Time-reversal of an unknown quantum state

For decades, researchers have sought to understand how the irreversibility of the surrounding world emerges from the seemingly time-symmetric, fundamental laws of physics. Quantum mechanics conjectured a clue that final irreversibility is set by the measurement procedure and that the time-reversal requires complex conjugation of the wave function, which is overly complex to spontaneously appear in nature. Building on this Landau-Wigner conjecture, it became possible to demonstrate that time-reversal is exponentially improbable in a virgin nature and to design an algorithm artificially reversing a time arrow for a given quantum state on the IBM quantum computer. However, the implemented arrow-of-time reversal embraced only the known states initially disentangled from the thermodynamic reservoir. Here we develop a procedure for reversing the temporal evolution of an arbitrary unknown quantum state. This opens the route for general universal algorithms sending temporal evolution of an arbitrary system backward in time. The concept of arrow of time expressing time asymmetric nature is intrinsically related to the second law of thermodynamics and increase of entropy. The authors show how a thermodynamic bath expected to add to entropy increase can be the key to time reversal for an unknown quantum state, paving the way to universal algorithms sending temporal evolution of an arbitrary system backward in time.

A n origin of the arrow of time, the concept coined for expressing one-way direction of time, is inextricably associated with the Second Law of Thermodynamics 1 , which declares that entropy growth stems from the system's energy dissipation to the environment 2-6 . Thermodynamic considerations [7][8][9][10][11][12][13][14][15][16][17] , combined with the quantum mechanical hypothesis that irreversibility of the evolution of the physical system is related to measurement procedure 18,19 , and to the necessity of the anti-unitary complex conjugation of the wave function of the system for time reversal 20 , led to understanding that the energy dissipation can be treated in terms of the system's entanglement with the environment 1, [21][22][23][24] . The quantum mechanical approach to the origin of the entropy growth problem was crowned by finding that in a quantum system initially not correlated with an environment, the local violation of the second law can occur 25 . Extending then the solely quantum viewpoint on the arrow of time and elaborating on the implications of the Landau-Neumann-Wigner hypothesis [18][19][20] , enabled to quantify the complexity of reversing the evolution of the known quantum state and realize the reversal of the arrow of time on the IBM quantum computer 26 .
In all these past studies, a thermodynamic reservoir at finite temperatures has been appearing as a high-entropy stochastic bath thermalizing a given quantum system and increasing thus its thermal disorder, hence entropy. We find that most unexpectedly, it is exactly the presence of the reservoir that makes it possible to prepare the high-temperature thermal states of an auxiliary quantum system governed by the same HamiltonianĤ as the Hamiltonian of a given system. This enables us to devise the operator of the backward-time evolutionÛ ¼ expðiĤtÞ reversing the temporal dynamics of the given quantum system. The necessary requirement is that the dynamics of the both, auxiliary and given, systems were governed by the same HamiltonianĤ. The time-reversal protocol comprises the cyclic sequential process of quantum computation on the combined auxiliary and the given systems and the thermalization process of the auxiliary system. A universal time-reversal procedure of an unknown quantum state defined through the density matrixρðtÞ of a quantum system S will be described as a reversal of the temporal system evolutionρðtÞ !ρð0Þ ¼ expðiĤt=_ÞρðtÞ expðÀiĤt=_Þ returning it to system's original stateρð0Þ. Importantly, we need not know the quantum state of this system in order to implement the arrow of time reversal. A dramatic qualitative advance of the new protocol is that it eliminates the need of keeping an exponentially huge record of classical information about the values of the state amplitudes. Moreover, the crucial step compared with the protocol of time reversal of the known quantum state 26 is that we now lift the requirement that initially the evolving quantum system must be a pure uncorrelated state. Here, we develop a procedure where the initial state can be a mixed state and, therefore, include correlations due to system's past interaction with the environment.

Results
Universal procedure. The calculations are organized as follows. First, we describe how the time reversal of an unknown state can be implemented in a universal manner and estimate its computational complexity. Next, we outline a somewhat more resourcedemanding procedure, where, however, one can relax the need of knowing the HamiltonianĤ. Then we show that if in addition to the quantum system S one is provided by an auxiliary system A, so that dim S ¼ dim A, whose dynamics is governed by the same HamiltonianĤ, one can deviseÛ y ðtÞ without knowing an exact form ofĤ. Finally, we discuss how the partial knowledge on the stateρðtÞ can reduce and optimize the complexity of the timereversal procedure. The starting point of the reversal procedure is drawn from the observation of S. Lloyd et al. 27 that having an ancilla system in a stateσ one can approximately construct a unitary operation expðÀiωσδtÞ acting on a system S simulating its evolution under HamiltonianĤ a ¼ _ωσ during the infinitesimal time interval δt. Here, ω refers to some arbitrary rate, which for a moment, we leave unspecified. Having N identical copies of ancillas, one generates a finite time evolution ρðtÞ ! ρðt þ τÞ ¼ e Àiωτσρ ðtÞe þiωτσ over the time interval τ = Nδt with the accuracy ∝ (ωτ) 2 /N (see "Methods"). The first step of the time-reversal procedure is then constructing the density matrixσ. Consider the density operator defined by the given finite-dimensional HamiltonianĤ having the maximal eigenvalue ϵ max :σ where Z ¼ ϵ max dim S À Tr fĤg is the normalization factor. Then the Lloyd (LMR) procedure maps the initial density matrixρðtÞ tô One sees that application of the LMR procedure with the specific density matrix σ realizes approximately the time-reversed evolution of the systemρ to a backward delay τ R = (ℏω/Z)τ. The accuracy δρðτÞ of such a time-reversal procedure is given by (see "Methods"), where jjÂjj is the operator norm: (3) and (4), one draws two important conclusions. First, the above time-reversal procedure for a backward delay τ R requires time τ to be completed. Therefore, while exercising the reversal, the system still maintain the forward evolution governed by its own Hamiltonian. Taking this into account, one has to modify Eq. (3) toρ which immediately poses the constraint on the operation rate ω of the LMR procedure: the actual time reversal occurs only for ℏω > Z. If this constraint is not satisfied, the time-reversal procedure only slows down the forward time evolution of the system. For a quantum system S, the threshold rate Z/ℏ is proportional to the Hilbert space dimension dim S: , which is typically an exponentially large number. In particular, in order to make the time reversal with the same rate as the forward time evolution, one has to demand ω > 2Z/ℏ. This brings straightforwardly the second conclusion: as far as ω is large, the infinitesimal time step δt of the procedure has to be small so that ωδt ≪ 1, therefore the number N has to be large. Indeed, fixing the backward delay τ R , the operation rate ω = 2Z/ℏ, and setting the reversal accuracy ϵ: jjδρðτ ¼ τ R Þjj ≤ ϵ one finds from Eq. (4): whereτ ¼ω À1 is the typical timescale of the system dynamics, and jjσjj / ðdim SÞ À1 ( jjρðtÞjj is assumed. Equation (6) implies that the computational complexity of the time-reversal procedure for an unknown quantum state is proportional to the square of the system's Hilbert space dimension. In contrast, the time reversal of a known pure quantum stateρðtÞ ¼ ψðtÞ j i ψðtÞ h j is proportional to the dimension of the Hilbert space, which is swept by the system in the course of its forward time evolution ψð0Þ j i! ψðtÞ j i 26 . As follows from Eq. (6), the time-reversal computational cost of an unknown pure state is maximal as long as jjρjj ¼ 1 in this case. For a mixed high-entropy stateρ, the reversal complexity is reduced: given a stateρ with the entropy S ρ ¼ ln ðdim SÞ À k ln ð2Þ, where only k ( log 2 ðdim SÞ bits of information is known, the upper estimate for the reversal complexity is given by (see "Methods") Having complete information about the HamiltonianĤ allows one to construct a corresponding quantum circuit realizing the forward time evolution operatorÛ ¼ expðÀiĤt=_Þ through a specific fixed set G of universal quantum gates:Û ¼Û 1 Á Á ÁÛ N , U i 2 G. As far as G is an universal set, for everyÛ i 2 G one can construct the inverse gateÛ y i . Therefore, the time-reversed evolution operatorÛ y can be constructed in a purely algorithmic way given the gate decomposition ofÛ. Thus, the above procedure may appear extremely ineffective for a practical time-reversal task. However, the situation turns completely different if we relax the requirement of the exact knowledge of H and assume that one, instead, is provided by the equivalent copy of the system S governed by the same HamiltonianĤ.
Auxiliary system. Let one be equipped with the thermodynamic bath at the temperature T = β −1 in addition to the ancilla. One can then thermalize the ancilla and prepare it in the equilibrium state σ β ¼ Z À1 β expðÀβĤÞ with Z β ¼ TrfexpðÀβĤÞg being a statistical sum. For high-enough temperature β → 0, one has βϵ max $ 1 and, therefore, σ β % Z À1 β ð1 À βĤÞ which gives the desired state of the ancilla to implement the reverse evolution through the LMR procedure. In this case, As can be seen from the above equation the actual time reversal requires the operation rate of the LMR procedure to exceed the threshold The approximation error δρ splits now to two contributions, δρ ¼ δρ 1 þ δρ 2 , where δρ 1 is the approximation error resulting from the LMR procedure, see Eq. (4) withσ !σ β , while the error δρ 2 describes the error due to the β expansion of the thermal statê σ β . Assuming ω = 2ω th , i.e. the backward evolution goes with the same rate as the forward time evolution one finds Then for jjσ β jj ( jjρðtÞjj one can estimate the net error as where τ β = ℏβ. The temperature dependence of two error contributions in Eq. (11) oppositely depends on the inverse temperature: the error due to thermal expansion (second term) reduces as β → 0, while the error due to LMR dynamics (first term) increases with decreasing β. For a given reverse time delay τ and the number of LMR iterations N ) Z 2 β % ðdim SÞ 2 , one has an optimal temperature and the corresponding net accuracy of the reversal procedure is then given by Comparing with the case of the known Hamiltonian time-reversal procedure, see Eq. (6), the reversal complexity here is again proportional to the square of the system's Hilbert space dimension, but, at the same time, has more adverse scaling with the reversal duration and the net accuracy. The above analysis does not need any prior information about the stateρ which would require very high temperature of the auxilliary thermostat in order to cover all the possible energy states of the system's Hilbert space that finally results in a tremendously high rate $ _β dim S of the LMR procedure. If, however, some information about the energy content of the statê ρ is available, one can appreciably reduce the reversal cost. Indeed, let the stateρ have the average energy E ¼ TrfρĤg with an energy variance ðδEÞ 2 ¼ TrfρððĤ À EÞ 2 Þg. Then one can present the density matrix as the result of the low-energy contribution,ρ < ¼PρP=TrfPρg and the high-energy remainder h j is a projection operator to the subspace with energies below some cutoff energy E max > E:ρ ¼ ð1 À ϵ E Þρ < þ ϵ Eρ > . The additional error due to truncating the system's Hilbert space to the lowenergy subspace is given by the constant ϵ E , which is a probability for the system to be found in the energy state E > E max , and, according to the Chebyshev inequality, is bound by Single-particle wave packet. Now, we consider an exemplary time-reversal procedure for a spreading single-particle wave packet with the quadratic spectrum. Let the packet at the time t = 0 be localized at the origin and have the Lorentzian shape with the width ξ 0 : A subsequent free evolution with quadratic HamiltonianĤ ¼ _ 2p2 =2m during the time interval τ > 0 broadens the particle's wave function into having the typical size ξ τ = ℏτ/mξ 0 or, equivalently, ξ τ =ξ 0 ¼ 4 Eτ=_, where E ¼ _ 2 =4mξ 2 0 is the average energy carried by the wave packet. The statistical sum Z β within the volume~ξ τ is given by where ν 1D ðEÞ ¼ ðm=2π 2 _ 2 EÞ 1=2 is one-dimensional density of states. Assuming E max $ E, the reversal complexity for the timereversal procedure with the accuracy ϵ is given by (see Eqs. (12) and (13)) The optimal inverse temperature of the thermostat is then given by Comparing this with the reversal complexity of a known state of the wave packet, N 0 ϵ $ ϵ À1 ðξ τ =ξ 0 Þ, see ref. 26 , one finds that the reversal of an unknown wave-packet state is a more laborious computational task.

Discussion
We have described the time-reversal procedure of an unknown mixed quantum state. The procedure relies on the ability to perform the LMR protocol and on the existence of an ancilla system whose dynamics is governed by the same Hamiltonian as the Hamiltonian of the reversed system, which is not required to be known to us. The reversal procedure is comprised of N ≫ 1 sequential applications of the LMR procedure to the joint state of the system and ancilla prepared in a thermal state. In contrast to the known state-reversal procedure, the introduced algorithm does not require to keep an information about all amplitudes of the reversed state. Yet the reversal complexity given by N scales typically as squared dimension of a Hilbert space spanned the unknown state. Moreover, the operation rate of the LMR procedure has to be sufficiently high to overrun the forward time evolution of the reversed system during the execution of the reversal protocol.
The experimental realization of such a protocol is a feasible yet challenging task. As a first step, it will require an upgrade of the existing design of quantum chips. In particular, one needs a set of interacting qubits (denoted by Q A ) capable to get thermalized ondemand being connected with the high-temperature environment. For superconducting qubits 28 , this can be implemented by coupling them with a transmission line, where the hightemperature thermal radiation is fed in, once one needs to set the qubits into a high-temperature state. Next, the second set of qubits Q B , dim Q B ¼ dim Q A is required, where one can store a quantum state prepared in the set Q A . Then the time-reversal procedure goes as follows. First, one prepares some state ψ A (0) of the qubits Q A , and lets it evolve according to an intrinsic Hamiltonian of the qubits Q A : ψ A ð0Þ ! ψ A ðτÞ ¼ e ÀiĤ A τ=_ ψ A ð0Þ. Second, at the end of the evolution, one swaps the states between Q A and Q B , ψ A ↔ ψ B . We assume that the set Q B can keep its quantum state untouched for a sufficient time. The procedure then continues as described above: one subsequently thermalizes the set Q A and implements the joint LMR evolution e ÀiωδtŜ ABρ A ρ B e iωδtŜ AB . As a result, the qubits Q B will undergo the time-reversed dynamics under the same HamiltonianĤ A . This procedure is to be implemented on the emergent quantum computers with the on-demand thermalizable qubits.
High-entropy-state-reversal complexity. Here, we derive the Eq. (7) for the time-reversal complexity of the stateρ with the entropy S ¼ ln dimðNÞ À k ln ð2Þ, where N ¼ dimðSÞ is the Hilbert space dimension of the system. The norm jjρjj is given by its maximum eigenvalue jjρjj ¼ p 1 > p i , i = 2, …N of the density operator. The von Neumann entropy can be decomposed into a sum wherep i ¼ p i =ð1 À p 1 Þ with P N i¼2pi ¼ 1, HðxÞ ¼ Àxln ðxÞ À ð1 À xÞln ð1 À xÞ ≤ ln ð2Þ. Let us find a maximal possible p 1 for a given S. One sees straightforwardly that p 1 is maximal if allp i , i = 2, …N are uniform and Eq. (25) is reduced to ln ðNÞ À k ln ð2Þ ¼ Hðp 1 Þ þ ð1 À p 1 Þln ðN À 1Þ: For N ≫ 1, one can assume p 1 ≪ 1 and get the approximate solution p 1 % k=log 2 ðNÞ that results in Eq. (7).

Data availability
Data sharing is not applicable to this article, as no data sets were generated or analyzed during this study.