolution equations for the relevant moments are linear in them, and the probability distribution function remains Gaussian if it is so initially—moreover, the probability distribution function is Gaussian at the NESSs. Also, the linearity of the evolution equations makes it possible to introduce normal modes, which further simplify the mathematical description of this non-equilibrium system. From an experimental perspective, the BG is one of the simplest candidates for designing a non-equilibrium heat engine at the mesoscopic level. In fact, it has recently been realised in actual experiments with different techniques <cit.>. 




 SST connections between two NESSs of the BG corresponding to different values of the parameters of the elliptic potential have been shown to be possible <cit.>, by employing inverse engineering techniques similar to the counterdiabatic method <cit.>. Here, we investigate the optimisation of the SST connection between two NESSs of the BG in the sense of minimising the connection time, i.e. we look into the brachistochrone problem for the BG. Moreover, we prove the existence of a speed-limit inequality for the connection between the two NESSs, which involves the connection time and the irreversible work. This inequality can be regarded as the extension to the connection between NESSs of similar inequalities for the connection between equilibrium states <cit.>. 

Specifically, we work with a BG in which the elliptical potential confining the particle is characterised by two controllable parameters (k,u), which respectively account for the diagonal terms k, i.e. we have identical stiffnesses in the (x,y) directions, and for the anti-diagonal terms u coupling the two degrees of freedom. Throughout our whole work, the temperatures (T_x,T_y) of the thermal baths are kept constant.

The problem we address in this paper is the following: we consider two NESSs of the BG, corresponding to different values of the parameters characterising the potential, (k_i,u_i) and (k_f,u_f) for the initial and final states, respectively. Then, we look for the time-dependent controls (k(t),u(t)) that make the connection time between these two states minimum.  In many time-optimisation problems <cit.>, the control parameters enter linearly in the dynamic evolution equations of the relevant physical observables, thus entailing that Pontryagin's Hamiltonian function is also linear in them. In that case, the optimal protocol leading to the brachistochrone between the initial and target states is usually of bang-bang type: it comprises different time windows for which the control parameters are constant and attain their extremum—i.e. maximum or minimum—values. 

For the BG, the control parameters are bounded by the non-holonomic constraints k^2-u^2≥ 0 and 0≤ k≤ k_max. The constraints k≥ 0 and k^2-u^2≥ 0, or k≥ 0 and |u|≤ k, are of fundamental nature, since they stem from the potential being confining—then having a minimum at the origin (x,y)=(0,0). The condition k≤ k_max is of practical nature, since  there is typically a maximum value of the stiffness of the trap in actual experiments. Therefore, we will consider that the constraints k≥ 0 and |u|≤ k always hold throughout our work, whereas the constraint k ≤ k_max will be relaxed, i.e. we will consider the limit k_max→∞ at certain points to make analytical progress. 

 Following the usual terminology of control theory, we refer to controls satisfying the non-holonomic restrictions |u|≤ k and 0 ≤ k≤ k_max as admissible controls. Our work shows that there appear two different scenarios for the admissible controls over the brachistochrone: (i) regular bang-bang protocols, in which the controls (k(t),u(t)) switch among the three points vertices of the triangular control set determined by the non-holonomic constraints k^2-u^2≥ 0 and 0≤ k≤ k_max: (0,0), (k_max,k_max) and (k_max,-k_max), and (ii) infinitely degenerate singular protocols, which belong to the boundaries of the control set but are more complex. We also analyse the behaviour of the resulting minimum connection time as a function of the chosen initial and target NESS. A very rich phenomenology emerges, including discontinuities in the connection time for incremental changes of the target states, sets of target states that can be instantaneously reached—in the limit k_max→∞, and sets of states that cannot be reached in a finite time.  

The paper is structured as follows. In Sec. <ref>, we introduce the dynamics of our BG model in detail, within the framework of non-equilibrium statistical mechanics. Section <ref> puts forward a derivation of a speed-limit inequality for the BG, which involves the connection time and the irreversible work done in the SST connection. In Sec. <ref>, we pose the problem of minimising the connection time between NESS by making use of PMP and present the main results of the paper. Section <ref> is devoted to building the optimal-time bang-bang protocols stemming from PMP. The minimum connection time as a function of the initial and target states is analysed in Sec. <ref>.
A brief recap of the main results, together with the main conclusions of this work, is presented in Sec. <ref>. Finally, we relegate most of the rigorous proofs and  technical details to the appendices.




§  MODEL


We consider a BG, which can be thought as an overdamped Brownian particle submitted to a two-dimensional confining potential U = U(r) and in contact with two thermal baths of temperatures T_x and T_y, which affect separately each degree of freedom. Hence, the system is described by the Fokker-Planck equation (FPE)

    γ ∂_t P = ∇_r^T[P  ∇_r U + k_B 𝕋 ∇_r P],   𝕋≡([ T_x   0;   0 T_y ]),

where P=P(r,t) is the probability distribution function for the position of the particle, γ the friction coefficient, k_B the Boltzmann constant, and we have introduced the notation

    r=[ x; y ],   r^T = (x,y),   ∇_r=[ ∂_x; ∂_y ],   ∇_r^T=(∂_x,∂_y),

i.e. the superindex T in a matrix denotes its transpose. The confining potential is given by

    U(r) = 1/2r^T𝕂r,   𝕂≡([ k u; u k ]),

which must satisfy the confining conditions

    k≥ 0,     (𝕂) = k^2-u^2≥ 0,

which ensure the existence of a NESS—for T_x T_y; if T_x=T_y=T, then the NESS reduces to the canonical equilibrium distribution at temperature T. 

The parameters (k,u) characterise the harmonic trap: the parameter k corresponds to the stiffness of the potential, which for the sake of simplicity we assume that is identical in both directions, while u accounts for the coupling between the two degrees of freedom. We assume that both parameters can be externally controlled, so they play the role of control functions in our problem. On experimental grounds, the BG model describes the behaviour of a colloidal particle trapped in an elliptical trap, and simultaneously in contact with two thermal baths at different temperatures in the two orthogonal directions. Such thermal baths have been realised in the laboratory with different experimental setups <cit.>.


Now let us introduce the normal modes q^T=(q_1,q_2) via the linear transformation

    q = 𝕄r,   𝕄≡1/√(2)([  1  1;  1 -1 ]).

Note that 𝕄 is a symmetric and orthogonal matrix, its orthogonality entails that the Jacobian of the coordinate transformation equals unity, and thus P(q,t) = P(r,t). Therefore, the FPE can now be easily rewritten in terms of the normal modes as

    γ ∂_t P = ∇_q^T[P  ∇_qU + k_B 𝕋̃ ∇_q P ],   𝕋̃≡𝕄𝕋𝕄 = 1/2([ T_x + T_y T_x - T_y; T_x - T_y T_x + T_y ]),

with 

    U(q) = 1/2q^T𝕂̃q,   𝕂̃≡𝕄𝕂𝕄 = ([ k + u     0;     0 k - u ]).

We have natural boundary conditions at infinity, i.e. P(q,t) vanishes fast enough for |q|→∞. Although the confining potential may be expressed as the sum of two independent potentials for each normal mode, the new FPE presents a cross-derivative term, which is related to the fact that the noises associated with the normal modes are now coupled. This entails that the long-time state for T_x≠ T_y is a NESS, with a non-zero torque proportional to the temperature difference |T_x-T_y| <cit.>.


Rather than working directly with the FPE, it is convenient to work with the dynamical equations for the moments of the probability distribution. At the NESS, the probability distribution is Gaussian <cit.>, and the linearity of the potential entails that the probability distribution remains Gaussian if it is so initially. Since we are interested in the connection between NESSs corresponding to different values of the potential parameters (k,u), the probability distribution is Gaussian for all times and thus it is completely characterised by its first  and second moments, i.e. 

    q=[ q_1; q_2 ],
         ℚ_2≡qq^T=[   q_1^2 q_1 q_2; q_2 q_1   q_2^2 ],

respectively. The evolution equations for the moments are

  
    γ d/dtq   = - 𝕂̃ q, 
    γ d/dtℚ_2    = -{𝕂̃,ℚ_2} + 2 k_B𝕋̃,
  

with {A,B }≡ AB + BA being the anticommutator between two matrices.  More explicitly, we have that

    
    γdq_1^2/dt    = -2(k+u)q_1^2+k_B (T_x+T_y),
    γdq_2^2/dt    = -2(k-u)q_2^2+k_B (T_x+T_y),
    γdq_1 q_2/dt    = -2 k q_1 q_2+k_B (T_x-T_y).


Here, it is where we can appreciate the main advantage of employing the normal modes: had we chosen to work with our original variables r^T = (x,y), the dynamical equations for their respective relevant moments would be coupled. However, the above equations for the second moments q_j q_k are not completely uncoupled, it has to be taken into account that the controls (k,u) appearing inside them are the same for all of them. For example, if we looked for an inverse engineering solution,  i.e. for the controls (k(t),u(t)) stemming from a given time evolution for the second moments, the three functions {q_1^2(t),q_2^2(t),q_1 q_2(t)}  would not be independent.
 

For time-independent values of (k,u), one readily obtains the values of the moments at the NESS from Eq. (<ref>): 

    q_ = 0,   q_1^2_ = k_B(T_x + T_y)/2(k+u),    q_2^2_ = k_B(T_x+T_y)/2(k-u),   q_1q_2_ = k_B(T_x - T_y)/2k.

As already stated, we would like to connect two NESSs, corresponding to different values of the potential parameters. i.e. we start from the NESS for (k_i,u_i) and would like to end up in the NESS for (k_f,u_f). Therefore, q_i ≡q(t=0)=0 and then Eq. (<ref>) implies that  <q>(t)=0 at all times.  As a consequence, the time evolution of the system is entirely determined by the dynamics of the  covariance matrix ℚ_2 in Eq. (<ref>), since the probability distribution reads

    P(q,t) = 1/2π√(ℚ_2(t))exp(-1/2q^Tℚ_2(t)^-1q),

with ℚ_2^-1 being the inverse of the matrix ℚ_2 defined in Eq. (<ref>). 




 

§ SPEED LIMIT FOR THE BROWNIAN GYRATOR



Let us consider the energetic balance, in average, for our system. In terms of the normal modes just introduced above, the average energy reads

    U=1/2(k+u)q_1^2+1/2(k-u)q_2^2.

The average energy changes in time because both the controls (k,u)  and the variances q_j^2 vary in time. The first contribution corresponds to the average work done on the system, whereas the second corresponds to the average heat exchanged with the thermal baths:

    d/dtU=1/2(k̇+u̇)q_1^2+1/2(k̇-u̇)q_2^2_Ẇ+1/2(k+u)dq_1^2/dt+1/2(k-u)dq_2^2/dt_Q̇.


We focus now on the work done on the system in a process where (k,u) are controlled during the time interval (0,t_f):

    W=∫_0^t_f dt [ 1/2(k̇+u̇)q_1^2+1/2(k̇-u̇)q_2^2]=ΔU-∫_0^t_f dt [ 1/2(k+u)dq_1^2/dt+1/2(k-u)dq_2^2/dt]

where ΔU≡U(t=t_f)-U(t=0). Bringing to bear Eqs. (<ref>), we have

    k+u    = k_B(T_x+T_y)/2q_1^2-γ/21/q_1^2dq_1^2/dt,    
        k-u    = k_B(T_x+T_y)/2q_2^2-γ/21/q_2^2dq_2^2/dt,

which, when inserted into Eq. (<ref>), gives

    W=Δ[ U-k_B(T_x+T_y)/4ln(q_1^2q_2^2) ]+γ/4∫_0^t_f dt [1/q_1^2(dq_1^2/dt)^2+1/q_2^2(dq_2^2/dt)^2].


Equation (<ref>) suggests the definitions

    ℱ   ≡U-k_B(T_x+T_y)/2ln(σ_1σ_2), 
    W_   ≡γ∫_0^t_f dt [(dσ_1/dt)^2+(dσ_2/dt)^2],

where

    σ_j≡√(q_j^2).

With these defitions,

    W=Δℱ+W_≥Δℱ.

The first contribution Δℱ to W is the variation of the function of state ℱ, which only depends on the initial and final values of the variances q_j^2. For T_x=T_y, ℱ equals Hemlhotz's free energy for a system of two oscillators; therefore, ℱ can be considered as its generalisation to a non-equilibrium situation for the BG. The second contribution W_  depends on the protocol employed to connect the initial and final states. In other words, it is a functional of the protocol, which is non-negative and only vanishes when dσ_j/dt=0 for all times—i.e. for an infinitely slow protocol in which the variances have their instantaneous equilibrium values at all times. Therefore, we physically interpret W_ as the irreversible contribution to the average work.

Interestingly, the above extension of the irreversible work to the connection between NESSs entails the emergence of a speed-limit for it. By using the Cauchy-Schwarz inequality, we have that

    | ∫_0^t_f dt  dσ_j/dt|^2 =| σ_j,f-σ_j,i|^2 ≤ t_f ∫_0^t_f dt  (dσ_j/dt)^2.

Therefore, we conclude that

    t_f W_≥∑_j=1^2 | σ_j,f-σ_j,i|^2 .

For the connection between NESSs in which we are interested in, the above inequality becomes

    t_f W_≥γ k_B(T_x+T_y)/2[ (1/√(k_f+u_f)-1/√(k_i+u_i))^2+(1/√(k_f-u_f)-1/√(k_i-u_i))^2] .

Equation (<ref>) is a speed limit inequality for the desired connection between two NESSs of our system. Since the right hand side only vanishes for (k_f=k_i,u_f=u_i), this speed limit hints at the existence of a minimum time for the connection. This is the problem we will address in the following. 






§  CONTROL PROBLEM FOR THE BRACHISTOCHRONE. MAIN RESULTS 


Let us pose the following control problem. Given an initial NESS of the BG, characterised by the control parameters (k_i,u_i), and a final NESS, characterised by the parameters (k_f,u_f), we aim at finding the optimal protocol (k^*(t),u^*(t)) that provides the fastest connection between them. In other words, we are interested in obtaining the protocol that connects two arbitrary NESSs in the shortest time t_f. 



 §.§ Dimensionless variables. Control set. Natural relaxation time



For our analytical calculations, it is handy to introduce dimensionless variables as follows:

    z_1,2≡k_iq_1,2^2/k_B(T_x + T_y),    z_3 ≡k_iq_1 q_2/k_B(T_x - T_y),    k^* ≡k/k_i,    u^* ≡u/k_i,    t^* ≡k_i/γt.

We are assuming that T_x T_y, which is the relevant situation for our purposes: if T_x=T_y=T,  the initial and final states would be equilibrium states at the common temperature T. From now on, asterisks are dropped not to clutter our formulas.  The z variables in Eq. (<ref>) correspond to the dimensionless relevant moments for the normal modes. 

In dimensionless variables, the evolution equations corresponding to Eq. (<ref>) are

    dz_j/dt = f_j(z,ω) ≡ -2ω_j z_j + 1,

where we have defined

    ω_1 = k + u,   ω_2 = k - u,   ω_3 = k.

We have introduced the compact notation z for the moments of the normal modes, such that z^T≡ (z_1,z_2,z_3). Equation (<ref>) is analogous to the dynamical equation for the variance of a one-dimensional Brownian particle confined in a harmonic trap, with each component ω_j of ω (with ω^T≡ (ω_1,ω_2,ω_3)) playing the role of an effective stiffness of the trap for each relevant moment z_j; in fact, equilibrium states fulfill z_j,=(2ω_j)^-1. We must highlight here an interesting feature: due to the linearity of the dynamical equations, regardless of the chosen protocol (k(t),u(t)) for the control variables, the time evolution of the system does not depend at all on the values of the bath temperatures T_x and T_y, when working with dimensionless variables.






In terms of the dynamical variables z_j, we want to drive our dynamical system (<ref>) with initial condition z_j(0) = (2ω_j,i)^-1 towards a final state with z_j(t_f) = (2ω_j,f)^-1, with ω_j,i≡ω_j(0) and  ω_j,f≡ω_j(t_f), in the fastest way. The connection is made by controlling the time dependence of (k,u), keeping the potential confining for all times. Therefore, we have the non-holonomic constraints

    -k ≤ u ≤ k,    0 ≤ k ≤ k_max.

The inequalities -k≤ u≤ k, equivalent to k^2-u^2≥ 0, and k≥ 0 ensure that the potential is confining. The additional inequality k≤ k_max accounts for the range of possible values of the strength of the confinement, stemming from experimental limitations on the amplitude of an externally induced electric or magnetic field, or the intensity of optical tweezers. In the usual terminology of optimal control theory <cit.>, Eq. (<ref>) defines the control set, i.e. the region in parameter space where all admissible controls (k(t),u(t)) must lie. In our case, we have the triangular control set depicted in Fig. <ref>. 


On general grounds, we expect that the fastest SST protocols achieve the desired connection in a finite time. This entails a huge improvement with respect to the direct STEP process, which consists in switching the control parameters from (1,u_i) to (k_f,u_f) instantaneously at t=0^+ and letting them constant afterwards. The relaxation to the final NESS corresponding to (k_f,u_f) is exponential in the relevant moments, Eq. (<ref>) tells us that the moment z_j has a characteristic relaxation time τ_j≡ (2ω_j,f)^-1.
 Therefore, we can define the characteristic relaxation time t_ for the STEP process of the BG as

    t_ = t_(k_f, u_f) = max[1/2(k_f-u_f),1/2(k_f+u_f)] = 1/2(k_f-|u_f|),

which gives the relaxation timescale for the slowest mode. From an experimental point of view, it is attractive to implement protocols that beat the natural timescale for relaxation, which one may estimate as 3  t_[An exponential relaxation of the form e^-t/t_ is completed up to 95% after t = 3t_.]; from a theoretical point of view, the system only reaches the target NESS for infinite time in the STEP process.



 §.§ Pontryagin's Maximum Principle for the time-optimal problem



Pontryagin's Maximum Principle (PMP) provides necessary conditions for the optimal protocols (k^*(t),u^*(t)) that minimise a certain functional of the system variables and the controls. For the brachistochrone, also known as the time-optimal control problem <cit.>, the functional to be minimised is 

    J[k,u] ≡∫_0^t_fdt  f_0(z,ω)_=1 ,

which corresponds to the total time for the protocol. The functional dependence on (k,u) takes place through the effective stiffnesses ω defined in Eq. (<ref>). For the SST connection, we have the following boundary conditions for the control and the variables


    (k(0),u(0)) = (1,u_i),       (k(t_f),u(t_f)) = (k_f,u_f), 
       z_j(0) = 1/2ω_j,i,       z_j(t_f) = 1/2ω_j,f,      j = 1, 2, 3.
   

This minimisation has to be done while keeping (k(t),u(t)) in the control set, i.e. (k,u) must fullfill the non-holonomic constraints (<ref>) for all times. Now, we define an additional variable z_0, such that  z_0(0) = 0 and 

    ż_0 = f_0(z,ω)   z_0(t_f) = ∫_0^t_fdt  f_0(z,ω) = t_f.

Next, we introduce the conjugate momenta (ψ_0,ψ^T) ≡ (ψ_0,ψ_1,ψ_2,ψ_3) and Pontryagin's Hamiltonian as

    Π(z,ψ_0,ψ,k,u)     = ψ_0 f_0(z,ω) + ψ^Tf(z,ω)=ψ_0+∑_j=1^3 ψ_j f_j(z,ω) 
       = ψ_0+ψ_1+ψ_2+ψ_3- 2 k (ψ_1 z_1+ψ_2 z_2+ψ_3 z_3) - 2 u (ψ_1 z_1 -ψ_2 z_2 ),

where f^T(z,ω) ≡ (f_1(z,ω),f_2(z,ω),f_3(z,ω)) is the right hand side of the dynamic equations (<ref>). Note that the Hamiltonian does not depend on z_0 by construction. 



Pontryagin's Maximum Principle (PMP) for our time-optimal problem states the following: let (k^*(t),u^*(t)) be the optimal control and z^*(t) the corresponding trajectory of the system variables. There exist a time-dependent vector ψ^*(t) and a constant ψ_0^*≤ 0 satisfying (ψ_0^*,ψ^T) (0,0,0,0) for all t∈(0,t_f) and having the following properties:

    
  * The variables z and their conjugate momenta ψ obey the canonical system
    
    
    ż_j    = ∂Π/∂ψ_j = f_j(z,ω)=-2ω_j z_j+1, 
        
    ψ̇_j    = - ∂Π/∂ z_j = 2ω_j ψ_j,

    
    for j=1,2,3, with the boundary conditions (<ref>).
    
  * For each fixed t∈(0,t_f), the Hamiltonian attains a global maximum at (k,u)=(k^*,u^*) as a function of the controls (k,u), i.e.
    
    Π(z^*(t),ψ_0^*,ψ^*(t),k^*(t),u^*(t))≥Π(z^*(t),ψ_0^*,ψ^*(t),k,u),

    for all (k,u) in the triangular control set defined by Eq. (<ref>).
    
  * Π(z^*(t),ψ_0^*,ψ^*(t),k^*(t),u^*(t))=0 for all t∈(0,t_f).



Some comments are in order: for the time-optimal problem, it is not necessary to write the evolution equations for z_0 and ψ_0, which also follow the canonical form, since

    ż_0=∂Π/∂ψ_0=1,   ψ̇_0=- ∂Π/∂ z_0=0

do not provide additional information—the statement of PMP already states that ψ_0 is constant, and the equation for z_0 simply tells us that z_0=t. For twice differentiable protocols, PMP is equivalent to solving the control problem using the ordinary tools of variational calculus <cit.>. In such scenario,  Pontryagin's Hamiltonian function can be mapped to a Lagrangian function that includes one Lagrange multiplier (the momenta ψ_j) for each of the time-dependent constraints given by the dynamical equations (<ref>) for the variables z_j. This variational framework gives rise to the corresponding Euler-Lagrange equations, which are equivalent to the canonical system of PMP. The advantage of PMP relies on the fact that it extends the variational framework by including the possibility of (i) introducing non-holonomic constraints, in the form of inequalities such as those we have in Eq. (<ref>), and (ii) considering less regular control functions, which in PMP's approach can have finite-jump discontinuities <cit.>. Neither (i) nor (ii) can be tackled with the simpler approach of variational calculus.

PMP entails that we can classify the time-optimal protocols into three distinct categories:

    
  * Euler-Lagrange protocols: They  correspond to situations for which the maximum of Π is attained inside the control set, so it is necessary that
    
    ∂Π/∂ k = 0,     ∂Π/∂ u = 0.

    These correspond to twice-differentiable functions that might also be determined via the variational calculus approach described above in the open interval (0,t_f). Still, within the PMP framework, sudden changes of the control parameters at both the initial and final times for the protocol, i.e. at t = 0^+ and t=t_f^-, respectively, are allowed.
    
  * Bang-bang protocols: They correspond to situations in which the maximum of Π is attained over the boundary of the control set, so that Eq. (<ref>) no longer holds. Specifically, we refer to a bang-bang protocol when the maximum is reached at the vertices of the control set, as explained below.  
    
    For the case of our concern, the boundaries of the control set correspond to the sides of the triangle depicted in Fig. <ref>. The specific point at which Π reaches its maximum value depends on the sign of the switching functions
    
    
    ϕ_OP(z,ψ)   ≡∂Π/∂ k + ∂Π/∂ u=-2(2ψ_1 z_1+ψ_3 z_3), 
            
    ϕ_ON(z,ψ)    ≡∂Π/∂ k - ∂Π/∂ u=-2(2ψ_2 z_2+ψ_3 z_3), 
            
    ϕ_NP(z,ψ)    ≡∂Π/∂ u=2(ψ_2 z_2-ψ_1 z_1),
   
    
    which give the component of ∇Π along the lines OP, ON, and NP, respectively, where O≡ (0,0), P≡ (k_max,k_max) and N≡ (k_max,-k_max) are the vertices of the triangle. The sign of ϕ_AB gives the direction in which Pontryagin's Hamiltonian Π increases, ϕ_AB>0 means that Π increases from A to B. Note that (i) the switching functions only depend on z and ψ, and not on (k,u), because the control functions enter linearly in Pontryagin's Hamiltonian, and (ii) the possible signs of the switching functions are restricted by the relation ϕ_OP-ϕ_ON=2ϕ_NP.
    
    At any given time, the sign of the switching functions will determine which of the three vertices of the control set gives the maximum value of Π. Specifically, for ϕ_OP<0 and ϕ_ON<0, the maximum is attained at  O, for ϕ_OP>0 and ϕ_NP>0,  the maximum is attained at  P, and for ϕ_ON>0 and ϕ_NP<0, the maximum is attained at  N. 
    Let us note that the sign of two switching functions determine which vertex of the triangle is chosen: for visualisation purposes, this is schematically represented in Fig. <ref>. The times at which these switching functions change their sign determine the switching times between the different vertices during a bang-bang protocol.  
      
    
  * Singular protocols: The above picture for bang-bang protocols breaks down if (at least) one of the switching functions in Eq. (<ref>) vanishes during a finite time interval [t_1,t_2] with 0≤ t_1 < t_2 ≤ t_f. We will refer to these protocols as singular protocols: any point along the singular branch—i.e. the branch of the control set over which the corresponding switching function identically vanishes in a finite interval—is a candidate for the optimal protocol, since Pontryagin's Hamiltonian is constant over the singular branch. Similarly to the bang-bang case, Fig. <ref> depicts a schematic representation of the three possible singular situations. 
    


 We must emphasise that, although our dynamical system is linear in the control variables, and thus so it is Pontryagin's Hamiltonian function, our system does not belong to the class of what is called a “linear system” in optimal control theory. Therein, for dynamical variables z and control χ,  a system is called linear if the dynamic equations have the form

    ż = 𝔸z + 𝔹χ,

where 𝔸 and 𝔹 are two matrices  of adequate dimensions with constant elements. It is for this kind of “linear systems” that there are theorems ensuring that the optimal-time control problem has a solution of bang-bang type, with as many bangs as dynamical variables z_j <cit.> [This was the case for the thermal brachistochrone of a system of oscillators in dimension d, in contact with a unique heat bath with controllable temperature T(t) <cit.>.]. 


In the BG, we do not have this structure, due to the products k z_j and u z_j in the dynamical equations (<ref>)—which are sometimes called bilinear <cit.>. Therefore, protocols belonging to any of the three categories above, Euler-Lagrange, bang-bang, and singular, are candidates for the brachistochrone. Also, optimal-time protocols could also combine different categories, concatenating time intervals corresponding to different classes of protocols. For example, we could have an optimal protocol starting as an Euler-Lagrange solution, then becoming bang-bang at some time t_1 ∈ (0,t_f), and then  becoming singular at some other time t_2 ∈ (t_1,t_f). In principle, PMP does not exclude these combinations for a general problem, so we need to thoroughly study the behaviour of the dynamical system and their conjugate variables in order to check which category combinations are possible in the BG's brachistochrone. 






 §.§  Main results


We now put forward the key findings stemming from the application of PMP to the time-optimal problem for the BG. In such presentation, we relegate the mathematical details to the appendices, since here we focus on emphasising the main underlying physical ideas. 

To start with, there is no brachistochrone of Euler-Lagrange type nor mixed protocols involving Euler-Lagrange time intervals.  Therefore, time-optimal protocols must always lie at the boundaries of the control set. As already stated, this is not a direct consequence of the linearity of  Pontryagin's Hamiltonian in the control functions (k,u). A detailed analysis of the possible solutions from Eq. (<ref>) is required in order to prove this statement, since the BG does not belong to the class of “linear-Pontryagin” systems, for which there are theorems ensuring the absence of Euler-Lagrange time-intervals <cit.>. The corresponding proof for the inexistence of Euler-Lagrange brachistochrones in the BG can be found in Appendix <ref>.

Optimal protocols achieving the fastest connection between NESSs can thus be split into two classes:

    
  * The first class of time-optimal protocols correspond to “pure”, regular, bang-bang protocols. The term “pure” reflects that these bang-bang protocols do not mix with Euler-Lagrange protocols nor with singular protocols. In this case, the bang-bang protocol extends to the whole interval (0,t_f). 
    
    Bang-bang protocols emerge as a combination of different time windows where the control parameters (k,u) switch between the values corresponding to the points O, P, and N. In principle, there are infinitely many ways of combining these points in order to construct a bang-bang protocol. However, as we show later in Sec. <ref>, bang-bang protocols involving three time windows suffice for connecting any two NESSs.

    
    
  * The second class of time-optimal protocols correspond to “pure” singular protocols, which, again,  extend to the entire time interval (0,t_f). The vanishing of one of the switching functions in the whole interval entails that there are infinitely many optimal protocols (k^*(t),u^*(t)) along such a branch, which satisfy the necessary conditions provided by PMP—see Appendix <ref> for details. 



In the remainder of the paper, we mainly work with regular bang-bang protocols, which can be completely determined using the tools provided by PMP. We will analyse the behaviour of the minimum connection time along these candidates to brachistochrone, as a function of the initial and target states—i.e. (1,u_i) and (k_f,u_f).  Similarly to what we did before with the dimensionless variables, in the following, we drop the asterisks for the optimal control functions and associated optimal variables and conjugate momenta to simplify our notation, since from now on all results refer to optimal protocols.





§  CONSTRUCTING OPTIMAL BANG-BANG PROTOCOLS


We devote this section to building optimal bang-bang protocols with different numbers of bangs—i.e. different number of switchings between the vertices O, P, N of the control set. Here, we use the general notation V to refer to one of these vertices.  We will show that (i) one-bang protocols only allow for connecting the initial point I≡ (1,u_i) with some isolated points (k_f,u_f) within the control set, (ii) two-bang protocols allow for connecting I with some specific curves in the (k_f,u_f) plane, (iii) three-bang protocols allow for connecting I with all the points of (k_f,u_f) within the control set. 



As the controls (k,u) take constant values at any of the vertices, the evolution equations (<ref>) can be analytically solved. Let us assume that the switching to a certain vertex V occurs at a certain time t_0, for which z_j(t_0)=z_j,0, and denote by ω_j, V the value of the j-th effective stiffness at the considered vertex; the time evolution of z_j is given by 

    ℰ^(τ)_ω_j,V(z_j,0) ≡ z_j(t) = 1/2ω_j, V + (z_j,0-1/2ω_j, V)e^-2τω_j, V,

with τ≡ t - t_0≥ 0. We have defined the operator ℰ^(τ)_ω_j,V(z_j,0), which generates the time evolution of the considered moment z_j during a time interval τ, with constant driving ω_j, V. This operator has a well-defined  limit for ω_j, V→ 0,

    ℰ_0^(τ)(z_j,0) = lim_ω_j, V→ 0ℰ^(τ)_ω_j,V(z_j,0) = z_j,0 + τ,

z_j linearly increases with time. This will be useful for our analysis, since at each of the vertices (O,P,N) at least one ω_j, V vanishes and then the corresponding moment linearly increases.  In fact, at the origin O we have  ω_j,O=0 and thus ℰ^(τ)_ω_j,O=ℰ_0^(τ), ∀ j.

A  bang-bang protocol V_1 V_2 V_3…, where V_1 V_2 V_3…is any permutation of the vertices of the control set, such that V_α+1V_α, is obtained by sequentially applying the time evolution operators corresponding to each of the vertices V_α during a certain time window τ_α. The total duration of the bang-bang protocol is thus given by t_f=∑_ατ_α, and the final value of the variables z_j is 

    z_j(t_f) = ( ⋯∘ℰ_ω_j,V_3^τ_3∘ℰ_ω_j,V_2^τ_2∘ℰ_ω_j,V_1^τ_1)_composition of  M  operators (z_j,i),    j=1,2,3.

The mathematical task ahead is looking for a consistent choice of the sequence V_1 V_2 V_3…with the right values of the corresponding time intervals τ_1, τ_2, τ_3, …to satisfy the boundary conditions for the moments (<ref>), i.e. to drive the system from the initial NESS with z_j(0)= (2ω_j,i)^-1 to the target NESS with z_j(t_f)= (2ω_j,f)^-1.

In order to simplify our analysis, we  consider the limit k_max→ +∞, i.e. the limit of having infinite capacity for compression—we recall that the constraint k≤ k_max was of practical nature, whereas the constraints k≥ 0 and k^2-u^2≥ 0 ensure that the potential is confining for all times. The infinite capacity for compression entails vanishing times τ→ 0 for the windows corresponding to the points P and N, as they involve exponential decays for the dynamic variables with characteristic timescales of the order of 1/k_max. Therefore, k_maxτ must remain finite, and this makes it possible to introduce a parameter ξ that substitutes τ for determining the optimal control. Specifically, we make 

    k_maxτ = -1/2lnξ ⇒ ξ = e^-2k_maxτ,

with ξ∈ [0,1] being the quenching factor corresponding to the relaxing dynamic variable. 
Let us consider the vertex P, thereat we have ω_1,P=2 k_max, ω_2,P=0, ω_3,P=k_max. Therefore, for k_max→∞, while keeping ξ constant, one has the following instantaneous evolutions, from t_0 to t_0^+=t_0+τ:

    z_1(t_0^+)= z_1,0 ξ^2 ,    z_2(t_0^+)= z_2,0,    z_3(t_0^+)=z_3,0 ξ.

If we define the operator

    ℰ̃_ξ(z_j,0) ≡ z_j,0ξ,

the above instantaneous evolutions can be rewritten as

    z_1(t_0^+)= ℰ̃_ξ^2(z_1,0) ,    z_2(t_0^+)= ℰ̃_1(z_2,0),    z_3(t_0^+)=ℰ̃_ξ(z_3,0).

For the vertex N, the situation is completely analogous, but the evolutions for z_1 and z_2 are exchanged because ω_1,N=0=ω_2,P and ω_2,N=2k_max=ω_1,P.

Bang-bang controls in the limit k_max→∞ are constructed analogously to Eq. (<ref>), but with the operators ℰ_ω_j,V^τ, with V being either P or N, being substituted with the corresponding operator from Eq. (<ref>). Our discussion above entails that we thus have a concatenation of operators ℰ_0^(τ), corresponding to the vertex O, and operators ℰ̃_ξ^2, ℰ̃_ξ, and ℰ̃_1, corresponding to the vertices P or N, in the time evolution of the moments.






  §.§.§  One-bang protocols


This is the most basic case, for which the optimal protocol remains at one of the vertices, O, P or N, during the entire time interval (0,t_f). In the following, we analyse the evolution corresponding to each of the three vertices separately. 

We start our analysis with the vertex O, for which ω_j,O=0 for all j. Therefore, we have 

    z_j(t_f) = 1/2ω_j,f =ℰ_0^(t_f)(1/2ω_j,i) = 1/2ω_j,i + t_f,    j=1,2,3.

Note that the above corresponds to an algebraic system of three equations with just one variable; the connection time. Thus, it will only be consistent for specific choices of both the initial and final conditions. Recalling the definition of the ω_j's, Eq. (<ref>), the above system presents the following solutions: (i) t_f = 0 for (k_f,u_f) = (1,u_i)—i.e. the initial and final states are the same, ad thus the system does not evolve, (ii) t_f → + ∞ for (k_f,u_f) = (0,0)—i.e. the target stationary state becomes the point O, which is reached in an infinite amount of time, as the optimal protocol becomes a direct step protocol, and (iii) t_f = k_f^-1 - 1 for u_f = u_i = 0 and k_f < 1. The latter case corresponds to connecting uncorrelated states of the BG, for which the dynamic variables degenerate into a unique one.

Now, me move onto the analysis of a one-bang protocol at the vertex P. In this case, we have the instantaneous evolution found in Eq. (<ref>),

    
    z_1(0^+)    = 1/2(k_f+u_f)=ℰ̃_ξ^2(1/2(1+u_i)) = ξ^2/2(1+u_i), 
    
            z_2(0^+)    = 1/2(k_f-u_f) = ℰ̃_1(1/2(1-u_i)) = 1/2(1-u_i),
            
    
            z_3(0^+)    = 1/2k_f = ℰ̃_ξ(1/2) = ξ/2.


Note that t_f=0 for this protocol, in the limit k_max→∞ we are considering. In the above, we have taken into account the explicit expressions for the initial and final values of the moments, as given by Eq. (<ref>) with the stiffnesses ω_j of Eq. (<ref>). Equation (<ref>) presents the following solutions: (i) ξ = 1 for (k_f, u_f) = (1, u_i)—i.e., again, the initial and final states are the same, and (ii) ξ = (1+u_i)/(1-u_i) for u_i<0, which corresponds to a final point P^* with coordinates

    k_P^*=1-u_i/1+u_i,      u_P^*=-u_i  k_P^*,    u_i<0.

The point P^* is the only non-trivial NESS, i.e. different from the initial state, reachable with a one-bang protocol at vertex P. We recall that |u_i|≤ k_i=1. 

Lastly, the one-bang N protocol is completely analogous to the P one; we have the same algebraic equations as in Eqs. (<ref>), but upon the variable changes z_1 ↔ z_2, u_i,f↔ -u_i,f. Thus, the corresponding solutions in this case are (i) the trivial solution ξ = 1, i.e. again (k_f, u_f) = (1, u_i), and (ii) ξ = (1-u_i)/(1+u_i) for u_i > 0,  which corresponds to a final point N^* such that

    k_N^*=1+u_i/1-u_i,      u_N^*=-u_i  k_N^*,    u_i>0.

Note that both P^* and N^* tend to the initial point in the limit u_i→ 0, i.e. for an uncorrelated initial state.

The symmetry between the P and N protocols will be further exploited when studying higher-order bang-bang protocols in the forthcoming sections. 




  §.§.§  Two-bang protocols


Optimal two-bang protocols involve a switching time t_1∈ (0,t_f) that splits the whole time interval into two different time windows, in which the control parameters correspond to one of the vertices, i.e. O, P, or N. Permutations of two points avoiding consecutive repetition give rise to six possible protocols: OP, ON, PO, NO, PN and NP. These protocols involve the action of two operators of the type (<ref>) or (<ref>) on each of the
dynamic variables. 

The two-bang protocols involve the action of two successive evolution operators, which in turn introduce two unknowns—(t_f,ξ) for the protocols OP, ON, PO and NO, and (ξ_1,ξ_2) for the protocols PN and NP. Since the boundary conditions for the target state give us three algebraic equations, one for each z_j,f, there will be solutions for the unknowns only for specific choices of the initial and target NESSs. However, for a given fixed initial condition, there will be curves of the form f(k_f,u_f)=0 consistent with each of the algebraic equations, instead of the isolated points found in the one-bang case. The analytical expressions for these curves are rather cumbersome and not especially illumating, thus we relegate them to Appendix <ref>.


Figure <ref> shows the curves in the (k_f,u_f)-plane that are reachable with two-bang protocols, for two different choices of the initial condition u_i. First of all, it is worth highlighting that we have fixed the initial condition u_i to be non-negative, without loss of generality. Due to the symmetry between the P and N vertices, the figure for u_i<0 would be obtained by reflecting the figure for u_i>0 through the u_f=0 axis, with the exchange N↔P in all protocols. This symmetry is particularly evident in the u_i = 0 case, whose figure is symmetric with respect to the u_f = 0 axis. We have also chosen the color code of each curve for the same reason: for example, the blue curves, corresponding to the protocols OP and ON, are exchanged upon the flipping of the figure, and they merge for the case u_i = 0. In fact, they become a one-bang protocol at vertex O, since for u_i=u_f=0, the operation of P or N during the two-bang protocol becomes the identity. The same colour code applies for the purple curves, corresponding to the protocols PO/NO, and the orange ones, accounting for the PN/NP protocols. Moreover, we have also marked with red dots the points reachable with a one-bang protocol, described in Sec. <ref>. On the left panel, we have the origin O, the initial state (1, u_i), and the point N^*, while on the right panel we also have the origin O, but the initial state and the point N^* have merged into a unique point. We note that these points constitute limiting values for all the two-bang protocols: OP starts from the origin O and ends at the initial state I, ON starts at O and ends at the point N^*, both PO and PN/NP start at I, and both NO and PN/NP start at the point N^*. These are not coincidences: the one-bang solutions must belong to a subgroup of the two-bang ones, they are only attained when (at least) one of the operators corresponding to one of the time windows of the considered two-bang protocol becomes the identity operator. For instance, let us consider the two-bang protocol ON. When the operator corresponding to the N time window becomes the identity, the ON protocol reduces to the O protocol that makes it possible to reach the origin; when the operator corresponding to the O time window becomes the identity, the ON protocol reduces to the N protocol that makes it possible to reach the point N^*.






  §.§.§  Three-bang protocols


The last case we look into is three-bang optimal protocols, which split the time interval (0,t_f) into three time windows with two switching times. In each time window, the controls (k,u) attain the values corresponding to one of the three vertices: O, P, and N. Permutations of these points—eliminating those having two identical consecutive vertices—give rise to twelve three-bang protocols: OPO, POP, ONO, NON, OPN, ONP, PON, NOP, PNO, NPO, PNP, NPN. For the sake of brevity, as we did in the previous section, the specific algebraic systems of equations associated to these candidates to optimal-time protocol are not given here—see Appendix <ref> for details. As clearly observed there, the symmetry between P and N entails that many of the aforementioned three-bang protocols have the same algebraic systems; specifically the pairs (OPN,ONP), (PNO,NPO), and (PNP,NPN). Thus, they allow to reach the same final states with the same connection time, so in practice we only need to deal with nine different three-bang protocols. Although the majority of these protocols constitute regular bang-bang protocols—i.e. their switching times are completely determined via the switching functions from Eq. (<ref>), the protocols POP and NON constitute pure singular bang-bang protocols, as proven in Appendix <ref>. 

As was the case for one- and two-bang protocols, the evolution equations provide three algebraic equations from the three boundary conditions at the final time for the moments z_j,f in the target NESS. Since the three-bang protocols introduce three unknown parameters, this algebraic system provides a solution for them and thus univocally characterises the three-bang protocols. However, the fact that the “quenching” parameters for the P and N vertices must verify 0≤ξ≤ 1 and that the lengths τ of the time windows for the O vertex must verify τ≥ 0 impose restrictions that make each V_1V_2V_3 protocol reach a certain region of the (k_f,u_f) plane—given an initial condition u_i. 

Figures <ref> and <ref> corroborate the insights above for the same values of the initial condition considered in Fig. <ref>: u_i = 0.5 for Fig. <ref>, and u_i = 0 for Fig. <ref>. In both figures, we plot the (k_f, u_f) regions attained via three-bang protocols where the connection time is minimum. This is an important clarification, since there are several regions corresponding to different three-bang protocols that intersect, as it is the case 
for the NOP and ONP/OPN protocols in Fig. <ref>, where the NOP region (green) also belongs to the ONP/OPN one (cyan), but NOP becomes the optimal protocol in that common region because its connection time is shorter. That is the main reason why the PNO/NPO protocols neither appear in Fig. <ref> nor in Fig. <ref>: they are always suboptimal, i.e. give longer connection times, with respect to other intersecting bang-bang protocols.



The curves corresponding to two-bang protocols from Fig. <ref> play for three-bang protocols a role similar to that played by the points corresponding to one-bang protocols for two-bang ones: the curves from  Fig. <ref> delimit the boundaries of the regions reachable by the different three-bang protocols. This is again logical: two-bang protocols are a limiting case for three-bang protocols [In terms of the algebraic systems of equations displayed in Appendix <ref> for the three-bang protocols, which are characterised by three variables, it is equivalent to the case in which at least one of such variables attains its lower or upper bound—if such variable corresponds to the length of a time window τ, its lower bound would be τ = 0, while if it corresponds to a quenching parameter ξ, its lower and upper bounds correspond to ξ = 0 and ξ = 1, respectively.]. However, there are additional dot-dashed curves that also delimit regions reached by three-bang protocols and do not correspond to two-bang protocols. At these dot-dashed curves, the algebraic systems for the three-bang protocols cease to have physical solutions because one (or more) of the unknown parameters become complex. 


It is worth emphasising that three-bang protocols already make it possible to reach any point in the (k_f, u_f)-plane—thus any target NESS of the BG— within the boundaries of the control set from Eq. (<ref>). In principle, we could keep constructing higher-order-bang protocols, but on a physical basis one expects the three-bang protocols to give the minimum connection time, since they comprise the minimum number of time windows. In this sense, it must be stressed that we have not been able to found shorter connections with a higher number of bangs [Still, from a strict mathematical viewpoint, we cannot assure that higher-order-bang protocols are suboptimal against our three-bang solutions.]


We devote the forthcoming section to the detailed study of the connection time for the optimal three-bang protocols presented in this section. Interestingly, these “simple” protocols already beat the characteristic relaxation timescale of the dynamics for certain sets of initial and final states.









§  MINIMUM CONNECTION TIME AS A FUNCTION OF THE TARGET STATE



In the previous section we have shown that, given a fixed initial state I=(1,u_i) in our dimensionless variables, we are able to reach any final NESS of the BG, characterised by one point in the (k_f,u_f)-plane within the control set, by means of a three-bang protocol. Now, we turn our attention into the original matter of concern, which is the minimum connection time between two NESSs. Therefore, we study here the behaviour of the connection time over the aforementioned three-bang protocols, candidates to be the brachistochrone, as a function of the target state.


Figure <ref> shows density maps for the connection time t_f over each of the optimal three-bang protocols built in Sec. <ref>. The behaviour of the minimum connection time t_f radically changes from the u_i ≠ 0 case to the u_i = 0 case. The most remarkable change refers to the final states lying on the line u_f = 0. On the one hand, for u_i = 0, points lying on the u_f = 0 line are reached in (i) a finite time for k_f < 1 and (ii) in a null connection time for k_f > 1 [If k_max were finite, the corresponding minimum connection time would be of the order of 1/k_max.].  The case u_i=u_f=0 is equivalent to connecting equilibrium states of two uncoupled oscillators with different bath temperatures T_x T_y, for which the minimum connection time is explicitly derived in Appendix <ref>. On the other hand, for u_i ≠ 0, states lying in the line u_f = 0 can only be reached in an infinite time. In Appendix <ref>, we show that

    t_f ∝ u_f^-2,      |u_f| ≪ 1,    u_i ≠ 0,

which correspond to the relevant OPN/ONP three-bang protocols for |u_f| ≪ 1.




The above has striking implications. From a physical standpoint, if we initially start from a state in which the two coordinates of the BG are uncoupled, u_i = 0, which is equivalent to having two independent oscillators at equilibrium with their respective baths at temperatures T_x and T_y, any NESS of the BG, with u_f 0 can be reached in a finite time. However, had we started from an initial NESS with u_i 0, it would take an infinite amount of time to reach any uncoupled final state with u_i=0. Hence, the candidates OPN/ONP to optimal-time SST protocol  would present no advantage with respect to the direct STEP protocol in this case. Recalling Eq. (<ref>), the relaxation time for the STEP process for u_f=0 results to be t_(k_f,0) = (2k_f)^-1, which is finite. 

Apart from the u_f = 0 line, the connection times also diverge along the u_f = ± k_f lines, which bound the control set of admissible controls, both for u_i=0 and u_i 0. Similarly to the previous analysis carried out for the u_f = 0 line, asymptotic expressions for the minimum connection time along the lines u_f = ± k_f can be derived from the algebraic systems that we explicitly write in Appendix <ref>. Specifically, along the u_f = k_f line, the OPO protocol gives

    t_f(k_f,u_f) = 1/2(k_f - u_f) - 1/2(1-u_i)∼1/2(k_f - u_f),      k_f-u_f ≪ 1-u_i

while for the u_f = -k_f line, the ONO protocol gives

    t_f(k_f,u_f) = 1/2(k_f + u_f) - 1/2(1+u_i)∼1/2(k_f + u_f),    k_f+u_f ≪ 1+u_i.

Let us note that the minimum times in Eqs. (<ref>) and (<ref>) have the same divergence of the characteristic relaxation time for the STEP process in Eq. (<ref>), which entails this property to be a consequence of the physical bounds of our model: the fact that k_f^2 - u_f^2 > 0 for the system to attain a NESS.

In order to improve our understanding on the global landscape of minimum connection times, several sections of Figs. <ref> for  different fixed values of k_f are plotted in Fig. <ref>. Again, this is done for the two choices of initial NESSs we have repeatedly employed: u_i = 0.5 and u_i = 0. Recall that, due to the symmetry between the P and N vertices, the results corresponding to the u_i = 0.5 would be equivalent to those for u_i = -0.5, upon reflection through the u_f = 0 axis. In Fig. <ref>, the solid lines correspond to the the minimum connection times already shown in Figs. <ref>, for the considered value of k_f. The background colour follows the same colour code as in Figs. <ref> and <ref>, thus showing which three-bang protocol the minimal time connection belongs to. The red dashed lines correspond to three times the characteristic relaxation time t_ for the STEP process as defined in Eq. (<ref>), thus allowing us to quantify the efficiency of the minimum-time protocols in beating the STEP one. 



As a general trend, the minimum connection time decreases with k_f. This is reasonable from a physical point of view: high values of k_f can only be reached by employing bang-bang protocols in which the P and N time windows dominate the dynamics over the O one. Since the former involve vanishing—i.e. of the order of 1/k_max— time windows, the resulting minimum connection time is small. 

The choice for the three values of k_f displayed in Fig. <ref> has not been arbitrary. For u_i ≠ 0 (left panels), three qualitatively different behaviors are observed for the connection time. For k_f < k_i=1 (top left), minimum connection times are non-zero and finite—with the exception of the asymptotic divergences when approaching u_f = 0 and u_f = ± k_f. For 1<k_f<k_N^* (left middle), where N^* is the only point reached by a one-bang N protocol—see Fig. <ref>,  the upper PN/NP line in the left panel of Fig. <ref> comes into play with a zero connection time. Lastly, for k_f>k_N^* (left bottom), the lower branch of the PN/NP line comes into play, which also involve a vanishing t_f—providing a second instantaneous brachistochrone. Recall that the PN/NP line of Fig. <ref> describes the set of final NESS attainable with a PN/NP two-bang protocol,


We may understand the difference between the phenomenology above for the cases k_f<k_i=1 and k_f>k_i=1 on a physical basis. Let us start by considering k_f < 1: this requires a decompression to attain a NESS with a target value of the stifness that is smaller than the initial one. Then, regardless of the specific target value of k_f, the corresponding bang-bang optimal protocol must include a time window in the O vertex, and therefore the connection time is non-vanishing. Next, let us move onto the case 1≤ k_f < k_N^*: there appear states with u_f>0 reachable by instantaneous—in the limit k_max→∞—two-bang PN/NP protocols, with the P window being the dominating one, because we are considering u_i>0 and u_P=k_max>0. These are the states over the positive branch, with u_f>0, of the PN/NP line. Finally, for k>k_N^*, there appear states with u_f<0 reachable by instantaneous PN/NP protocols, with the N window being dominating now. This is reasonable: N^* is the only point that can be reached from the initial one with a one-bang N protocol, so states close to it over the negative branch, with u_f<0, of the PN/NP line are expected to be dominated by the N time window.

The central OPN/ONP region, which includes u_f=0, in the left panels of Fig. <ref> deserves further attention, as it presents interesting features. First, the minimum connection time t_f diverges for u_f→ 0, which implies that, from a practical perspective, optimal bang-bang protocols become inefficient with respect to the direct STEP protocol, because the 3t_ curve lies below t_f over most the  region OPN/ONP. Second,  the minimum connection time t_f presents discontinuities at the boundaries of the region OPN/NOP: on the left, for u_f<0, when the region NON emerges; on the right, for u_f>0, when the region NOP emerges.  Such discontinuities stem from the emergence of the NON and NOP regions, since the OPN/ONP protocols continue to exist in such regions but give longer connection times than the NON and NOP protocols. Therefore, the latter become the optimal ones.


The right panels of Fig. <ref>, for which u_i = 0, are simpler to understand. There are fewer three-bang regions stemming from the scheme in Fig. <ref>, since the existing OPN/ONP regions for u_i 0 disappear in the case u_i=0: as we have already commented, this vanishing stems from the degeneration of the PN/NP lines into a unique line u_f=0, k_f≥ 1, as seen in Fig. <ref>. Therefore, there are only two cases of interest: (i) k_f < 1, for which the minimum connection time is always non-zero, due to the presence of time window corresponding to the vertex O in the optimal protocols, and (ii) k_f > 1, for which the minimum connection time vanishes along the u_f = 0 line—i.e. the PN/NP line, with a vanishing time in the limit k_max→∞. In this case, the minimum connection time from optimal control systematically beats the characteristic relaxation time of the STEP protocol.

 In Appendix <ref>, we briefly discuss the bound for the minimum connection time stemming from the speed limit inequality (<ref>). Therein, it is shown that W_ linearly increases with k_max in the limit k_max→∞ we have considered to simplify our discussion. This entails that the bound given by Eq. (<ref>) becomes very loose, since it vanishes as k_max^-1. This behaviour is similar to the one found in other systems for thermal control, when the upper bound of bath temperature goes to infinity and the corresponding bang become instantaneous <cit.>.










§ SUMMARY AND DISCUSSION


A relevant question within the field of shortcuts, or SST, concerns the brachistochrone—i.e. the protocol involving the minimum connection time—between two states of a physical system. For systems at the mesoscopic level of description, the latter is significant from a theoretical point of view, since it brings together concepts from the field of non-equilibrium statistical mechanics, stochastic thermodynamics <cit.>, and optimal control theory <cit.>. From a practical perspective, it is relevant for the design of nanodevices, for which fluctuations are non-negligible and may affect their performance.

 In this work, we have tackled, both numerically and analytically, the above question for a paradigmatic model in non-equilibrium statistical mechanics: the Brownian Gyrator (BG) <cit.>. The SST connection we would like to build aims at connecting an initial NESS, corresponding to potential parameters (k_i,u_i), with a target NESS, with parameters (k_f,u_f) in the minimum possible time, while keeping the potential confining for all times. Therefore, the controls (k,u) belong to the triangular control set illustrated in Fig. <ref>, which has three vertices O=(0,0), P=(k_max,k_max), N=(k_max,-k_max), where k_max is the maximum value of the stiffness—typically, in experiments one cannot have a harmonic trap with an arbitrarily large value of the elastic constant. 


 A first main result of our paper is that the protocols providing the minimum time connection between two NESSs  are of bang-bang type, i.e. the controls (k,u) lie on the boundaries of the triangular control set of Fig. <ref>. In other words, the optimal time protocols do not have time windows with Euler-Lagrange solutions, for which (k,u) would belong in the interior of the control set. Bang-bang protocols are divided into two classes: (i) regular protocols, in which the maximum of Pontryagin's Hamiltonian is reached at the vertices of control set, and thus the controls switch between the values corresponding to the three vertices 0, N, and P of Fig. <ref>, and (ii) singular protocols,  Pontryagin's Hamiltonian Π is constant along one of the edges of the control set and the maximum of Π is reached precisely over that edge, being thus degenerate. To make analytical progress, we have considered mainly the limit k_max→∞ throughout the paper, since it simplifies the mathematical description.


 We have focused on three-bang regular protocols, i.e. protocols with three time windows, with each time window corresponding to the controls being given by one of the vertices of the control set. We have denoted them by the sequence of vertices involved: e.g. the protocol ONP means that the first time window corresponds to the vertex O, the second to the vertex N, and the third to the vertex P. These three-bang protocols (i) can be univocally determined as a function of the target state (k_f,u_f), for an arbitrary given initial state (k_i,u_i), and (ii) connect all possible initial states with all possible final states. Simpler one-bang and two-bang protocols only make the connection to specific target points and lines, respectively, of the control set. Over the optimal three-bang protocols, we have studied the behaviour of the final connection time t_f as a function of the target state (k_f,u_f), for a given initial state (k_i = 1,u_i). The connection time landscape radically changes from the u_i ≠ 0 to the u_i = 0 cases, with the former being much more complicated than the latter.

There are some connections that imply an infinite time: in particular, an infinite time is needed to reach the boundary lines of the control set k_f=± u_f. This was expected on a physical basis, since over these lines one of the characteristic relaxation times of the normal modes diverge. More surprising is the divergence of the connection time for reaching uncoupled target states, with u_f=0, from initial states that are coupled, with u_i 0, since all the characteristic relaxation times of the normal modes remain finite over the line u_f=0. From a physical point of view, this is related to needing an infinite time at vertex O to “kill” the correlations between the original variables (x,y) at the final state.


 There are some connections that are instantaneous—in the limit k_max→∞ we are considering, in which the time windows corresponding to the P and N vertices involve no time. Therefore, any bang-bang protocol containing only the vertices P and N give a null connection time. On the one hand, we have the one-bang P and N protocols, which only reach a specific point P^* and N^* of the control set, for u_i<0 and u_i>0, respectively. On the other hand, we have the two-bang PN/NP protocols, which reach two specific lines of the control set, as illustrated in Fig. <ref>. For the case u_i=0, the PN and NP lines merge into the {u_f=0,k_f>1} line [Three-bang PNP/NPN protocols reach the same lines as the PN/NP lines, see Appendix <ref> for details.]. Therefore, at variance with the u_i 0 case, in which the connection time at the u_f=0 line diverges, for u_i=0 not only are states along the u_f = 0 line reachable, but instantaneously reachable. 


 It must also be remarked that there appear discontinuities in the minimum connection time as a function of (k_f,u_f), for an initial state with u_i 0—i.e. an actual NESS in which the two degrees of freedom are coupled. We have illustrated these discontinuities in Fig. <ref>, in which we have plotted the minimum connection time as a function of u_f, for several fixed values of k_f. For k_f>1, a finite-jump discontinuity of t_f emerges when the optimal protocol changes from the central OPN/ONP region, which includes the u_f=0 line, to the NOP or NON regions [For the shown case u_i>0, for u_i<0 it would be the PON and POP regions.. In the NOP and NON regions, the OPN and ONP protocols continue to exist—and would give a continuous connection time—but the connection time for the NOP and NON protocols are shorter in their respective domains.].




This work also opens the door for future research. First, the speed limit inequality derived here for the connection between NESSs of the BG is analogous to others for connections between equilibrium states, which also involve the connection time and the irreversible work <cit.>. Also, speed limit inequalities based on thermodynamic geometry quantities like the Fisher information have been obtained for the connection between arbitrary states <cit.>, and applied to the connection between two NESSs of a uniformly heated granular system <cit.>. Still, it must be remarked that the thermodynamic geometry quantities in the latter speed limit inequalities have a less transparent physical meaning than the irreversible work. The possible generalisation of the speed limit inequality for the BG to a more general out-of-equilibrium system  thus deserves further research. 

Second, the current state of the art in optical trapping allows for the experimental implementation of the optimal protocols developed here for a BG, following the techniques employed in its recent practical realisations <cit.>. The fact that bang-bang protocols involve instantaneous switchings of the control variables may be implemented by engineering the potential to vary over a timescale much shorter than that corresponding to the timescale of the dynamics of the BG. 

Finally, due to the infinite degeneracy of singular protocols, we may further consider the additional optimisation of other figures of merit, such as the irreversible work or the entropy production <cit.>, the information cost or thermodynamic length <cit.>,  in order to further constrain the subspace of optimal protocols.  This seems particularly relevant, due to the competition between the connection time and irreversible work expressed by our speed limit inequality. These perspectives would become significant for the design of optimal heat engines with Brownian particles <cit.>, which constitutes an active field of research. 



§ ACKNOWLEDGEMENTS


We thank Marco Baldovin for fruitful discussions. We acknowledge financial support from Grants PID2021-122588NB-I00, funded by MCIN/AEI/ 10.13039/501100011033/ and by “ERDF A way of making Europe”, and ProyExcel_00796, funded by Junta de Andalucía's PAIDI 2020 programme. CAP acknowledges the funding received from European Union’s Horizon Europe–Marie Skłodowska-Curie 2021 program through the Postdoctoral Fellowship
with Reference 101065902 (ORION). A. Patrón also acknowledges support from the FPU programme through Grant FPU2019-4110.



§ DATA AVAILABILITY STATEMENT


The Mathematica notebooks employed for generating the data and figures that support the findings of this study are openly available in the https://github.com/fine-group-us/Brownian-gyratorGitHub page of University of Sevilla's FINE research group.











§  ON THE INEXISTENCE OF EULER-LAGRANGE OPTIMAL-TIME PROTOCOLS



In this section we explicitly show that Euler-Lagrange solutions—i.e those for which the optimal control (k^*(t),u^*(t)) lies in the interior of the control set—do not exist for the time-optimisation problem of concern. We recall that our dynamical system does not belong to the class of “linear systems” of optimal control theory, since the evolution equations do not have the form (<ref>). Therefore, the question of whether we may rule out or not Euler-Lagrange solutions is non-trivial.



 §.§ Pure Euler-Lagrange solutions

We start by considering “pure” Euler-Lagrange solutions, i.e. those for which the control variables lie in the interior of the control set for all t ∈ (0,t_f). Therefore,  Eq. (<ref>) must be satisfied at all times: specifically, we have that
    
    ∂Π/∂ k   = -2(ψ_1z_1 + ψ_2z_2 + ψ_3z_3) = 0,
           ∂Π/∂ u    = -2(ψ_1z_1 - ψ_2z_2) = 0.

Since these two equalities hold ∀ t ∈ (0,t_f), their time derivatives must also vanish. On the one hand, the first time derivative gives

    
    d/dt(ψ_1z_1 + ψ_2z_2 + ψ_3z_3 ) = ψ_1 + ψ_2 + ψ_3 = 0,
       d/dt(ψ_1z_1 - ψ_2z_2) = ψ_1 - ψ_2 = 0.


Let us start with Eq. (<ref>). Together with Eq. (<ref>) and the fact that Π=0 over the optimal solution, they imply that ψ_1=ψ_2  0 and thus z_1 = z_2—we cannot have ψ_1=ψ_2=0, because then Eq. (<ref>) implies that ψ_3=0, and Π=0 in turn implies that ψ_0=0 [One of the conditions in PMP is that (ψ_0,ψ^T) (0,0,0,0), the so-called non-triviality condition <cit.>.]. Moreover, their initial and final values must coincide as well,


    z_1(0)    = 1/2(1+u_i) = 1/2(1-u_i) = z_2(0)    ⇔  u_i    = 0,
        
    
        z_1(t_f)    = 1/2(k_f+u_f) = 1/2(k_f-u_f) = z_2(t_f)    ⇔   u_f    = 0.


Thus, regardless of the form of the optimal control (k,u), Euler-Lagrange protocols only allow to connect uncorrelated states. Now, Eq. (<ref>) provides further insights. Recalling again that Π(t) = 0 over the optimal solution, we have

    0 = Π   = ψ_0 + ψ_1[-2(k+u)z_1 + 1] + ψ_2[-2(k-u)z_2 + 1] + ψ_3[-2kz_3 + 1] 
       = ψ_0 + 0(ψ_1 + ψ_2 + ψ_3) -2k0(ψ_1z_1 + ψ_2z_2 + ψ_3z_3 ) -2u0(ψ_1z_1 - ψ_2z_2) .

Therefore, ψ_0 = 0 for Euler-Lagrange protocols. Although this does not contradict PMP, it will become useful later in this section to disregard “mixed” protocols. 

The second time derivatives of Eq. (<ref>) must vanish too:

    
    d/dt(ψ_1 + ψ_2 + ψ_3 ) = 2k0(ψ_1 + ψ_2 + ψ_3) + 2u0(ψ_1-ψ_2) = 0, 
       d/dt(ψ_1 - ψ_2) = 2u(ψ_1 + ψ_2) = 0.


Equation (<ref>) identically vanishes due to the relations (<ref>) and (<ref>) and, thus, does not provide any further information. On the other hand, Eq. (<ref>) leaves us with two different cases:

    
  * ψ_1 + ψ_2=0. This again leads to  ψ_0=ψ_1=ψ_2=ψ_3=0, which makes us ignore this possibility.
    
  * u=0, for all t∈(0,t_f). Recalling the dynamical system from Eq. (<ref>), together with the fact that u_i = u_f = 0, we have that all the dynamical variables z_j degenerate onto the same one, and the structure of our optimisation problem radically changes, as we only need to study the behaviour of one dynamical variable now. In such scenario, it can be shown that optimal-time protocols belong to the subclass of one-bang protocols, for which the control parameter k takes either the value 0 or k_max—depending on the choice of the target state—during the entire time window. See Appendix <ref> for further details.






 §.§ Mixed Euler-Lagrange solutions


Once we have disregarded “pure” Euler-Lagrange solutions, we consider here “mixed” solutions involving at least one time interval of Euler-Lagrange type. In the following, we investigate whether such Euler-Lagrange intervals can be mixed with regular bang-bang ones—we do not consider mixing with singular intervals, the impossibility of which is shown in Appendix <ref>. 



Let us consider a bang-bang window starting from O at some time t_0. Then, the switching functions must verify ϕ_OP(t_0)<0 and ϕ_ON(t_0)<0—as shown in Fig. <ref>. Making use of Eq. (<ref>), we have the conditions

    2ψ_1(t_0)z_1(t_0) + ψ_3(t_0)z_3(t_0) >0,    2ψ_2(t_0)z_2(t_0) + ψ_3(t_0)z_3(t_0) >0.

Taking into account that both k and u vanish at the vertex O, Pontryagin's Hamiltonian function at t=t_0 is Π = ψ_0 + ψ_1(t_0) + ψ_2(t_0) + ψ_3(t_0) = 0, which entails that

    ψ_0 = -ψ_1(t_0) - ψ_2(t_0) - ψ_3(t_0) 
        < z_3(t_0)/2z_1(t_0)ψ_3(t_0) + z_3(t_0)/2z_2(t_0)ψ_3(t_0) - ψ_3(t_0) = ψ_3(t_0) (z_3(t_0)/2z_1(t_0) + z_3(t_0)/2z_2(t_0) - 1 ),

where we have substituted the inequalities from Eq. (<ref>). Now we have two possibilities: (i) either t_0=0, implying that we start the optimal protocol at O, or (ii) t_0  0, corresponding to the scenario for which we had a previous Euler-Lagrange window from which we switched towards O. In the former, we substitute the initial state into Eq. (<ref>), giving

    ψ_0 < ψ_3(t_0)(1+u_i/2 + 1-u_i/2 - 1 ) = 0  ⇒ ψ_0 < 0.

In the latter, we take into account that z_1(t_0)=z_2(t_0)=z_3(t_0) in the initial Euler-Lagrange window, so Eq. (<ref>) now implies

    ψ_0 < ψ_3(t_0)(1/2 + 1/2 - 1 ) = 0  ⇒ ψ_0 < 0.

In both cases,  ψ_0 < 0:  this is inconsistent with the fact that, during an Euler-Lagrange window, ψ_0 must be zero. Since ψ_0 is constant and cannot change during the entire protocol, this rules out the possibility of having mixed Euler-Lagrange protocols with O time windows. 

We proceed in a similar manner for the possible switching between P/N and Euler-Lagrange windows. Let us now consider a P window, starting again at some time t_0. Then, the switching functions must verify ϕ_OP(t_0)>0 and ϕ_NP(t_0)>0, which, using  again Eq. (<ref>), imply the conditions

    2ψ_1(t_0)z_1(t_0) + ψ_3(t_0)z_3(t_0) <0,   ψ_1(t_0)z_1(t_0) - ψ_2(t_0)z_2(t_0) < 0.

Now, Pontryagin's Hamiltonian at time t_0 is Π= ψ_0 + ψ_1(t_0) + ψ_2(t_0) + ψ_3(t_0) - 2k_max(2ψ_1(t_0)z_1(t_0) + ψ_3(t_0)z_3(t_0)) = 0, which entails

    ψ_0    = -ψ_1(t_0) - ψ_2(t_0) - ψ_3(t_0) + 2k_max(2ψ_1(t_0)z_1(t_0) + ψ_3(t_0)z_3(t_0))
       < -ψ_1(t_0)( 1+z_1(t_0)/z_2(t_0)) - ψ_3(t_0)+ 2k_max(2ψ_1(t_0)z_1(t_0) + ψ_3(t_0)z_3(t_0)),

where we have substituted the inequalities from Eq. (<ref>). On the one hand, had we initially started at vertex P, i.e. t_0 = 0, then we would have

    ψ_0 < 2(k_max-1)(2ψ_1(t_0)z_1(t_0) + ψ_3(t_0)z_3(t_0) ) < 0  ⇒ ψ_0 < 0,

since k_i=1<k_max. On the other hand, had we previously applied an Euler-Lagrange window, then

    ψ_0 < (2k_maxz_1(t_0)-1)(2ψ_1(t_0) + ψ_3(t_0)) < 0  ⇒ ψ_0 < 0,

 since 2k_maxz_1(t_0)-1 >0—due to the evolution equation for u=0. Once again, as in both cases we have that ψ_0 < 0, then we may disregard mixed protocols involving Euler-Lagrange and P windows. Let us remark that, due to the symmetry between the P and N points, the above proof extends to N time windows as well, and thus, we may conclude that there cannot be mixed protocols involving Euler-Lagrange and bang-bang windows. 






§  CHARACTERISATION OF SINGULAR PROTOCOLS



As mentioned in the main text, singular protocols are those for which at least one of the switching functions ϕ_OP, ϕ_ON or ϕ_NP identically vanishes during a time window within (t_1,t_2)∈(0,t_f)  and, in addition, the maximum of Pontryagin's Hamiltonian is attained over the corresponding edge of the triangular control set. In this section we show that singular protocols, if they exist, extend to the whole time interval (0,t_f), i.e. switchings between regular and singular protocols is not possible.

Let us start by considering the edge OP, where k=u, and its corresponding switching function ϕ_OP, which is given by Eq. (<ref>).  Since this function is identically zero in (t_1,t_2), its time derivatives must also cancel:

    ϕ̇_OP   = -2(2ψ_1 + ψ_3) = 0,
           ϕ̈_OP   = -4k(4ψ_1 + ψ_3) = 0.

Since k(t) does not identically vanish over the edge OP, the only available solution is ψ_1(t) = ψ_3(t) = 0, ∀ t∈(t_1,t_2). However, recalling Hamilton's canonical equations (<ref>), we have that  ψ̇_1=4kψ_1 and ψ̇_3=2kψ_3. Therefore, if ψ_1,3 vanishes at some specific time, ψ_1,3(t) must be zero ∀ t. Hence, ψ_1(t) = ψ_3(t) = 0, ∀ t∈(0,t_f), and  ϕ_OP(t) also vanishes over the whole time interval: the singular protocol over the edge OP extends to all times in (0,t_f). Now, invoking the third condition of PMP, we have that

    0 = Π = ψ_0 + 0ψ_1(-4kz_1+1) + ψ_2 + 0ψ_3(-2kz_3+1)  ⇒ ψ_2 = -ψ_0 > 0,  ∀ t∈ (0,t_f).

That is, ψ_2(t) should be constant for singular protocols, and moreover, it cannot cancel.  This is consistent with the evolution equations: over the OP edge, the evolution equation for ψ_2 reduces to ψ̇_2=0. 



It is straightforward to extend the above analysis to the ON edge and the ϕ_ON switching function, due to the symmetry between the P and N points. In such a case, the same results apply, upon exchanging the behaviour found above for the ψ_1 and ψ_2 momenta.  Wrapping things up, we have shown that singular protocols over the OP and ON edges may appear, but they have to extend to the whole time interval (0,t_f).

Let us now consider the last edge NP and the corresponding switching function, given by Eq. (<ref>). Again, since ϕ_NP identically vanishes in (t_1,t_2), taking time derivatives we get 

    ϕ̇_NP   = -2(ψ_1 - ψ_2) = 0, 
           ϕ̈_NP   = -4u(ψ_1 + ψ_2) = 0.

 Similarly, since u(t) does not identically vanish over the edge NP, we have that ψ_1(t)=ψ_2(t)=0 for all t∈(t_1,t_2). Again, from the evolution equations for the momenta, we conclude that ψ_1(t)=ψ_2(t)=0 for all t∈(0,t_f), the singular time window extends to the whole interval. Invoking again that Pontryagin's Hamiltonian vanishes over the time optimal protocol,

    0 = Π   = ψ_0 + 0ψ_1(-2(k_max+u)z_1 + 1) + 0ψ_2(-2(k_max-u)z_1 + 1) + ψ_3(-2k_maxz_3+1) ,

we have

    ψ_3 = ψ_0/2k_maxz_3-1<0,

where we have taken into account the inequality 2k_maxz_3(t)-1 ≥ 0, which stems from the evolution equation for z_3. Again, Eq. (<ref>) is consistent with the evolution equations:

    ψ̇_3=-2k_maxψ_0/(2k_maxz_3-1)^2ż_3=-2k_maxψ_0/(2k_maxz_3-1)^2(-2 k_max z_3+1)=2k_maxψ_0/2k_maxz_3-1=2k_maxψ_3.

 

    
Following the preceding discussion, we conclude that the singular protocols corresponding to the OP, ON, and NP edges share similar features: (i) they correspond to “pure” protocols that last for the whole interval (0,t_f), (ii) they involve the vanishing of two conjugate momenta during the entire protocol, and (iii) they are characterised by the action of an unique control variable—either k(t), for the OP and ON edges, or u(t), for the  NP edge. 

An additional interesting feature is that, for each singular protocol, the connection time is always fixed—regardless of the choice of k(t) or u(t) over the considered edge. The physical reason for this behaviour is that the time evolution of one of the variances sets the connection time, independently of the others. More specifically, for the OP edge, we have that ż_2 = 1, thus giving

    t_f^(OP) = z_2(t_f) - z_2(t_0) = 1/2(1/k_f-u_f -1/1-u_i).

Similarly, for the OP branch, we have ż_̇1̇ = 1, implying that

    t_f^(ON) = z_1(t_f) - z_1(t_0) = 1/2(1/k_f+u_f -1/1+u_i).

And finally, for the NP, we have ż_3 = -2k_maxz_3+1, which gives

    t_f^(NP) = 1/2k_maxln(k_f k_max-1/k_max-k_f).

Each of the singular branches has still two dynamical variables whose behaviour is determined via one control parameter k(t) or u(t). In principle, since there are no further restrictions on the values of k or u, there is an infinite number of possible optimal protocols achieving the desired connection in the given connection time t_f. One could take such degeneracy as an opportunity for optimising an additional figure of merit, such as the irreversible work or the information cost.










§ EXPLICIT EQUATIONS FOR OPTIMAL BANG-BANG PROTOCOLS


Here we will explicitly write the systems of algebraic equations for each of the two-bang and three-bang protocols, in the k_max→ +∞ limit, that are presented in Sec. <ref>. These sets of equations provide the solutions that are depicted in Figs. <ref> - <ref>. Throughout this section, in order to simplify the presentayion, we directly make the substitutions z_j(0) = (2ω_j(0))^-1 and z_j(t_f) = (2ω_j(t_f))^-1.






 §.§ Two-bang protocols


There are six possible permutations of the points (O, P, N) giving rise to two-bang protocols. The symmetry between the P and N points translate into the PN and NP protocols providing the same algebraic system of equations.

    
  * OP: Variables (t_f, ξ).
    
        
    1/2(k_f + u_f)   = (ℰ̃_ξ^2∘ℰ_0^(t_f))(1/2(1+u_i))= (1/2(1+u_i) + t_f)ξ^2,
                
    1/2(k_f - u_f)   = (ℰ̃_1∘ℰ_0^(t_f))(1/2(1-u_i)) = 1/2(1-u_i) + t_f,
                
    1/2k_f   = (ℰ̃_ξ∘ℰ_0^(t_f))(1/2)= (1/2 + t_f)ξ.

    
    
  * ON: Variables (t_f, ξ).
    
        
    1/2(k_f + u_f)   = (ℰ̃_1∘ℰ_0^(t_f))(1/2(1+u_i)) = 1/2(1+u_i) + t_f,
                
    1/2(k_f - u_f)   = (ℰ̃_ξ^2∘ℰ_0^(t_f))(1/2(1-u_i))= (1/2(1-u_i) + t_f)ξ^2,
                
    1/2k_f   = (ℰ̃_ξ∘ℰ_0^(t_f))(1/2)= (1/2 + t_f)ξ.

    
    
  * PO: Variables (t_f, ξ).
    
        
    1/2(k_f+u_f)   = (ℰ_0^(t_f)∘ℰ̃_ξ^2)(1/2(1+u_i))= ξ^2/2(1+u_i) + t_f,
                
    1/2(k_f-u_f)   = (ℰ_0^(t_f)∘ℰ̃_1)(1/2(1-u_i))= 1/2(1-u_i) + t_f,
                
    1/2k_f   =(ℰ_0^(t_f)∘ℰ̃_ξ)(1/2)= ξ/2 + t_f.

    
    
  * NO: Variables (t_f, ξ).
    
        
    1/2(k_f+u_f)   = (ℰ_0^(t_f)∘ℰ̃_1)(1/2(1+u_i))= 1/2(1+u_i) + t_f,
                
    1/2(k_f-u_f)   = (ℰ_0^(t_f)∘ℰ̃_ξ^2)(1/2(1-u_i))= ξ^2/2(1-u_i) + t_f,
                
    1/2k_f   =(ℰ_0^(t_f)∘ℰ̃_ξ)(1/2)= ξ/2 + t_f.

    
    
  * PN and NP: Variables (ξ_1, ξ_2), and thus a null connection time.
    
        
    1/2(k_f+u_f)   = (ℰ̃_1∘ℰ̃_ξ_1^2)(1/2(1+u_i)) = ξ_1^2/2(1+u_i), 
                
    1/2(k_f-u_f)   = ( ℰ̃_ξ_2^2∘ℰ̃_1)(1/2(1-u_i)) = ξ_2^2/2(1-u_i),
                
    1/2k_f   = (ℰ̃_ξ_2∘ℰ̃_ξ_1)(1/2) = ξ_1ξ_2/2,

    







 §.§ Three-bang protocols


In principle, there are twelve possible permutations of the vertices (O, P, N). Although the protocols POP and NON are actually singular protocols—as mentioned in the main text and thoroughly studied in Appendix <ref>, we  consider them here because they have the three-bang structure as well. Similarly to what occurred for the two-bang protocols, the symmetry between the P and N points reduces our possible combinations from twelve to nine, as the pairs of protocols (PNO, NPO), (PNP, NPN) and (OPN, ONP) give the same algebraic equations. 

    
  * OPO: Variables (τ_1, τ_2, ξ), with τ_1 + τ_2 = t_f.
    
        
    1/2(k_f + u_f)   = (ℰ_0^(τ_2)∘ℰ̃_ξ^2∘ℰ_0^(τ_1))(1/2(1+u_i))= (1/2(1+u_i) + τ_1)ξ^2 +τ_2,
                
    1/2(k_f - u_f)   = (ℰ_0^(τ_2)∘ℰ̃_1∘ℰ_0^(τ_1))(1/2(1-u_i)) = 1/2(1-u_i) + τ_1 + τ_2,
                
    1/2k_f   = (ℰ_0^(τ_2)∘ℰ̃_ξ∘ℰ_0^(τ_1))(1/2)= (1/2 + τ_1)ξ + τ_2.

    
    
  * ONO: Variables (τ_1, τ_2, ξ), with τ_1 + τ_2 = t_f.
    
        
    1/2(k_f + u_f)   = (ℰ_0^(τ_2)∘ℰ̃_1∘ℰ_0^(τ_1))(1/2(1+u_i)) = 1/2(1+u_i) + τ_1 + τ_2,
                
    1/2(k_f - u_f)   = (ℰ_0^(τ_2)∘ℰ̃_ξ^2∘ℰ_0^(τ_1))(1/2(1-u_i))= (1/2(1-u_i) + τ_1)ξ^2 +τ_2,
                
    1/2k_f   = (ℰ_0^(τ_2)∘ℰ̃_ξ∘ℰ_0^(τ_1))(1/2)= (1/2 + τ_1)ξ + τ_2.

    
    
  * POP (Singular): Variables (t_f, ξ_1, ξ_2).
    
        
    1/2(k_f+u_f)   = (ℰ̃_ξ_2^2∘ℰ_0^(t_f)∘ℰ̃_ξ_1^2)(1/2(1+u_i))= (ξ_1^2/2(1+u_i) + t_f) ξ_2^2,
                
    1/2(k_f-u_f)   = (ℰ̃_1∘ℰ_0^(t_f)∘ℰ̃_1)(1/2(1-u_i))= 1/2(1-u_i) + t_f,
                
    1/2k_f   =(ℰ̃_ξ_2∘ℰ_0^(t_f)∘ℰ̃_ξ_1)(1/2)= (ξ_1/2 + t_f) ξ_2.

    
    
  * NON (Singular): Variables (t_f, ξ_1, ξ_2).
    
        
    1/2(k_f+u_f)   = (ℰ̃_1∘ℰ_0^(t_f)∘ℰ̃_1)(1/2(1+u_i))= 1/2(1+u_i) + t_f,
                
    1/2(k_f-u_f)   = (ℰ̃_ξ_2^2∘ℰ_0^(t_f)∘ℰ̃_ξ_1^2)(1/2(1-u_i))= (ξ_1^2/2(1-u_i) + t_f) ξ_2^2,
                
    1/2k_f   =(ℰ̃_ξ_2∘ℰ_0^(t_f)∘ℰ̃_ξ_1)(1/2)= (ξ_1/2 + t_f) ξ_2.

    
    
  * OPN and ONP: Variables (t_f, ξ_1, ξ_2).
    
        
    1/2(k_f + u_f)   = (ℰ̃_1∘ℰ̃_ξ_1^2∘ℰ_0^(t_f))(1/2(1+u_i))= (1/2(1+u_i) + t_f)ξ_1^2,
                
    1/2(k_f - u_f)   = (ℰ̃_ξ_2^2∘ℰ̃_1∘ℰ_0^(t_f))(1/2(1-u_i)) = (1/2(1-u_i) + t_f)ξ_2^2,
                
    1/2k_f   = (ℰ̃_ξ_2∘ℰ̃_ξ_1∘ℰ_0^(t_f))(1/2)= (1/2 + t_f)ξ_1ξ_2.

    
    
  * PON: Variables (t_f, ξ_1, ξ_2).
    
        
    1/2(k_f+u_f)   = (ℰ̃_1∘ℰ_0^(t_f)∘ℰ̃_ξ_1^2)(1/2(1+u_i))= ξ_1^2/2(1+u_i) + t_f,
                
    1/2(k_f-u_f)   = (ℰ̃_ξ_2^2∘ℰ_0^(t_f)∘ℰ̃_1)(1/2(1-u_i))= (1/2(1-u_i) + t_f)ξ_2^2,
                
    1/2k_f   =(ℰ̃_ξ_2∘ℰ_0^(t_f)∘ℰ̃_ξ_1)(1/2)= (ξ_1/2 + t_f)ξ_2.

    
    
  * NOP: Variables (t_f, ξ_1, ξ_2).
    
        
    1/2(k_f+u_f)   = (ℰ̃_ξ_2^2∘ℰ_0^(t_f)∘ℰ̃_1)(1/2(1+u_i))= (1/2(1+u_i) + t_f)ξ_2^2,
                
    1/2(k_f-u_f)   = (ℰ̃_1∘ℰ_0^(t_f)∘ℰ̃_ξ_1^2)(1/2(1-u_i))= ξ_1^2/2(1-u_i) + t_f,
                
    1/2k_f   =(ℰ̃_ξ_2∘ℰ_0^(t_f)∘ℰ̃_ξ_1)(1/2)= (ξ_1/2 + t_f)ξ_2.

    
    
  * PNO and NPO: Variables (t_f, ξ_1, ξ_2).
    
        
    1/2(k_f+u_f)   = (ℰ_0^(t_f)∘ℰ̃_1∘ℰ̃_ξ_1^2)(1/2(1+u_i)) = ξ_1^2/2(1+u_i) + t_f, 
                
    1/2(k_f-u_f)   = (ℰ_0^(t_f)∘ℰ̃_ξ_2^2∘ℰ̃_1)(1/2(1-u_i)) = ξ_2^2/2(1-u_i) + t_f,
                
    1/2k_f   = (ℰ_0^(t_f)∘ℰ̃_ξ_2∘ℰ̃_ξ_1)(1/2) = ξ_1ξ_2/2 + t_f.

    
    
  * PNP and NPN: Variables (ξ_1, ξ_2, ξ_3), and thus a null connection time.
    
        
    1/2(k_f+u_f)   = (ℰ̃_ξ_3^2∘ℰ̃_1∘ℰ̃_ξ_1^2)(1/2(1+u_i)) = ξ_1^2ξ_3^2/2(1+u_i), 
                
    1/2(k_f-u_f)   = (ℰ̃_1∘ℰ̃_ξ_2^2∘ℰ̃_1)(1/2(1-u_i)) = ξ_2^2/2(1-u_i),
                
    1/2k_f   = (ℰ̃_ξ_3∘ℰ̃_ξ_2∘ℰ̃_ξ_1)(1/2) = ξ_1ξ_2ξ_3/2,

    

We highlight that the PNP and NPN protocols present an undetermined system of equations, as the variables ξ_1 and ξ_3 always appear together as ξ_1 ξ_3. Therefore, we could just refer to that product as a new quenching variable ξ̃_1 ≡ξ_1 ξ_3. Thus, the resulting system is equivalent to that for the PN and NP protocols in Eq. (<ref>) with ξ_1→ξ̃_1.







§ SINGULAR BANG-BANG PROTOCOLS



Let us start by considering the three-bang protocol POP—due to the symmetry between the P and N points, the results obtained here are also valid for the protocol NON. We do not consider the limit k_max→ +∞ in this section, since it becomes a singular limit when studying the derivatives of Pontryagin's Hamiltonian function. It suffices to proceed with the forthcoming analysis for k_max finite, and then extend our results to the k_max→ +∞ regime, which is the one we employ throughout the majority of this work. 

Let us recall that the switching function ϕ_OP(t) is given by Eq. (<ref>), it accounts for the derivative of Pontryagin's Hamiltonian function along the OP edge of the triangular control set. Since the POP protocol involves two switchings—one switch from P to O at some time t_1 ∈ (0,t_f) followed by another one from O to P at t_2 ∈ (t_1,t_f), ϕ_OP(t) must have at least two zeros on the entire interval (0,t_f). The protocols starts from vertex P, so we have 

    ϕ_OP(0) = -2[2ψ_1(0) z_1(0) + ψ_3(0) z_3(0)] = -2(ψ_1,0/1+u_i + ψ_3,0/2) > 0.

Now, by integrating Hamilton's canonical equations (<ref>) at vertex P, we  write our switching function explicitly as

    ϕ_OP(t) = ϕ_OP(0) - ψ_1,0/k_max(e^4k_maxt-1) -ψ_3,0/k_max(e^2k_maxt-1),    0≤ t ≤ t_1.

This function constitutes a second-degree polynomial for the variable exp(2k_maxt), and thus, it presents two roots at most—i.e. up to two candidates for the switching time t_1, for which ϕ_OP(t_1)=0. The fact that t_1 must be positive sets bounds on the sign of the ratio ψ_1,0/ψ_3,0, which must be consistent with that in Eq. (<ref>). Once we switch to the point O, by integrating once again the canonical equations at vertex O, we get

    ϕ_OP(t) = 0ϕ_OP(t_1) -2(2ψ_1,0e^4k_maxt_1 + ψ_3,0e^2k_maxt_1)(t-t_1),    t ≥ t_1.

However, this corresponds to a linear function starting from zero, which does not have any roots apart from the trivial one at t = t_1, and thus, the switching back to the last time window at point P is forbidden.  Therefore, a regular three-bang POP protocol is not possible. Still, a three-bang PON protocol would be indeed feasible, since the switching to vertex N is ruled by the switching function ϕ_ON(t), for which we do not have any restriction.

Another possibility to overcome the found issue could be to consider ϕ_OP(t) = 0 for t≥ t_1, which would set the value of the ratio ψ_1,0/ψ_3,0 from the condition 2ψ_1,0e^4k_maxt_1 + ψ_3,0e^2k_maxt_1=0. In this case, the time window at point O would become a singular time interval. However, by virtue of our analysis of singular protocols in Appendix <ref>, if ϕ_OP(t) = 0 for some time window, then it must be zero fon the whole interval (0,t_f). Henceforth, the protocols POP and NON—and any other higher order protocol involving the POP and/or NON combinations—belong in the class of singular protocols depicted in Fig. <ref>.





§ BRACHISTOCHRONES FOR UNCOUPLED OSCILLATORS


Let us consider the u(t)=0 case for the Brownian Gyrator, which corresponds to connecting arbitrary stationary states for two uncorrelated overdamped oscillators coupled to two different thermal baths. Such states are thus characterised uniquely by the final value of the control parameter k(t), i.e. k_f. As u_i = u_f = 0, all the dynamical variables of our original system degenerate towards a unique one z_j(t) = z(t), j=1,2,3, whose dynamical evolution is given by

    ż = -2k(t)z + 1,

with k(t) ∈ [0,k_max] once again. Due to the simplicity of this situation—since we are dealing only with one dynamical variable—we derive the minimum connection times by other means. 

Equation (<ref>) can be integrated analytically, thus providing an integral form for the final connection time

    t_f = ∫_z_i^z_fdz/1-2 k(z)z,

with z_i ≡ z(t=0) = 1/2—recall, once more, that k_i = 1 in our scaled units— and z_f = z(t=t_f) = 1/(2k_f). Now, t_f does not diverge as long as the denominator 1-2k(z)z ≠ 0  ∀ z ∈ [z_i,z_f]. This implies that the evolution of z must be monotonic for the optimal time connection: given an initial z_i, if z_f > z_i (z_f < z_i), then we need ż > 0 (ż < 0). Thus, we have two possibilities:

    
  * 2k(z)z > 1: In this case, the integrand becomes negative during the entire integration interval. The connection time will only be positive thus if z_f < z_i, implying that k_f > 1. The integral from Eq. (<ref>) is thus bounded from below by
    
    t_f = ∫_z_i^z_fdz/1-2 k(z)z = ∫_z_f^z_idz/2 k(z)z-1≥∫_z_f^z_idz/2 k_maxz-1 = 1/2k_maxln(k_f k_max-1/k_max-k_f),

    which corresponds to the final connection time for a bang-bang protocol with k(t) = k_max.
    
  * 2k(z)z < 1: Here, the integrand is positive, thus implying that z_f > z_i and k_f < 1. In this case, the integral from Eq. (<ref>) is bounded from below by
    
    t_f = ∫_z_i^z_fdz/1-2 k(z)z≥∫_z_i^z_fdz = z_f - z_i = 1/2(1/k_f - 1 ),

    which corresponds to the final connection time for a bang-bang protocol with k(t) = 0.


Given the above, we have proved that optimal protocols achieving the fastest connection between stationary states for uncorrelated oscillators are also the bang-bang type, specifically corresponding to one-bang protocols of the type k(t) = 0, for k_f<1, and k(t) = k_max, for k_f>1, for all t ∈ (0,t_f). 




§ BOUND ON THE CONNECTION FROM THE SPEED LIMIT INEQUALITY (<REF>)


In this appendix, we evaluate the contribution to the irreversible work W_ defined in Eq. (<ref>) from the bangs at the different vertices O, P, and N. Throughout the appendix, we consider that the bang starts at a certain time t_1 and ends at a certain time t_2≥ t_1, so that the duration of the bang is τ=t_2-t_1. It is useful to write W_ as

    W_=W_^(1)+W_^(2),   W_^(j)=γ∫_0^t_fdt (dσ_j/dt)^2.


Consistently with the non-dimensionalisation introduced in Eq. (<ref>), we introduce dimensionless standard deviations σ_j^* as:

    σ_j=√(k_B(T_x+T_y)/k_i)σ_j^* σ_j^*=√(z_j).

Therefore, we obtain

    W_^(j)=γk_i/γk_B(T_x+T_y)/k_i∫_0^t_f^* dt^* (dσ_j^*/dt^*)^2 ,

which suggest the following definition for the non-dimensionless work:

    W_^(j)=k_B(T_x+T_y)W_^(j)^* W_^(j)^*=∫_0^t_f^* dt^* (dσ_j^*/dt^*)^2 .

As in the remainder of the paper, we drop the asterisks in the following to simplify our notation.



 §.§ Vertex O


At the vertex O, we have that k=u=0, so the evolution equations for the variances z_j=σ_j^2 are ż_j=1 and thus

    z_j(t)=z_j(t_1)+(t-t_1) σ_j(t)=√(σ_j^2(t_1)+(t-t_1)),    t_1≤ t≤ t_2.

Therefore, we have that

    W_^(j)= ∫_t_1^t_2dt [σ_j^2(t_1)+(t-t_1)]^-1=ln[σ_j^2(t_2)/σ_j^2(t_1)],

and, finally

    W_=2ln[σ_1(t_2)σ_2(t_2)/σ_1(t_1)σ_2(t_1)].




 §.§ Vertex P


At the vertex P, k=u=k_max, so the evolution equation for the variances are

    dz_1/dt=-4k_maxz_1+1,   dz_2/dt=1.

The equation for z_2 is the same as for the vertex O, so we do not need to derive the contribution W_^(2). We thus focus on the evolution of z_1—or σ_1—and its corresponding contribution W_^(1):

    σ_1(t)=√(1/4k_max+(σ_1^2(t_1)-1/4k_max)e^-4k_maxt),

from which we derive

    W_^(1)=k_max{σ_1^2(t_1)-σ_1^2(t_2)+1/4k_maxln[σ_1^2(t_2)/σ_1^2(t_1)]}.

In the limit k_max→∞ we have considered, the bang at vertex P becomes instantaneous, τ=t_2-t_1→ 0, but ξ=e^-2k_maxt/γ<1 remains finite—as discussed in Sec. <ref>. Then, we get

    W_^(1)∼ k_max σ_1^2(t_1)(1-ξ^2),

and this contribution to the irreversible work diverges linearly with k_max. An analogous divergent behaviour of the cost of the connection has been found for the thermodynamic geometry cost in other systems <cit.>—although, therein, the parameter that was controlled was the bath temperature instead of the confining potential.




 §.§ Vertex N


At the vertex N, k=-u=k_max, so the evolution equation for the variances are

    dz_1/dt=1,   dz_2/dt=-4k_maxz_2+1.

As repeatedly commented along the paper, there is a symmetry between the P and N vertices. For the calculation of the contribution to the irreversible work we are considering here, we see that we have only to exchange the labels 1↔ 2. Therefore, for the vertex N, it is W_^(1) that coincides with our calculation for vertex O, whereas

    W_^(2)=k_max{σ_2^2(t_1)-σ_2^2(t_2)+1/4k_maxln[σ_2^2(t_2)/σ_2^2(t_1)]},

which, in the limit k_max→∞, reduces to

    W_^(2)∼ k_max σ_2^2(t_1)(1-ξ^2).


As a consequence of the above results, the irreversible work for any bang-bang protocol involving at least one P or N vertex—i.e. any bang-bang protocol different from the one-bang at vertex O—diverges linearly with k_max.







 Paper Title: Non-linear inhibitory responses enhance performance in collective decision-making.




david.march@upc.edu
Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain
Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain.


Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain



Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain.
Institute of Complex Systems (UBICS), Universitat de Barcelona, 08028 Barcelona, Spain


§ ABSTRACT 
    The precise modulation of activity through inhibitory signals ensures that both insect colonies and neural circuits operate efficiently and adaptively, highlighting the fundamental importance of inhibition in biological systems. Modulatory signals are produced in various contexts and are known for subtly shifting the probability of receiver behaviors based on response thresholds. Here we propose a non-linear function to introduce inhibitory responsiveness in collective decision-making inspired by honeybee house-hunting. We show that, compared with usual linear functions, non-linear responses enhance final consensus and reduce deliberation time. This improvement comes at the cost of reduced accuracy in identifying the best option. Nonetheless, for value-based tasks, the benefits of faster consensus and enhanced decision-making might outweigh this drawback.


Non-linear inhibitory responses enhance performance in collective decision-making
    M. Carmen Miguel
    August 5, 2024
=================================================================================



The behavioral and signaling patterns of social animal groups <cit.> have sparked extensive research into collective behavior and decision-making, primarily to understand the underlying mechanisms that drive these emergent properties <cit.>. Inhibitory signals, in particular, play an essential role in social insects, fine-tuning collective decision-making and coordinating critical tasks such as house-hunting and foraging <cit.>. These inhibitory signals, often communicated through vibrations or tactile interactions, allow colonies to efficiently allocate resources and labor. For instance, in honeybees, stop signals can prevent the recruitment of additional foragers to poor or perilous food sources, thereby optimizing foraging efforts <cit.>. Similarly, during nest site selection, bees use stop signals to halt the promotion of less suitable sites, ensuring that the colony converges on the best available option <cit.>. By integrating these stop signals, social insects enhance their ability to make adaptive and robust decisions, ultimately supporting the survival and success of the colony. The fascinating social behavior of honeybees, including their intricate recruiting signaling patterns such as the waggle dance <cit.>, has inspired the design of decentralized decision-making algorithms <cit.>, and their application to robotic systems <cit.>. 

According to Nieh <cit.> and Pastor et al. <cit.>, during foraging tasks, honeybees' stop signals can alter the probability of waggle dancers ceasing their dance and leaving the nest, thereby reducing recruitment. However, dancers do not exhibit an immediate response to these signals. This feature is characteristic of modulatory signals, which 

are known for subtly shifting the probability of receiver behaviors based on their response thresholds. Lau et al. <cit.> further suggested that, depending on receiver response thresholds, stop signals do not exert a strong colony-wide effect until signaling levels are sufficiently elevated. A similar mechanism has for long also been observed in brain neuronal activity <cit.>, where balance between excitation and inhibition is critical for processes such as sensory perception, motor control or cognitive functions. Recent efforts have been made in establishing the similarities between individual decision-making in primate brains and collective decision-making in social insect colonies <cit.>. 

Field experiments on honeybee house-hunting <cit.> introduced the term cross-inhibition, as it was demonstrated that stop signals were predominantly exchanged between agents promoting competing options.
Cross-inhibition has proven essential for resolving deadlocks in decisions between very similar alternatives <cit.>. 
However, as argued in Ref. <cit.>, cross-inhibition trades accuracy for stability. This means the system can confidently make a decision for any option, regardless of whether it is the highest quality one or not. In value-based tasks, this trade-off may not necessarily be detrimental, as the system prioritizes making a choice that yields a sufficiently high reward within a limited time, thus balancing the speed-value trade-off <cit.>. 
Furthermore, depending on the intensity of cross-inhibition, this mechanism may pause the decision-making process if the qualities of the available options are not deemed high enough, allowing the system to wait for a potentially better option to appear <cit.>. Such a system transitions from indecision to decision through pitchfork or saddle-node bifurcations <cit.>, controlled by the model parameters. 


In honeybee-inspired collective decision-making models, the cross-inhibition rate has usually been considered a linear function of the population sending the stop signals. This choice represents the simplest modeling assumption, where the abandonment of one’s opinion is linearly proportional to the accumulation of stop signals received from peers with opposing options. 
However, similar to the foraging behavior of bees discussed earlier, Seeley et al. also suggested that nest-site scout waggle dances are likely terminated when stop-signal inhibition surpasses a certain threshold <cit.>.
Motivated by this experimental evidence, here we investigate the impact of non-linear inhibitory responsiveness <cit.> within honeybee-inspired decision-making models.
The response depends on the amount of stop signals received and diminishes or becomes negligible when stop signals are sparse, see Fig. <ref>(a).
This approach also aligns with the concept of complex social contagion models <cit.>, which posits that multiple exposures to a given opinion are required to trigger a shift in belief. Similarly, our model assumes that a minimum threshold of stop signals must be reached before 

cross-inhibition acts.


Focusing on binary decision tasks, we demonstrate that our  approach enhances the consensus formation capabilities of decentralized systems compared to linear cross-inhibition models, particularly when dealing with options of similar qualities. The benefits are twofold: first, the final decision is achieved with virtually no bees committed to the less favored option; second, the time to reach a stationary state is significantly reduced. 




Model – We use a simplified version of the honeybee inspired LES model <cit.>, which features the typical transition rates of collective decision making models <cit.>. In the LES model, a swarm of N scout bees

evaluates k potential nest sites, indexed by α = 1,…, k. Each site α is characterized by an intrinsic quality q_α≥ 0 and a spontaneous discovery probability π_α≥ 0.

Bees can be in any of k+1 states: uncommitted or committed to one of the k available sites. 
The transitions from uncommitted to committed state are governed by discovery and recruitment rates, representing individual and social behavior, and balanced by the interdependence parameter, λ. 

Likewise, the transitions from committed to uncommitted states are governed by abandonment and cross-inhibition rates, which reflect individual and socially motivated behaviors, respectively. 

The model's mean-field rate equations for the fractions of agents committed to each site, f_α(t), can be derived using the master equation formalism <cit.>. Including cross-inhibition, these equations are:

    ḟ_α(t)    =     f_0(t) [(1-λ)π_α + λ  f_α(t)] 
       -    r_α f_α(t) - λ' f_α(t) ∑_β≠ασ(f_β),   α=1,…,k

where f_0(t) = 1 - ∑_α = 1^k f_α(t) is the fraction of uncommitted bees. The discovery rate, (1-λ)π_α, refers to the rate at which uncommitted bees discover and commit to site α, and the recruitment rate λ f_α represents the rate at which uncommitted bees are recruited by peers already committed to option α. The rate r_α at which bees stop advertising a site is inversely proportional to its quality, r_α=1/q_α. Finally, the cross-inhibition rate, λ' f_ασ(f_β) (β≠α), is the rate at which bees abandon their options after receiving stop signals from those advocating for competing options. Here, λ' regulates the intensity of cross-inhibition interactions. For the purposes of this study, we will set λ' = 1 in the following.

The stationary points of the system can be determined by numerically solving the equations obtained by setting ḟ_α(t) = 0 <cit.>. 
Without cross-inhibition (λ' = 0), the system simplifies to the expressions derived in <cit.>, thoroughly analyzed in <cit.>.





Cross-inhibition response – The function σ (f_β) determines the actual strength of the cross-inhibition based on the fraction of bees f_β sending stop signals. Traditionally, cross-inhibition, similarly to recruitment interactions, has been modeled as a proportional response to the fraction of adversary population, i.e. σ(f_β) = f_β.  Here we consider a non-linear cross-inhibition response. Specifically, we propose two sigmoid-like test functions, where the cross-inhibition strength remains weak for small values of the inhibiting population:

    σ_1(f_β; x_0, a) = 1/1+e^-a(f_β-x_0) ,
    σ_2(f_β; x_0, a) = f_β/1+e^-a(f_β-x_0).

The parameter a controls the steepness of these functions, and x_0 is a threshold controlling the sigmoid's ascent position.
Some instances of these functions

are depicted in Fig. <ref>(a). The function σ_1 captures the scenario where the cross-inhibition strength increases sharply, similar to a step function, once the threshold population x_0 is approached. On the other hand, σ_2 assumes that the cross-inhibition strength grows sub-linearly below this threshold and transitions to a limiting linear regime above it.




In the following, we will focus on the simplest case of a binary decision between two sites that differ in quality (q_1 < q_2).
The system's dynamics display a different number of stable points for different values of the model's parameters.
For instance, increasing the strength of the social interactions leads to an (unfolded) pitchfork bifurcation <cit.>. 

This behavior is shown in Fig. <ref>(b) for linear cross-inhibition (black-circles), and has been previously observed in similar models <cit.>. 
When switching to non linear cross-inhibition, a bifurcation still occurs, but its position depends on the particular response choice. Two examples of this are also shown in Fig. <ref>(b). The curves with blue squares and red triangles represent results for a sharp sigmoid function σ_1(f_β;0.3,500) and a smooth linearly-bounded sigmoid function σ_2(f_β;0.3,10), respectively.
Results obtained for other non-linear functions displayed in Fig <ref>(a) are shown in 
Supplemental Figure SF1 <cit.>.
Reducing the strength of cross-inhibition, or slightly varying the threshold parameter x_0, produces qualitatively similar results, though the positions of the bifurcations are shifted. For bifurcation plots at λ'=0.5, 

see Supplemental Figure SF2 <cit.>.

Performance measure – 
To assess the model's performance with non-linear cross-inhibition interactions, we numerically evaluate the stationary fixed point values,  focusing on the occupation fraction for the best-quality site, f_2^*. This quantity represents the decision accuracy of the system. However, as previously discussed, decision accuracy alone is not the only relevant variable a system seeks to maximize, especially in value-based decisions <cit.>. 

In scenarios where the available sites are similar in quality, it may be preferable to make a quick decision rather than spending a large amount of time to choose a slightly better site. Therefore, in addition to accuracy, we use agent-based stochastic simulations to measure two additional performance metrics: (i) the probability P(f_2^*) of reaching the best option;
and (ii) the time t_ss required to settle into this stationary state. These complementary quantities provide a comprehensive evaluation of the system's decision accuracy and speed performance.

Figure <ref> represents the behavior of these three quantities as a function of the interdependence  λ for close values of the sites' qualities, q_1=9 and q_2=10, for the different cross-inhibition responses. 

In the non-linear case, we use the same parameters as in Fig. <ref>. 
We can observe that all non-linear cross-inhibition functions tested outperform the linear cross-inhibition in terms of pure consensus accuracy. However, the linear approach provides a higher probability of selecting the better option. These differences are particularly relevant for small to moderate values of the interdependence.
In this regime, not triggering cross-inhibition unless an option has gained some representation allows the system to build a stronger consensus, albeit at the risk of less reliably choosing the better option. 
Nonetheless, this comes with the benefit of making a decision in a much shorter time, as shown in  Fig. <ref>(c). This can be a significant advantage when choosing between similarly valued options. As reported in <cit.>, quicker consensus can be achieved by allowing the system to first build sub-populations of comparable sizes before triggering competition between them. In those works, this is achieved by time-varying social interaction rates, including recruitment and cross-inhibition. In contrast, we propose a simpler time-invariant mechanism that weakens the perception of cross-inhibition signals unless they are received from a significant portion of the population. This approach allows both populations to grow without interference from stop signals. Once the populations reach substantial sizes, cross-inhibition is triggered, and a faster decision is made.

Each type of sigmoid function is tested with both a sharp response (high a=500 value), where the cross-inhibition rapidly shifts from no effect to maximum or linear bound, and a smooth response (low a ∈ [10,20] value), where the transition to the final bound is more gradual. Interestingly, our results for the best-quality site occupation fraction show a remarkable insensitivity to the specific details of the sigmoid functions (see Fig. <ref>(a)). Moreover, these results are significantly higher for low and moderate interdependence compared to those of the linear cross-inhibition model.
On the other hand, the probability of reaching the best option is considerably reduced for sharper cross-inhibitory responsiveness, independently of the function selected, Fig. <ref>(b). This is due to the indiscriminate action of inhibition on the option that first reaches the activation threshold x_0, irrespective of its quality. While the smooth sigmoid also yields probabilities similar to the sharp functions, due to the over-representation of the inhibiting population when the threshold is trespassed, approaching smoothly the linear bound grants an intermediate result.





In order to encapsulate the effect of these three measures in a single quantity 

we define the objective performance, 

    ψ_σ = f_2^*  P(f_2^*)/t_ss,

weighting the three quantities at stake. To assess how non-linear cross-inhibition compares to linear cross-inhibition, we also introduce the performance ratio χ = ψ_σ/ψ_lin. 
Fig. <ref> depicts this performance ratio for the same quality pair as in Fig. <ref>. We observe a performance ratio χ > 1 for nearly all values of λ. Moreover, as depicted in Supplemental Figure SF3 <cit.> this ratio increases as the site qualities become closer, indicating a more significant performance improvement when using non-linear cross-inhibition.



The performance improvement peaks around λ∼ 0.2, corresponding to the point where the difference in decision times between linear and nonlinear models is the greatest. The subsequent performance decrease, on increasing λ,
is due to the combined effect of interdependence and cross-inhibition driving the losing population to very low fractions, while the winning population dominates. 
In this scenario, the cross-inhibition strength exerted by the winning population on its adversary becomes similar to that in the linear model, regardless of the specific non-linear response chosen. The advantage of the non-linear response is mainly due to the weaker effect of the losing population's cross-inhibition. Furthermore, as noted in <cit.>, when λ→ 1, the system can make a strong decision without cross-inhibition, although incorporating 
it significantly reduces decision time.

Comparing different non-linear cross-inhibition functions, we find that their performances are relatively close, with the smooth, linearly bounded sigmoid being the only one that underperforms. The effectiveness of a strong, sudden activation of cross-inhibition was previously reported by Talamali et al. <cit.>, mainly for the choosing accuracy. 


In contrast, our approach demonstrates a comprehensive enhancement in both overall consensus accuracy and decision time.




So far, we have tested consensus dynamics by fixing the discovery probabilities and varying the interdependence. Increasing interdependence reduces the amount of individual exploration by prioritizing peers' options. This strategy has been shown to optimize consensus accuracy, even in the absence of cross-inhibition <cit.>, although it may extend decision time <cit.>. 

When the discovery probabilities increase (with fixed λ), the system more readily incorporates environmental information. This reduces decision time but leads to poorer final consensus, especially when options are of similar quality <cit.>. Thus, π_α can also be viewed as a noise parameter. In such scenarios, cross-inhibition is crucial to avoid deadlocks, as reported in various case studies <cit.>. 

Fig. <ref> shows the performance ratio of non-linear cross-inhibition responses as discovery probabilities increase.
The corresponding performance variables are plotted in Supplemental Figure SF4 <cit.>. As the noise in the system increases, non-linear cross-inhibition yields better performance.  
Interestingly, examining the individual quantities f_2^* and t_ss on increasing π_1 = π_2 = π_1,2, we observe opposing trends for the linear and non-linear model. The linear response yields decreasing f_2^* while increasing t_ss; in contrast, non-linear responses reverse this trend. Consequently, weakening the stop signals from the losing population becomes essential in this context.
Comparing the different non-linear responses, we observe that the performance is consistently higher for the standard sigmoid functions than for the linearly bounded sigmoids. In each case, the sharp response also grants better performance.



Conclusions – 
In this letter, we investigate non-linear cross-inhibition interactions in decentralized decision making models inspired by house-hunting honeybees. The primary design goal is to weaken an individual's response to stop signals when they are received from a small fraction of the population. We model this behavior using two non-linear functions, tested with different parameters (Fig. <ref>).
Focusing on binary decision tasks, we demonstrate that non-linear cross-inhibition results in higher consensus (the fraction of the population committed to the chosen option) and quicker decisions. These two benefits come at the cost of reducing accuracy in reliably choosing the best quality option. Nonetheless, in decisions made among options with close qualities, a stronger and quicker decision for a "good enough" option may be more beneficial than a weaker consensus or a slower decision process that yields the absolute best option <cit.>.
Our results thereby open promising avenues for future research in decentralized collective decision-making and practical applications in swarm robotics.


We acknowledge financial support from projects PID2022-137505NB-C21 and PID2022-137505NB-C22 funded by MICIU/AEI/10.13039/501100011033, and by “ERDF A way of making Europe”.














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 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.





