Abstract
Uncovering the origin of the “arrow of time” remains a fundamental scientific challenge. Within the framework of statistical physics, this problem was inextricably associated with the Second Law of Thermodynamics, which declares that entropy growth proceeds from the system’s entanglement with the environment. This poses a question of whether it is possible to develop protocols for circumventing the irreversibility of time and if so to practically implement these protocols. Here we show that, while in nature the complex conjugation needed for time reversal may appear exponentially improbable, one can design a quantum algorithm that includes complex conjugation and thus reverses a given quantum state. Using this algorithm on an IBM quantum computer enables us to experimentally demonstrate a backward time dynamics for an electron scattered on a twolevel impurity.
Introduction
A fundamental question of the origin of irreversibility of time emerged already in classical statistical physics^{1,2,3,4,5} and has been remaining ever since a subject of an continuous attention^{6,7,8}. Intense researches revealed several aspects of this problem. One of them is a statistical mechanics view discussing the irreversibility problem in the context of the fluctuation theorem^{9,10,11,12,13,14,15,16}. In particular, it was quantitatively described and shown experimentally that in a finite temporal interval the time reversed dynamics can emerge^{17}. The quantum systems were discussed in^{18} where the positive entropy production rate was experimentally demonstrated on a single spin1/2 particle, while in^{19} the negative entropy production rate in the presence of a Maxwell’s Demon was observed for spin1/2 quantum system. Moreover, the full quantum treatment have shown theoretically^{20,21} and later experimentally^{22} that the presence of initial mutual correlations between subparts of a quantum system may lead to a local violation of thermodynamical laws and hence to the thermodynamic arrow of time reversal. Even in a quantum system initially not correlated with an environment, the local violation of the Second Law can occur, as it was demonstrated, with the mathematical rigor^{23}, in the framework of the quantum channel theory^{24}. Most of the above works were based in a good part on thermodynamic considerations. From the slightly different perspective this question was discussed in the seminal work by Zurek^{25}, who looked at the irreversibility issue from the angle of the loss of predictability with the time. A solely quantum mechanical aspect of the problem was stressed by Landau^{26} and von Neumann^{27} who related irreversiblity to the process of a macroscopic measurement. In^{28} the arrow of time dilemma was addressed from the point of view systemobserver considerations, but later this approach was criticized in^{29}. Here, in the spirit of quantum mechanics, we elaborate on the implications of the Wigner’s result^{30} that time reversal operation is antiunitary because it requires complex conjugation. We demonstrate that this emerging antiunitarity predicates that the universal time reversal operation does not spontaneously appear in nature. To make the time reversal possible, one would need a supersystem manipulating the quantum system in question. In most of the cases, such a supersystem cannot materialize spontaneously. As an illustration, we use the simplest systems of a single or two particles subject to electromagnetic fluctuations. We show that even the evolution of these single or twoparticle states in a free space generates the complexity that renders spontaneous time reversal either highly improbable or actually impossible. We expect that if irreversibility emerges even in the systems that simple, than, even, more it should appear in the more complex systems. In what follows, we quantify the complexity of the preparation of the timereversed quantum state and the probability of its spontaneous emergence. We show that the timereversal complexity of the developed quantum state scales linearly with the dimension of the Hilbert space swept by the system in the course its forward time evolution, but that one can devise an administering supersystem artificially. This is implemented experimentally by modeling a real system, the electron scattered on the twolevel systems, on the IBM quantum computer. In this respect we utilize the conjectures by Lloyd^{6}.
Further, a principal possibility of occurring of the time reversal was discussed in^{20}.
Reversal of The Spreading Wave Packet
That in quantum mechanics in order to execute a time reversal operation one has to perform complex conjugation of the wave function, implies that the time reversal operator \(\hat{{\mathscr{T}}}\) is a product of a complex conjugation operator \(\hat{{\mathscr{K}}}\) and a unitary rotation \({\hat{U}}_{R}\), i.e. \(\hat{{\mathscr{T}}}={\hat{U}}_{R}\,\hat{{\mathscr{K}}}\), where for any \({\rm{\Psi }}\), \(\hat{{\mathscr{K}}}{\rm{\Psi }}={{\rm{\Psi }}}^{\ast }\). This operation not only reflects velocities like in the classical physics, but also reverses phases of the wave function components. A general universal operation that can reverse any arbitrary wave function, does not exist in nature. Yet, some special \({\rm{\Psi }}\)dependent operation such that \({\hat{U}}_{{\rm{\Psi }}}{\rm{\Psi }}={{\rm{\Psi }}}^{\ast }\) can exist and below we explicitly construct such an operation for a system of qubits. To that end, one has to design a supersystem that is external with respect to the system of interest and which is capable to implement the purposeful manipulating on the given system. In nature, in the simplest case of a single particle, the role of such a supersystem can be taken up, for example, by the fluctuating electromagnetic field. To gain an insight into how this works, let us consider a wave packet corresponding to the particle with the square energy dispersion, \(\varepsilon ={p}^{2}/2m\), where p is the particle momentum and m is the particle mass, propagating in space, see Fig. 1. The electromagnetic field is assumed to be predominantly weak except for rare fluctuations. Thus, the spreading of the wave packet is coherent. At large times \(\tau \) the wave packet spreads as
where f(q) is a Fourier image of the initial spatial wave function. The phase of \({\rm{\Psi }}\) changes as a result of the action of the fast fluctuation of an external potential, i.e. with the potential that changes on the times much shorter than the characteristic time of the phase change. To set the fluctuation that complex conjugates \({\rm{\Psi }}\), let us divide the coordinate space into a large number of the elemental cells δx_{n} where a wavefunction’s phase \(\phi (x,\tau )\) changes slowly and look for a fast electromagnetic potential fluctuation \(V(x,t)\) which is smooth on the cell’s scale and reverts the phase of the wavefunction: \(\int \,dt\,eV({x}_{n},t)/\hslash =\,2\phi ({x}_{n},\tau )\). If during the \(\tau \) the wave packet (1) has spread from the size L_{0} to the size \({L}_{\tau }=\hslash \tau /m{L}_{0}\), it would require \(N\sim {\epsilon }^{1/2}({L}_{\tau }/{L}_{0})\) elementary cells to approximately revert the quantum state \({\rm{\Psi }}(x,\tau )\to {\tilde{{\rm{\Psi }}}}^{\ast }(x,\tau )\) with the probability \(1\epsilon \): \(\langle {\tilde{{\rm{\Psi }}}}^{\ast }(x,\tau ){{\rm{\Psi }}}^{\ast }(x,\tau )\rangle {}^{2}=1\epsilon \), see Supplementary Information (SI). Then the probability of the spontaneous reversal, i.e. the probability of the appearance of the required electromagnetic potential fluctuation, estimates as 2^{−N}. Now we determine the typical time scale \(\tau \) on which the spontaneous time reversal of a wavepacket can still occur within the universe lifetime \({t}_{U}\sim 4.3\times {10}^{17}\) sec. The latter is obtained from the estimate \({2}^{N}\simeq \tau /{t}_{U}\), where the number of cells \(N\sim {\epsilon }^{1/2}\,(\langle E\rangle \tau /\hslash )\) is expressed through the average particle energy \(\langle E\rangle ={\hslash }^{2}/m{L}_{0}^{2}\). As a typical average energy of the wavepacket we take the energy corresponding to the current universe temperature 2.72 K, and arrive at \(\tau \simeq 6\times {10}^{11}\) sec. One thus sees that even in the discussed simplest possible example of a single quantum particle the time reversal is already a daunting task where even with the GHz rate of attempts, the required fluctuation is not observable within the universe lifetime. The above arguments reveal that, in quantum mechanics, time irreversibility emerges already on the level of a single evolving particle.
Now we consider a more complex example and demonstrate that a separable state
of two particles can not be reverted by classical field fluctuations in the case where particle’s wave functions overlap. Let all particles have the same electric charge q and interact with a classical electric potential v(x, t). The potential fluctuations produce phase shifts \(\int \,dtqv(x,t)/\hslash \). Accordingly the proper fluctuations capable to reverse the quantum state should satisfy the condition \({\phi }_{1}({x}_{1})+{\phi }_{2}({x}_{2})\) + \(\int \,dt[qv({x}_{1},t)+qv({x}_{2},t)]/\hslash \) = \(\,{\phi }_{1}({x}_{1}){\phi }_{2}({x}_{2})\). For \({x}_{1}={x}_{2}\) it implies \(\int \,dtqv(x,t)/\hslash =\,{\phi }_{1}(x){\phi }_{2}(x)\), and therefore at \({x}_{1}\ne {x}_{2}\) one has to satisfy the condition \({\phi }_{2}({x}_{2})+{\phi }_{1}({x}_{1})={\phi }_{2}({x}_{1})+{\phi }_{1}({x}_{2})\) which, in general, does not hold.
Quantum entanglement introduces the next level of complexity for the timereversal procedure. Consider a twoparticle state \({\rm{\Psi }}({x}_{1},{x}_{2})={\rm{\Psi }}({x}_{1},{x}_{2}){e}^{i\phi ({x}_{1},{x}_{2})}\) with the nonseparable phase function \(\phi ({x}_{1},{x}_{2})={a}_{1}({x}_{1}){b}_{1}({x}_{2})+\)\({a}_{2}({x}_{1}){b}_{2}({x}_{2})\). In this situation even for the nonoverlapping particles with \({\rm{\Psi }}({x}_{1},{x}_{2})=0\) for \({x}_{1}={x}_{2}\) the twoparticle state can not be reversed by an interaction with classical fields. Let one access the particles by different fields which induce separate phase shifts \({\rm{\Psi }}({x}_{1},{x}_{2})\to {\rm{\Psi }}({x}_{1},{x}_{2}){e}^{i({\varphi }_{1}({x}_{1})+{\varphi }_{2}({x}_{2}))}\). The induced phase shifts should satisfy the relation: \({\varphi }_{1}({x}_{1})+{\varphi }_{2}({x}_{2})=\,2\phi ({x}_{1},{x}_{2})\), therefore for any three points \({x}_{1}\ne {x}_{2}\ne {x}_{3}\) the following conditions should hold
Subtracting these relations one gets \({\varphi }_{2}({x}_{2}){\varphi }_{2}({x}_{3})\) = \(\,2{a}_{1}({x}_{1})\,({b}_{1}({x}_{2}){b}_{1}({x}_{3}))\) − \(2{a}_{2}({x}_{1})\,({b}_{2}({x}_{2}){b}_{2}({x}_{3}))\) where the left hand side does not depend on x_{1} and therefore one has to assume a_{1} and a_{2} to be constant. This, however, contradicts the nonseparability assumption for \(\phi ({x}_{1},{x}_{2})\).
An entangled twoparticle state with a nonseparable phase function can naturally emerge as a result of scattering of two localized wavepackets^{31}. However, as we have seen, the generation of the timereversed state, where a particle gets disentangled in the course of its forward time evolution, requires specific twoparticle operations which, in general, cannot be reduced to a simple twoparticle scattering.
The above consideration enables us to formulate important conjectures about the origin of the arrow of time: (i) For the time reversal one needs a supersystem manipulating the system in question. In the most of the cases, such a supersystem cannot spontaneously emerge in nature. (ii) Even if such a supersystem would emerge for some specific situation, the corresponding spontaneous time reversal typically requires times exceeding the universe lifetime.
A matterofcourse supersystem of that kind is implemented by the socalled universal quantum computer. It is capable to efficiently simulate unitary dynamics of any physical system endowed with local interactions^{32}. A system’s state is encoded into the quantum state of the computer’s qubit register and its evolution is governed by the quantum program, a sequence of the universal quantum gates applied to the qubit register. There exists a panoply of ways by which a quantum state of a system can be encoded into the states of the quantum computer. Indeed, choosing a proper dimension of the quantum computer register one can swap its state \({\psi }_{0}{\rangle }_{{\rm{reg}}}\) with the system’s quantum state, \({\rm{\Psi }}{\rangle }_{{\rm{sys}}}\), by the unitary operation \({\hat{U}}_{{\rm{SWAP}}}{\psi }_{0}{\rangle }_{{\rm{reg}}}\otimes {\rm{\Psi }}{\rangle }_{{\rm{sys}}}=\psi {\rangle }_{{\rm{reg}}}\otimes {{\rm{\Psi }}}_{0}{\rangle }_{{\rm{sys}}}\), where the mapping \({\rm{\Psi }}{\rangle }_{{\rm{sys}}}\to \psi {\rangle }_{{\rm{reg}}}\) completes the encoding task. Such an encoding procedure is universal i.e. it does not require the knowledge of the system state \({\rm{\Psi }}{\rangle }_{{\rm{sys}}}\). However, nonphysical encodings might be suggested which can not be accomplished by unitary transformation. One of the ways to do that was proposed in^{33} where the real and the imaginary components of the system’s wave function were separately mapped onto the different Hilbert subspaces of the auxiliary system, i.e. quantum computer. Within this representation of the initial quantum system, the complex conjugation can be formulated as a universal unitary rotation of the wave function of the auxiliary system. However, the mapping itself is not a universal unitary operation as follows from the superposition principle arguments. This means that the approach of^{33} merely lifts the problem of the nonunitarity of the quantum conjugation hiding it in the nonunitarity of the mapping procedure. At variance, in what follows we address the time reversal of the original physical system without nonphysical mapping it on some completely different system unrelated to the original one. We start with formulating general principles of constructing timereversal algorithms on quantum computers and, in the next section, present a practical implementation of a fewqubit algorithm that enabled experimental time reversal procedure on the public IBM quantum computer.
General Time Reversal Algorithms
Consider a quantum system initially prepared in the state \({\rm{\Psi }}(t=0)\) and let it evolve during the time \(\tau \) into the state \({\rm{\Psi }}(\tau )=\exp (\,\,iH\tau /\hslash ){\rm{\Psi }}(0)\). Let us find a minimal size of a qubit register needed to simulate the dynamics of a system \({\rm{\Psi }}(0)\to {\rm{\Psi }}(\tau )\) with a given fidelity \(1\epsilon \). Let us choose a finite set of time instances \({t}_{i}\in [0,\tau ]\), \(i=0,\ldots N^{\prime} \) subject to a condition \(\langle {\rm{\Psi }}({t}_{i}){\rm{\Psi }}({t}_{i+1}\rangle ){}^{2}=1\epsilon \) with \({t}_{0}=0\) for some small \(\epsilon > 0\). Then at any time instant \(t\in [0,\tau ]\) a state \({\rm{\Psi }}(t)\) can be approximated by the discrete set of states \(\{{\rm{\Psi }}({t}_{i}),i=0,\ldots {\mathscr{N}}^{\prime} \}\) with the fidelity \(1\epsilon \). The set of states \(\{{\rm{\Psi }}({t}_{i})\}\) spans the Hilbert subspace \({\mathscr{S}}\) of the dimension \({\mathscr{N}}\le {\mathscr{N}}^{\prime} \). Therefore, \({\mathscr{N}}\) basis vectors \({e}_{i}\rangle \in {\mathscr{S}}\) can be represented by \({\mathscr{N}}\) orthogonal states of the qubit register, \({e}_{i}\rangle \to {\overrightarrow{b}}_{i}\rangle \equiv {b}_{0}{b}_{1}\ldots \rangle \). The corresponding qubit Hamiltonian \(\hat{H}\) which mimics the original Hamiltonian \(\hat{ {\mathcal H} }\) is then defined by the relation \({(\hat{H})}_{ij}\equiv \langle {\overrightarrow{b}}_{i}\hat{H}{\overrightarrow{b}}_{j}\rangle =\langle {e}_{i}\hat{ {\mathcal H} }{e}_{j}\rangle \).
Below we introduce two encoding procedures \({e}_{i}\rangle \to {\overrightarrow{b}}_{i}\rangle \). In the first, sparse coding approach, one assigns a separate qubit to each state \({e}_{i}\rangle \), \(i\in [0,{\mathscr{N}}1]\) and encodes the state \(\psi (\tau )\) into the \({\mathscr{N}}\)qubit state \(\psi \rangle ={\sum }_{i=0}^{{\mathscr{N}}1}\,{\psi }_{i}\,{0}_{0}\ldots {1}_{i}\ldots {0}_{{\mathscr{N}}1}\rangle \). The second approach is a dense coding scheme where one records the state \(\psi (\tau )\) into a state of \(n=\,{\rm{int}}\,[{\mathrm{log}}_{2}\,({\mathscr{N}})]+1\) qubits \(\psi \rangle ={\sum }_{i=0}^{{\mathscr{N}}1}\,{\psi }_{i}\,i\rangle \), where int[x] is the closest upper integer to x: \(x\le \,{\rm{int}}\,[x]\), \(i\rangle \equiv {b}_{0}\ldots {b}_{n1}\rangle \) is a computational basis state corresponding a binary representation of the number \(i={\sum }_{k=0}^{n1}\,{b}_{k}{2}^{n1k}\).
A timereversal operation \(\hat{{\mathscr{T}}}\) of the qubit register can be presented as a product \(\hat{{\mathscr{T}}}={\hat{U}}_{R}\hat{{\mathscr{K}}}\) of the complex conjugation operator \(\hat{{\mathscr{K}}}\), \(\hat{{\mathscr{K}}}({\psi }_{i}i\rangle )\equiv {\psi }_{i}^{\ast }i\rangle \), and some unitary operator \({\hat{U}}_{R}\), whose form is defined by the Hamiltonian \(\hat{H}\), \({\hat{U}}_{R}={\hat{U}}_{H}^{\dagger }{\hat{U}}_{H}^{\ast }\), where \(\hat{H}={\hat{U}}_{H}^{\dagger }\,{\rm{diag}}\,\{{E}_{1}\ldots {E}_{n}\}{\hat{U}}_{H}\) see SI. Therefore, ?in order to implement the timereversal operation \(\hat{{\mathscr{T}}}\) one needs to know the Hamiltonian \(\hat{H}\) explicitly. Note, that quantum computer is able to simulate unitary dynamics governed by an arbitrary Hamiltonian including those that do not correspond any physical system (for example, some nonlocal Hamiltonian). It is known, that the joint transformation of the charge conjugation, parity inversion, and time reversal is considered as an exact symmetry of all known laws of physics, and, therefore, the qubit Hamiltonian \(\hat{H}\), which corresponds to a real physical system, has to honor this symmetry as well. Therefore, the unitary operation describing evolution of the physical system \({\hat{U}}_{R}\) is generally known and represents a transformation which is inherited from the timereversal symmetry of the original Hamiltonian \(\hat{ {\mathcal H} }\). In particular, if the qubit Hamiltonian \(\hat{ {\mathcal H} }\) is real, then the corresponding evolution operator \(\hat{U}(\tau )\) is symmetric that entails \({\hat{U}}_{R}={\bf{1}}\).
In the following we assume the unitary \({\hat{U}}_{R}\) to be known and focus on the unitary implementation of a complex conjugation operation \(\hat{{\mathscr{K}}}\), \(\hat{{\mathscr{K}}}\to {\hat{U}}_{\psi }\). In particular, we quantify a complexity of such implementation as a number of elementary quantum gates or/and auxiliary qubits needed to implement \({\hat{U}}_{\psi }\). For a sparse coding scheme, the complex conjugation of the \({\mathscr{N}}\)qubit state \(\psi \rangle ={\sum }_{i=0}^{{\mathscr{N}}1}\,{\psi }_{i}{e}^{i{\phi }_{i}}\,{0}_{0}\ldots {1}_{i}\ldots {0}_{{\mathscr{N}}1}\rangle \) can be accomplished by the unitary operation \({\hat{U}}_{\psi }^{(1)}={\prod }_{i=0}^{{\mathscr{N}}1}\,\otimes {\hat{T}}_{i}(\,\,2{\phi }_{i})\) where \({\hat{T}}_{i}(\phi )\) is the single qubit operation: \({\hat{T}}_{i}(\phi ){0}_{i}\rangle ={0}_{i}\rangle \) and \({\hat{T}}_{i}(\phi ){1}_{i}\rangle ={e}^{i\phi }{1}_{i}\rangle \). Consequently, the sparse coding scheme does not require the most “expensive” twoqubit gates at all but do require a large number \({\mathscr{N}}\) of qubits. For the dense coding scheme the situation is the opposite: this scheme involves only a logarithmically smaller number n of qubits but instead requires implementation of 2^{n} nqubit conditional phase shift operations: \(\hat{{\mathscr{K}}}\to {\hat{U}}_{\psi }^{(2)}={\sum }_{j=0}^{{2}^{n}1}\,j\rangle \langle j{e}^{2i{\phi }_{j}}\) which add proper phases to each component of the state \(\psi \rangle \): \({\hat{U}}_{\psi }^{(2)}\psi \rangle ={\psi }^{\ast }\rangle \). Therefore, \({\hat{U}}_{\psi }^{(2)}\) must involve twoqubit gates, i.e. conditionalNOT (CNOT) gates. We quantify the complexity of the dense coding scheme by a number \({N}_{\oplus }\) of CNOT gates needed to implement it. Each phase shift operation \({\hat{{\rm{\Phi }}}}_{i}(\phi )\equiv i\rangle \langle i{e}^{i\phi }\) can be build with the help of \(n1\) ancillary qubits and \(\mathrm{2(}n\mathrm{1)}\) Toffoli gates, as shown in Fig. 2A. In total, it requires \({N}_{\oplus }[{\hat{U}}_{\psi }^{(2)}]=12(n1){2}^{n}\sim 12{\mathscr{N}}\,{\mathrm{log}}_{2}({\mathscr{N}})\) CNOT gates. However, such an arrangement is nonoptimal as it involves an excess usage of Toffoli gates. Indeed, let us consider two states, \(j\rangle ={b}_{0}{b}_{1}\ldots {b}_{n1}\rangle \) and \(j^{\prime} \rangle ={b^{\prime} }_{0}{b^{\prime} }_{1}\ldots {b^{\prime} }_{n1}\rangle \), with coincident two older bits \({b}_{0}={b^{\prime} }_{0}\), \({b}_{1}={b^{\prime} }_{1}\). The separate usage of phase shifts \({\hat{{\rm{\Phi }}}}_{j}({\phi }_{j})\) and \({\hat{{\rm{\Phi }}}}_{j^{\prime} }({\phi }_{j^{\prime} })\) involves the double check of b_{0} and b_{1} values. The better implementation checks b_{0} and b_{1} only once and conjugates phases of all states within a set \({b}_{0}{b}_{1}{b}_{2}\ldots {b}_{n1}\rangle \) within a separate circuit block. In fact, such optimization can be done for all subsequent junior bits b_{3}, b_{4}, see Fig. 2B, that can minimize the usage of Toffoli gates and reduce the reversal complexity to be linear in \({\mathscr{N}}\): \({{\mathscr{N}}}_{\oplus }\sim 24{\mathscr{N}}\), see SI. We thus arrive at the conclusion that The number of elementary operations needed for the exact time reversal procedure of the dynamics of a quantum system which in the course its evolution sweeps a Hilbert space of a dimension \({\mathscr{N}}\) is bounded from above by some number \({\mathscr{O}}({\mathscr{N}})\). If now we consider typical systems emerging in nature, then the entanglement generates the dimensionality, \({\mathscr{N}}\), that is exponentially large with respect to the number of particles involved.
Time Reversal Experiment
Now we are equipped to carry out an experiment implementing two and threequbit timereversal procedures utilizing the public IBM quantum computer. We model a one dimensional particle scattering on a twolevel impurity (TLI). The dynamics of the impurity is governed by a Hamiltonian, \({\hat{H}}_{{\rm{i}}}=\hslash \omega (\cos \,\alpha \,{\hat{\sigma }}_{z}+\,\sin \,\alpha \,{\hat{\sigma }}_{x})\). The scattering potential seen by the particle depends on the state of the TLS. The corresponding scattering operator has the form \({\hat{S}}_{i}=0\rangle \langle 0\otimes {\hat{S}}_{0}+1\rangle \langle 1\otimes {\hat{S}}_{1}\), where \({\hat{S}}_{0}\) and \({\hat{S}}_{1}\) are symmetric unitary scattering matrices of the TLI in a state \(\mathrm{0}\rangle \) or \(\mathrm{1}\rangle \). This scattering problem is modeled by the evolution of the qubit register \({\hat{U}}_{n{\rm{bit}}}{q}_{i}\rangle \otimes ({q}_{1}\rangle \otimes \cdots \otimes {q}_{n}\rangle )\), where \({q}_{i}\rangle \) qubit describes the state of the TLI and the remaining qubits describe the state of scattered particles. The basis states \({0}_{i}\rangle \) and \({1}_{i}\rangle \), \(i=1,\ldots n\) correspond to the left and right incoming/outgoing state of the ith particle. We consider the processes in which one or two incoming particles are scattered on the freely evolving TLI. We assume that both particles incident from the left being in a well localized ballistically propagating states and arrive at the impurity where they experience instantaneous respective scatterings at separate time instants \(t=\tau \) and \(t=2\tau \). The corresponding evolutions are described by unitary rotations \({\hat{S}}_{{\rm{i}}}^{(1)}\cdot {\hat{U}}_{{\rm{i}}}(\tau )\) and \({\hat{S}}_{{\rm{i}}}^{\mathrm{(2)}}\cdot {\hat{U}}_{{\rm{i}}}(\tau )\cdot {\hat{S}}_{{\rm{i}}}^{\mathrm{(1)}}\cdot {\hat{U}}_{{\rm{i}}}(\tau )\) for one and twoparticle situation, where \({\hat{U}}_{{\rm{i}}}(\tau )=\exp (\,\,i{\hat{H}}_{{\rm{i}}}\tau /\hslash )\) describes the free evolution of the TLI between the particles arrivals and \({\hat{S}}_{{\rm{i}}}^{(i)}\), \(i=1,2\) is the scattering operator for the ith particle. Next, in both cases we let the system freely evolve during a time \(\tau \) that makes the resulting 2qubit and 3qubit evolution operators: \({\hat{U}}_{{\rm{2bit}}}={\hat{U}}_{{\rm{i}}}(\tau )\cdot {\hat{S}}_{{\rm{i}}}^{\mathrm{(1)}}\cdot {\hat{U}}_{{\rm{i}}}(\tau )\) and \({\hat{U}}_{{\rm{3bit}}}={\hat{U}}_{{\rm{i}}}(\tau )\cdot {\hat{S}}_{{\rm{i}}}^{\mathrm{(2)}}\cdot {\hat{U}}_{{\rm{i}}}(\tau )\cdot {\hat{S}}_{{\rm{i}}}^{\mathrm{(1)}}\cdot {\hat{U}}_{{\rm{i}}}(\tau )\) more symmetric, that simplifies the form of \({\hat{U}}_{R}\) unitary rotation entering in the timereversal procedure. The 2qubit scattering model is endowed with the symmetric evolution operator \({\hat{U}}_{{\rm{2bit}}}\) and, therefore, its time reversal requires only the complex conjugation operation \(\hat{{\mathscr{T}}}=\hat{{\mathscr{K}}}\). At variance, the evolution operator \({\hat{U}}_{{\rm{3bit}}}\) of the 3qubit model is non symmetric and its time reversal requires an additional unitary rotation \(\hat{{\mathscr{T}}}={\hat{U}}_{R}\hat{{\mathscr{K}}}\). It follows from the relation \({{\rm{SWAP}}}_{12}\cdot {\hat{U}}_{{\rm{3bit}}}\cdot {{\rm{SWAP}}}_{12}={\hat{U}}_{{\rm{3bit}}}^{t}\), where \({{\rm{SWAP}}}_{12}{q}_{1}\rangle \otimes {q}_{2}\rangle ={q}_{2}\rangle \otimes {q}_{1}\rangle \) is the swap operation, that the required unitary operation \({\hat{U}}_{R}={{\rm{SWAP}}}_{12}\). The corresponding quantum circuits realizing \({\hat{U}}_{{\rm{2bit}}}\) and \({\hat{U}}_{{\rm{3bit}}}\) are shown on Fig. 3A and C, see the detail in SI.
According to the results of the Section 2, the unitary implementation of the complex conjugation for a 2 or 3qubit register will require 48 or 144 CNOT gates. These numbers are beyond of the present capability of the IBM public quantum computer due to the finite error rate 1.5–2.5% of its CNOT gates. Here we utilize an alternative to the Section 2 approach (see SI for details), which is based on the arithmetic representation of the nbit AND Boolean function^{34},
where \({b}_{0}\wedge {b}_{1}\wedge \ldots \) is 1 if and only if all \({b}_{0}={b}_{1}=\cdots =1\) and \({b}_{0}\oplus {b}_{1}\oplus \ldots \) is modulo 2 addition. Consider, for instance, a 2qubit situation. The complex conjugation \({\hat{U}}_{\psi }={\sum }_{k=0}^{3}\,{\hat{{\rm{\Phi }}}}_{k}({\phi }_{k})\) can be alternatively represented as
where \({\bar{b}}_{i}\) denotes a negation of the bit value. Summing up all four components in the exponent according to the Eq. (5) one decomposes \({\hat{U}}_{\psi }\) operator into a product of 1qubit and 2qubit phase shifts operations:
where a_{i}, β_{j} are linear combinations of the phase shifts \({\phi }_{k}\). The modulo 2 addition \({b}_{0}\oplus {b}_{1}\) can be effectively implemented with only two CNOT gates. This approach can be generalized for arbitrary number of qubits and turns out to be more efficient at small n since it does not need an ancillary qubits at all and requires \((n{\mathrm{1)2}}^{n1}\) CNOT gates for the complex conjugation of an arbitrary nqubit state that wins over the approach discussed in Section 2 for \(n\le 48\). In particular, at \(n=2\) and 3 one needs only two or eight CNOT gates, respectively. The corresponding 2 and 3qubit quantum circuits are shown on Fig. 3B and 3D.
The timereversal experiment runs in several steps: (i) The qubit register that is initially set into the state \(\psi (0)\rangle =0\ldots 0\rangle \) accomplishes the forward time unitary evolution \({\psi }_{0}\rangle \to {\psi }_{1}\rangle ={\hat{U}}_{n{\rm{bit}}}{\psi }_{0}\rangle \). Next, (ii′) the unitary complex conjugation operation \(\hat{{\mathscr{K}}}={\hat{U}}_{\psi }\) is applied \({\psi }_{1}\rangle \to {\psi }_{1}^{\ast }\rangle ={\hat{U}}_{\psi }{\psi }_{1}\rangle \) followed by (ii″) the unitary transformation \({\hat{U}}_{R}\), \({\psi }_{1}^{\ast }\rangle \to \hat{{\mathscr{T}}}{\psi }_{1}\rangle ={\hat{U}}_{R}{\psi }_{1}^{\ast }\rangle \). As a result, the timereversed state \(\hat{{\mathscr{T}}}{\psi }_{1}\rangle \) is generated. Finally, at step (iii) one applies the same forward time unitary evolution \(\hat{{\mathscr{T}}}{\psi }_{1}\rangle \to {\hat{U}}_{n{\rm{bit}}}\hat{{\mathscr{T}}}{\psi }_{1}\rangle \) and measures the resulting state of the register in the computational basis. In practice, the step 2″ is only needed for the 3qubit model where \({\hat{U}}_{R}={{\rm{SWAP}}}_{12}\) requires three additional CNOT gates. In order to save this number of CNOTs we replace the forward evolution operator \({\hat{U}}_{{\rm{3bit}}}\) at step (iii) by the new evolution operation obtained from \({\hat{U}}_{{\rm{3bit}}}\) via the physical interchange of two particle qubits, rather than to implement the SWAP_{12} operation at step (ii″). Generally, to arrive to the same initial state one has to apply the inverse timereversal operation \({{\mathscr{T}}}^{1}=\hat{{\mathscr{K}}}{\hat{U}}_{R}^{\dagger }\) to the final state \({\hat{U}}_{n{\rm{bit}}}\hat{{\mathscr{T}}}{\psi }_{1}\rangle \). However, if the initial state \(\psi \mathrm{(0)}\rangle \) was a product state \(0\ldots 0\rangle \) this operation is in fact not needed. Indeed, the complex conjugation just changes the overall phase of the qubit register while \({\hat{U}}_{R}^{\dagger }\) swaps the same qubit states in 2particle scattering experiment.
The above time reversal experiment sets the qubit register again into the initial state \(\mathrm{0}\ldots 0\rangle \) with the probability unity, provided all quantum gates are prefect and no decoherence and relaxation processes are present. The exemplary outcome probabilities \({P}_{ij}=\langle {b}_{i}{b}_{j}{\tilde{\psi }}_{0}\rangle {}^{2}\) and \({P}_{ijk}=\langle {b}_{i}{b}_{j}{b}_{k}{\tilde{\psi }}_{0}\rangle {}^{2}\), \(i,j,k=0,1\) obtained in a real experiment for the 2 and 3qubit models are shown on the Fig. 3E. One can see that the probability for observing the correct final state \(\mathrm{0}\ldots 0\rangle \) is less than 100% and for 2 and 3qubit experiment are given by \(85.3\pm 0.4 \% \) and \(49.1\pm 0.6 \% \) correspondingly. This considerable distinction from the perfect scenario comes from the three main sources: (i) The finite coherence time T_{2} of qubits; (ii) The errors of CNOT gates and (iii) The readout errors of the final state of the qubit register.
The observed outcome probabilities were obtained after 8192 runs of each experiment at the same state of the ‘ibmqx4’ 5qubit quantum processor, see details in SI. For the 2qubit experiment two processor’s qubit lines q_{1} and q_{2} with the coherence times 41.0 μs and 43.5 μs and readout errors \({\epsilon }_{r1}=3.3 \% \) and \({\epsilon }_{r2}=2.9 \% \) were involved. For the 3qubit experiment, the additional q_{0} qubit line with \({T}_{2}=39.4\,\mu {\rm{s}}\) and the readout error \({\epsilon }_{r0}=4.8 \% \) was used. The 2qubit experiment requires six CNOT_{q2,q1} gates with the gate error \({\epsilon }_{g21}=2.786 \% \), while the 3qubit experiment acquires, in addition, six \({{\rm{CNOT}}}_{q\mathrm{2,}q0}\) and four \({{\rm{CNOT}}}_{q\mathrm{1,}q0}\) gates with the corresponding gate errors \({\epsilon }_{g20}=2.460 \% \) and \({\epsilon }_{g10}=1.683 \% \). This numbers give us a rough estimate of the net error rates: \({\epsilon }_{{\rm{2bit}}}=1{\mathrm{(1}{\epsilon }_{g21})}^{6}\)\(\mathrm{(1}{\epsilon }_{r1})\,\mathrm{(1}{\epsilon }_{r2})\approx \mathrm{15.6 \% }\) and \({\epsilon }_{{\rm{3bit}}}=1{\mathrm{(1}{\epsilon }_{g21})}^{6}{\mathrm{(1}{\epsilon }_{g20})}^{6}{\mathrm{(1}{\epsilon }_{g10})}^{4}\)\((1{\epsilon }_{r0})(1{\epsilon }_{r1})(1{\epsilon }_{r2})\)\(\approx \mathrm{34.4 \% }\). One can see, that while this estimate agrees with an observed error of a 2qubit experiment, the error probability for the 3qubit experiment is underestimated. We argue that a time duration of a single 3qubit experiment is about 7.5 μs is comparable with T_{2} times, while a single 2qubit experiment takes less time about 3 μs. Hence, the decoherence effects are more prominent in a 3qubit case that might explain the underestimated value of the error rate. The more experimental data for the different system parameters and processor states are discussed in SI. We note, that at the present date the more accurate computation can be made within NMR quantum computation paradigm^{35}, where much more accurate twoqubit gates can be achieved.
Conclusion
Our findings suggest several directions for investigating time reversal and the backward time flow in real quantum systems. One of the directions to pursue, is the time dependence of the reversal complexity \({\mathscr{N}}\) of an evolving quantum state. In our work, we have shown that an isolated ddimensional quantum particle with quadratic spectrum exhibits a polynomial complexity growth \({\mathscr{N}}(\tau )={\tau }^{d}\). Uncovering the \({\mathscr{N}}(\tau )\) dependence for realistic situations, accounting for the interactions will establish a mechanism and the corresponding timescale on which timereversed states can spontaneously emerge. Another fundamental question is whether it is possible at all to design a quantum algorithm that would perform timereversal more efficiently than using \({\mathscr{O}}({\mathscr{N}})\) elementary gates. So far, our timereversal schemes were scrolling one by one through the state components but did not exploit a quantum parallelism in its full power. On a practical side the timereversal procedure might be helpful for the quantum program testing. Having in hands a multiqubit quantum computer it is hard to verify that it really has computed the desired result. Indeed, the full tomography of the computed state is an exponentially hard task. Alternatively, making the timereversal of the anticipated computed state and running the same evolution drives the computer back to its initial state if and only if the computer really made a correct computation. The initial state is typically nonentangled and therefore its verification is an easy task.
Data Availability
All data generated or analyzed during this study are included in this published article and its Supplementary Information file.
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Acknowledgements
The work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (V.M.V.), by the Swiss National Foundation via the National Centre of Competence in Research in Quantum Science and Technology (NCCR QSIT) (A.V.L.) and by the RFBR Grants No. 170200396A (G.B.L.) and 180200642A (A.V.L. and G.B.L.). A.V.L. and M.V.S. acknowledges the support from the Ministry of the Education and Science of the Russian Federation 16.7162.2017/8.9. G.B.L. was supported by the Government of the Russian Federation (Agreement 05.Y09.21.0018), Foundation for the Advancement of Theoretical Physics “BASIS”, the Pauli Center for Theoretical Physics. All data are available in the online supplementary materials.
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G.B.L., I.A.S., M.V.S., A.V.L. and V.M.V. conceived the work, carried out calculations, and wrote the manuscript.
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Lesovik, G.B., Sadovskyy, I.A., Suslov, M.V. et al. Arrow of time and its reversal on the IBM quantum computer. Sci Rep 9, 4396 (2019). https://doi.org/10.1038/s41598019407656
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