IT IS NOT REAL TIME TRAVEL BUT WILL BE A GIANT LEAP ON QUANTUM COMPUTING: ARROW OF TIME AND ITS REVERSAL ON THE IBM QUANTUM COMPUTER.

Scientific Reportsvolumeย 9, Articleย number:ย 4396ย (2019)ย |ย D

Abstract

Uncovering the origin of the โ€œarrow of timeโ€ 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 systemโ€™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 two-level impurity.

Introduction

A fundamental question of the origin of irreversibility of time emerged already in classical statistical physics1,2,3,4,5 and has been remaining ever since a subject of an continuous attention6,7,8. 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 theorem9,10,11,12,13,14,15,16. In particular, it was quantitatively described and shown experimentally that in a finite temporal interval the time reversed dynamics can emerge17. The quantum systems were discussed in18 where the positive entropy production rate was experimentally demonstrated on a single spin-1/2 particle, while in19 the negative entropy production rate in the presence of a Maxwellโ€™s Demon was observed for spin-1/2 quantum system. Moreover, the full quantum treatment have shown theoretically20,21 and later experimentally22 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 rigor23, in the framework of the quantum channel theory24. 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 Zurek25, 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 Landau26 and von Neumann27 who related irreversiblity to the process of a macroscopic measurement. In28 the arrow of time dilemma was addressed from the point of view system-observer considerations, but later this approach was criticized in29. Here, in the spirit of quantum mechanics, we elaborate on the implications of the Wignerโ€™s result30 that time reversal operation is anti-unitary because it requires complex conjugation. We demonstrate that this emerging anti-unitarity 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 two-particle 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 time-reversed quantum state and the probability of its spontaneous emergence. We show that the time-reversal 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 two-level systems, on the IBM quantum computer. In this respect we utilize the conjectures by Lloyd6.

Further, a principal possibility of occurring of the time reversal was discussed in20.

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 ๐’ฏฬ‚ T^ is a product of a complex conjugation operator ๐’ฆฬ‚ K^ and a unitary rotation ๐‘ˆฬ‚ ๐‘…U^R, i.e. ๐’ฏฬ‚ =๐‘ˆฬ‚ ๐‘…๐’ฆฬ‚ T^=U^RK^, where for any ฮจฮจ, ๐’ฆฬ‚ ฮจ=ฮจโˆ—K^ฮจ=ฮจโˆ—. 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 ฮจฮจ-dependent operation such that ๐‘ˆฬ‚ ฮจฮจ=ฮจโˆ—U^ฮจฮจ=ฮจโˆ— 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, ๐œ€=๐‘2/2๐‘šฮต=p2/2m, 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 ๐œฯ„ the wave packet spreads asฮจ(๐‘ฅ,๐œ)โ‰ƒ๐‘“(๐‘ฅ๐‘š/โ„๐œ)2๐œ‹โ„๐œ/๐‘šโ€พโ€พโ€พโ€พโ€พโ€พโ€พโˆšexp(๐‘–๐‘š๐‘ฅ22โ„๐œ),ฮจ(x,ฯ„)โ‰ƒf(xm/โ„ฯ„)2ฯ€โ„ฯ„/mexp(imx22โ„ฯ„),(1)

where f(q) is a Fourier image of the initial spatial wave function. The phase of ฮจฮจ 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 ฮจฮจ, let us divide the coordinate space into a large number of the elemental cells ฮดxn where a wavefunctionโ€™s phase ๐œ™(๐‘ฅ,๐œ)ฯ•(x,ฯ„) changes slowly and look for a fast electromagnetic potential fluctuation ๐‘‰(๐‘ฅ,๐‘ก)V(x,t) which is smooth on the cellโ€™s scale and reverts the phase of the wavefunction: โˆซ๐‘‘๐‘ก๐‘’๐‘‰(๐‘ฅ๐‘›,๐‘ก)/โ„=โˆ’2๐œ™(๐‘ฅ๐‘›,๐œ)โˆซdteV(xn,t)/โ„=โˆ’2ฯ•(xn,ฯ„). If during the ๐œฯ„ the wave packet (1) has spread from the size L0 to the size ๐ฟ๐œ=โ„๐œ/๐‘š๐ฟ0Lฯ„=โ„ฯ„/mL0, it would require ๐‘โˆผ๐œ–โˆ’1/2(๐ฟ๐œ/๐ฟ0)Nโˆผฯตโˆ’1/2(Lฯ„/L0) elementary cells to approximately revert the quantum state ฮจ(๐‘ฅ,๐œ)โ†’ฮจฬƒ โˆ—(๐‘ฅ,๐œ)ฮจ(x,ฯ„)โ†’ฮจ~โˆ—(x,ฯ„) with the probability 1โˆ’๐œ–1โˆ’ฯต: |โŸจฮจฬƒ โˆ—(๐‘ฅ,๐œ)|ฮจโˆ—(๐‘ฅ,๐œ)โŸฉ|2=1โˆ’๐œ–|โŸจฮจ~โˆ—(x,ฯ„)|ฮจโˆ—(x,ฯ„)โŸฉ|2=1โˆ’ฯต, 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 2โˆ’N. Now we determine the typical time scale ๐œฯ„ on which the spontaneous time reversal of a wave-packet can still occur within the universe lifetime ๐‘ก๐‘ˆโˆผ4.3ร—1017tUโˆผ4.3ร—1017 sec. The latter is obtained from the estimate 2โˆ’๐‘โ‰ƒ๐œ/๐‘ก๐‘ˆ2โˆ’Nโ‰ƒฯ„/tU, where the number of cells ๐‘โˆผ๐œ–โˆ’1/2(โŸจ๐ธโŸฉ๐œ/โ„)Nโˆผฯตโˆ’1/2(โŸจEโŸฉฯ„/โ„) is expressed through the average particle energy โŸจ๐ธโŸฉ=โ„2/๐‘š๐ฟ20โŸจEโŸฉ=โ„2/mL02. As a typical average energy of the wave-packet we take the energy corresponding to the current universe temperature 2.72โ€‰K, and arrive at ๐œโ‰ƒ6ร—10โˆ’11ฯ„โ‰ƒ6ร—10โˆ’11โ€‰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.

Figure 1
Figure 1

Now we consider a more complex example and demonstrate that a separable stateฮจ(๐‘ฅ1,๐‘ฅ2)=|๐œ“1(๐‘ฅ1)๐œ“2(๐‘ฅ2)|exp[๐‘–(๐œ™1(๐‘ฅ1)+๐œ™2(๐‘ฅ2))]ฮจ(x1,x2)=|ฯˆ1(x1)ฯˆ2(x2)|exp[i(ฯ•1(x1)+ฯ•2(x2))](2)

of two particles can not be reverted by classical field fluctuations in the case where particleโ€™s wave functions overlap. Let all particles have the same electric charge q and interact with a classical electric potential v(xt). The potential fluctuations produce phase shifts โˆซ๐‘‘๐‘ก๐‘ž๐‘ฃ(๐‘ฅ,๐‘ก)/โ„โˆซdtqv(x,t)/โ„. Accordingly the proper fluctuations capable to reverse the quantum state should satisfy the condition ๐œ™1(๐‘ฅ1)+๐œ™2(๐‘ฅ2)ฯ•1(x1)+ฯ•2(x2) + โˆซ๐‘‘๐‘ก[๐‘ž๐‘ฃ(๐‘ฅ1,๐‘ก)+๐‘ž๐‘ฃ(๐‘ฅ2,๐‘ก)]/โ„โˆซdt[qv(x1,t)+qv(x2,t)]/โ„ = โˆ’๐œ™1(๐‘ฅ1)โˆ’๐œ™2(๐‘ฅ2)โˆ’ฯ•1(x1)โˆ’ฯ•2(x2). For ๐‘ฅ1=๐‘ฅ2×1=x2 it implies โˆซ๐‘‘๐‘ก๐‘ž๐‘ฃ(๐‘ฅ,๐‘ก)/โ„=โˆ’๐œ™1(๐‘ฅ)โˆ’๐œ™2(๐‘ฅ)โˆซdtqv(x,t)/โ„=โˆ’ฯ•1(x)โˆ’ฯ•2(x), and therefore at ๐‘ฅ1โ‰ ๐‘ฅ2×1โ‰ x2 one has to satisfy the condition ๐œ™2(๐‘ฅ2)+๐œ™1(๐‘ฅ1)=๐œ™2(๐‘ฅ1)+๐œ™1(๐‘ฅ2)ฯ•2(x2)+ฯ•1(x1)=ฯ•2(x1)+ฯ•1(x2) which, in general, does not hold.

Quantum entanglement introduces the next level of complexity for the time-reversal procedure. Consider a two-particle state ฮจ(๐‘ฅ1,๐‘ฅ2)=|ฮจ(๐‘ฅ1,๐‘ฅ2)|๐‘’๐‘–๐œ™(๐‘ฅ1,๐‘ฅ2)ฮจ(x1,x2)=|ฮจ(x1,x2)|eiฯ•(x1,x2) with the non-separable phase function ๐œ™(๐‘ฅ1,๐‘ฅ2)=๐‘Ž1(๐‘ฅ1)๐‘1(๐‘ฅ2)+ฯ•(x1,x2)=a1(x1)b1(x2)+๐‘Ž2(๐‘ฅ1)๐‘2(๐‘ฅ2)a2(x1)b2(x2). In this situation even for the non-overlapping particles with ฮจ(๐‘ฅ1,๐‘ฅ2)=0ฮจ(x1,x2)=0 for ๐‘ฅ1=๐‘ฅ2×1=x2 the two-particle state can not be reversed by an interaction with classical fields. Let one access the particles by different fields which induce separate phase shifts ฮจ(๐‘ฅ1,๐‘ฅ2)โ†’ฮจ(๐‘ฅ1,๐‘ฅ2)๐‘’๐‘–(๐œ‘1(๐‘ฅ1)+๐œ‘2(๐‘ฅ2))ฮจ(x1,x2)โ†’ฮจ(x1,x2)ei(ฯ†1(x1)+ฯ†2(x2)). The induced phase shifts should satisfy the relation: ๐œ‘1(๐‘ฅ1)+๐œ‘2(๐‘ฅ2)=โˆ’2๐œ™(๐‘ฅ1,๐‘ฅ2)ฯ†1(x1)+ฯ†2(x2)=โˆ’2ฯ•(x1,x2), therefore for any three points ๐‘ฅ1โ‰ ๐‘ฅ2โ‰ ๐‘ฅ3×1โ‰ x2โ‰ x3 the following conditions should hold๐œ‘1(๐‘ฅ1)+๐œ‘2(๐‘ฅ2)=โˆ’2(๐‘Ž1(๐‘ฅ1)๐‘1(๐‘ฅ2)+๐‘Ž2(๐‘ฅ1)๐‘2(๐‘ฅ2)),ฯ†1(x1)+ฯ†2(x2)=โˆ’2(a1(x1)b1(x2)+a2(x1)b2(x2)),(3)๐œ‘1(๐‘ฅ1)+๐œ‘2(๐‘ฅ3)=โˆ’2(๐‘Ž1(๐‘ฅ1)๐‘1(๐‘ฅ3)+๐‘Ž2(๐‘ฅ1)๐‘2(๐‘ฅ3)).ฯ†1(x1)+ฯ†2(x3)=โˆ’2(a1(x1)b1(x3)+a2(x1)b2(x3)).(4)

Subtracting these relations one gets ๐œ‘2(๐‘ฅ2)โˆ’๐œ‘2(๐‘ฅ3)ฯ†2(x2)โˆ’ฯ†2(x3) = โˆ’2๐‘Ž1(๐‘ฅ1)(๐‘1(๐‘ฅ2)โˆ’๐‘1(๐‘ฅ3))โˆ’2a1(x1)(b1(x2)โˆ’b1(x3)) โˆ’ 2๐‘Ž2(๐‘ฅ1)(๐‘2(๐‘ฅ2)โˆ’๐‘2(๐‘ฅ3))2a2(x1)(b2(x2)โˆ’b2(x3)) where the left hand side does not depend on x1 and therefore one has to assume a1and a2 to be constant. This, however, contradicts the non-separability assumption for ๐œ™(๐‘ฅ1,๐‘ฅ2)ฯ•(x1,x2).

An entangled two-particle state with a non-separable phase function can naturally emerge as a result of scattering of two localized wave-packets31. However, as we have seen, the generation of the time-reversed state, where a particle gets disentangled in the course of its forward time evolution, requires specific two-particle operations which, in general, cannot be reduced to a simple two-particle 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 questionIn the most of the casessuch a supersystem cannot spontaneously emerge in nature. (ii) Even if such a supersystem would emerge for some specific situationthe corresponding spontaneous time reversal typically requires times exceeding the universe lifetime.

A matter-of-course supersystem of that kind is implemented by the so-called universal quantum computer. It is capable to efficiently simulate unitary dynamics of any physical system endowed with local interactions32. A systemโ€™s state is encoded into the quantum state of the computerโ€™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 |๐œ“0โŸฉreg|ฯˆ0โŸฉreg with the systemโ€™s quantum state, |ฮจโŸฉsys|ฮจโŸฉsys, by the unitary operation ๐‘ˆฬ‚ SWAP|๐œ“0โŸฉregโŠ—|ฮจโŸฉsys=|๐œ“โŸฉregโŠ—|ฮจ0โŸฉsysU^SWAP|ฯˆ0โŸฉregโŠ—|ฮจโŸฉsys=|ฯˆโŸฉregโŠ—|ฮจ0โŸฉsys, where the mapping |ฮจโŸฉsysโ†’|๐œ“โŸฉreg|ฮจโŸฉsysโ†’|ฯˆโŸฉreg completes the encoding task. Such an encoding procedure is universal i.e. it does not require the knowledge of the system state |ฮจโŸฉsys|ฮจโŸฉsys. However, non-physical encodings might be suggested which can not be accomplished by unitary transformation. One of the ways to do that was proposed in33 where the real and the imaginary components of the systemโ€™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 of33 merely lifts the problem of the non-unitarity of the quantum conjugation hiding it in the non-unitarity 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 time-reversal algorithms on quantum computers and, in the next section, present a practical implementation of a few-qubit 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 ฮจ(๐‘ก=0)ฮจ(t=0) and let it evolve during the time ๐œฯ„ into the state ฮจ(๐œ)=exp(โˆ’๐‘–๐ป๐œ/โ„)ฮจ(0)ฮจ(ฯ„)=expโก(โˆ’iHฯ„/โ„)ฮจ(0). Let us find a minimal size of a qubit register needed to simulate the dynamics of a system ฮจ(0)โ†’ฮจ(๐œ)ฮจ(0)โ†’ฮจ(ฯ„) with a given fidelity 1โˆ’๐œ–1โˆ’ฯต. Let us choose a finite set of time instances ๐‘ก๐‘–โˆˆ[0,๐œ]tiโˆˆ[0,ฯ„], ๐‘–=0,โ€ฆ๐‘โ€ฒi=0,โ€ฆNโ€ฒ subject to a condition |โŸจฮจ(๐‘ก๐‘–)|ฮจ(๐‘ก๐‘–+1โŸฉ)|2=1โˆ’๐œ–|โŸจฮจ(ti)|ฮจ(ti+1โŸฉ)|2=1โˆ’ฯต with ๐‘ก0=0t0=0 for some small ๐œ–>0ฯต>0. Then at any time instant ๐‘กโˆˆ[0,๐œ]tโˆˆ[0,ฯ„] a state ฮจ(๐‘ก)ฮจ(t) can be approximated by the discrete set of states {ฮจ(๐‘ก๐‘–),๐‘–=0,โ€ฆ๐’ฉโ€ฒ}{ฮจ(ti),i=0,โ€ฆNโ€ฒ} with the fidelity 1โˆ’๐œ–1โˆ’ฯต. The set of states {ฮจ(๐‘ก๐‘–)}{ฮจ(ti)} spans the Hilbert subspace ๐’ฎS of the dimension ๐’ฉโ‰ค๐’ฉโ€ฒNโ‰คNโ€ฒ. Therefore, ๐’ฉN basis vectors |๐‘’๐‘–โŸฉโˆˆ๐’ฎ|eiโŸฉโˆˆS can be represented by ๐’ฉNorthogonal states of the qubit register, |๐‘’๐‘–โŸฉโ†’|๐‘โ†’๐‘–โŸฉโ‰ก|๐‘0๐‘1โ€ฆโŸฉ|eiโŸฉโ†’|bโ†’iโŸฉโ‰ก|b0b1โ€ฆโŸฉ. The corresponding qubit Hamiltonian ๐ปฬ‚ H^ which mimics the original Hamiltonian ๎ˆดฬ‚ H^ is then defined by the relation (๐ปฬ‚ )๐‘–๐‘—โ‰กโŸจ๐‘โ†’๐‘–|๐ปฬ‚ |๐‘โ†’๐‘—โŸฉ=โŸจ๐‘’๐‘–|๎ˆดฬ‚ |๐‘’๐‘—โŸฉ(H^)ijโ‰กโŸจbโ†’i|H^|bโ†’jโŸฉ=โŸจei|H^|ejโŸฉ.

Below we introduce two encoding procedures |๐‘’๐‘–โŸฉโ†’|๐‘โ†’๐‘–โŸฉ|eiโŸฉโ†’|bโ†’iโŸฉ. In the first, sparse coding approach, one assigns a separate qubit to each state |๐‘’๐‘–โŸฉ|eiโŸฉ, ๐‘–โˆˆ[0,๐’ฉโˆ’1]iโˆˆ[0,Nโˆ’1] and encodes the state ๐œ“(๐œ)ฯˆ(ฯ„) into the ๐’ฉN-qubit state |๐œ“โŸฉ=โˆ‘๐’ฉโˆ’1๐‘–=0๐œ“๐‘–|00โ€ฆ1๐‘–โ€ฆ0๐’ฉโˆ’1โŸฉ|ฯˆโŸฉ=โˆ‘i=0Nโˆ’1ฯˆi|00โ€ฆ1iโ€ฆ0Nโˆ’1โŸฉ. The second approach is a densecoding scheme where one records the state ๐œ“(๐œ)ฯˆ(ฯ„) into a state of ๐‘›=int[log2(๐’ฉ)]+1n=int[log2(N)]+1 qubits |๐œ“โŸฉ=โˆ‘๐’ฉโˆ’1๐‘–=0๐œ“๐‘–|๐‘–โŸฉ|ฯˆโŸฉ=โˆ‘i=0Nโˆ’1ฯˆi|iโŸฉ, where int[x] is the closest upper integer to x: ๐‘ฅโ‰คint[๐‘ฅ]xโ‰คint[x], |๐‘–โŸฉโ‰ก|๐‘0โ€ฆ๐‘๐‘›โˆ’1โŸฉ|iโŸฉโ‰ก|b0โ€ฆbnโˆ’1โŸฉ is a computational basis state corresponding a binary representation of the number ๐‘–=โˆ‘๐‘›โˆ’1๐‘˜=0๐‘๐‘˜2๐‘›โˆ’1โˆ’๐‘˜i=โˆ‘k=0nโˆ’1bk2nโˆ’1โˆ’k.

A time-reversal operation ๐’ฏฬ‚ T^ of the qubit register can be presented as a product ๐’ฏฬ‚ =๐‘ˆฬ‚ ๐‘…๐’ฆฬ‚ T^=U^RK^ of the complex conjugation operator ๐’ฆฬ‚ K^, ๐’ฆฬ‚ (๐œ“๐‘–|๐‘–โŸฉ)โ‰ก๐œ“โˆ—๐‘–|๐‘–โŸฉK^(ฯˆi|iโŸฉ)โ‰กฯˆiโˆ—|iโŸฉ, and some unitary operator ๐‘ˆฬ‚ ๐‘…U^R, whose form is defined by the Hamiltonian ๐ปฬ‚ H^, ๐‘ˆฬ‚ ๐‘…=๐‘ˆฬ‚ โ€ ๐ป๐‘ˆฬ‚ โˆ—๐ปU^R=U^Hโ€ U^Hโˆ—, where ๐ปฬ‚ =๐‘ˆฬ‚ โ€ ๐ปdiag{๐ธ1โ€ฆ๐ธ๐‘›}๐‘ˆฬ‚ ๐ปH^=U^Hโ€ diag{E1โ€ฆEn}U^H see SI. Therefore, ?in order to implement the time-reversal operation ๐’ฏฬ‚ T^ one needs to know the Hamiltonian ๐ปฬ‚ 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 non-local 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 ๐ปฬ‚ 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 ๐‘ˆฬ‚ ๐‘…U^R is generally known and represents a transformation which is inherited from the time-reversal symmetry of the original Hamiltonian ๎ˆดฬ‚ H^. In particular, if the qubit Hamiltonian ๎ˆดฬ‚ H^ is real, then the corresponding evolution operator ๐‘ˆฬ‚ (๐œ)U^(ฯ„)is symmetric that entails ๐‘ˆฬ‚ ๐‘…=1U^R=1.

In the following we assume the unitary ๐‘ˆฬ‚ ๐‘…U^R to be known and focus on the unitary implementation of a complex conjugation operation ๐’ฆฬ‚ K^, ๐’ฆฬ‚ โ†’๐‘ˆฬ‚ ๐œ“K^โ†’U^ฯˆ. In particular, we quantify a complexity of such implementation as a number of elementary quantum gates or/and auxiliary qubits needed to implement ๐‘ˆฬ‚ ๐œ“U^ฯˆ. For a sparse coding scheme, the complex conjugation of the ๐’ฉN-qubit state |๐œ“โŸฉ=โˆ‘๐’ฉโˆ’1๐‘–=0|๐œ“๐‘–|๐‘’๐‘–๐œ™๐‘–|00โ€ฆ1๐‘–โ€ฆ0๐’ฉโˆ’1โŸฉ|ฯˆโŸฉ=โˆ‘i=0Nโˆ’1|ฯˆi|eiฯ•i|00โ€ฆ1iโ€ฆ0Nโˆ’1โŸฉ can be accomplished by the unitary operation ๐‘ˆฬ‚ (1)๐œ“=โˆ๐’ฉโˆ’1๐‘–=0โŠ—๐‘‡ฬ‚ ๐‘–(โˆ’2๐œ™๐‘–)U^ฯˆ(1)=โˆi=0Nโˆ’1โŠ—T^i(โˆ’2ฯ•i) where ๐‘‡ฬ‚ ๐‘–(๐œ™)T^i(ฯ•) is the single qubit operation: ๐‘‡ฬ‚ ๐‘–(๐œ™)|0๐‘–โŸฉ=|0๐‘–โŸฉT^i(ฯ•)|0iโŸฉ=|0iโŸฉ and ๐‘‡ฬ‚ ๐‘–(๐œ™)|1๐‘–โŸฉ=๐‘’๐‘–๐œ™|1๐‘–โŸฉT^i(ฯ•)|1iโŸฉ=eiฯ•|1iโŸฉ. Consequently, the sparse coding scheme does not require the most โ€œexpensiveโ€ two-qubit gates at all but do require a large number ๐’ฉN 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 2n n-qubit conditional phase shift operations: ๐’ฆฬ‚ โ†’๐‘ˆฬ‚ (2)๐œ“=โˆ‘2๐‘›โˆ’1๐‘—=0|๐‘—โŸฉโŸจ๐‘—|๐‘’โˆ’2๐‘–๐œ™๐‘—K^โ†’U^ฯˆ(2)=โˆ‘j=02nโˆ’1|jโŸฉโŸจj|eโˆ’2iฯ•j which add proper phases to each component of the state |๐œ“โŸฉ|ฯˆโŸฉ: ๐‘ˆฬ‚ (2)๐œ“|๐œ“โŸฉ=|๐œ“โˆ—โŸฉU^ฯˆ(2)|ฯˆโŸฉ=|ฯˆโˆ—โŸฉ. Therefore, ๐‘ˆฬ‚ (2)๐œ“U^ฯˆ(2) must involve two-qubit gates, i.e. conditional-NOT (CNOT) gates. We quantify the complexity of the dense coding scheme by a number ๐‘โŠ•NโŠ• of CNOT gates needed to implement it. Each phase shift operation ฮฆฬ‚ ๐‘–(๐œ™)โ‰ก|๐‘–โŸฉโŸจ๐‘–|๐‘’๐‘–๐œ™ฮฆ^i(ฯ•)โ‰ก|iโŸฉโŸจi|eiฯ• can be build with the help of ๐‘›โˆ’1nโˆ’1 ancillary qubits and 2(๐‘›โˆ’1)2(nโˆ’1) Toffoli gates, as shown in Fig. 2A. In total, it requires ๐‘โŠ•[๐‘ˆฬ‚ (2)๐œ“]=12(๐‘›โˆ’1)2๐‘›โˆผ12๐’ฉlog2(๐’ฉ)NโŠ•[U^ฯˆ(2)]=12(nโˆ’1)2nโˆผ12Nlog2(N) CNOT gates. However, such an arrangement is non-optimal as it involves an excess usage of Toffoli gates. Indeed, let us consider two states, |๐‘—โŸฉ=|๐‘0๐‘1โ€ฆ๐‘๐‘›โˆ’1โŸฉ|jโŸฉ=|b0b1โ€ฆbnโˆ’1โŸฉ and |๐‘—โ€ฒโŸฉ=|๐‘โ€ฒ0๐‘โ€ฒ1โ€ฆ๐‘โ€ฒ๐‘›โˆ’1โŸฉ|jโ€ฒโŸฉ=|bโ€ฒ0bโ€ฒ1โ€ฆbโ€ฒnโˆ’1โŸฉ, with coincident two older bits ๐‘0=๐‘โ€ฒ0b0=bโ€ฒ0, ๐‘1=๐‘โ€ฒ1b1=bโ€ฒ1. The separate usage of phase shifts ฮฆฬ‚ ๐‘—(๐œ™๐‘—)ฮฆ^j(ฯ•j) and ฮฆฬ‚ ๐‘—โ€ฒ(๐œ™๐‘—โ€ฒ)ฮฆ^jโ€ฒ(ฯ•jโ€ฒ) involves the double check of b0 and b1 values. The better implementation checks b0and b1 only once and conjugates phases of all states within a set |๐‘0๐‘1๐‘2โ€ฆ๐‘๐‘›โˆ’1โŸฉ|b0b1b2โ€ฆbnโˆ’1โŸฉ within a separate circuit block. In fact, such optimization can be done for all subsequent junior bits b3b4, see Fig. 2B, that can minimize the usage of Toffoli gates and reduce the reversal complexity to be linear in ๐’ฉN: ๐’ฉโŠ•โˆผ24๐’ฉNโŠ•โˆผ24N, 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 ๐’ฉN is bounded from above by some number ๐’ช(๐’ฉ)O(N). If now we consider typical systems emerging in nature, then the entanglement generates the dimensionality, ๐’ฉN, that is exponentially large with respect to the number of particles involved.

Figure 2
Figure 2

Time Reversal Experiment

Now we are equipped to carry out an experiment implementing two- and three-qubit time-reversal procedures utilizing the public IBM quantum computer. We model a one dimensional particle scattering on a two-level impurity (TLI). The dynamics of the impurity is governed by a Hamiltonian, ๐ปฬ‚ i=โ„๐œ”(cos๐›ผ๐œŽฬ‚ ๐‘ง+sin๐›ผ๐œŽฬ‚ ๐‘ฅ)H^i=โ„ฯ‰(cosฮฑฯƒ^z+sinฮฑฯƒ^x). The scattering potential seen by the particle depends on the state of the TLS. The corresponding scattering operator has the form ๐‘†ฬ‚ ๐‘–=|0โŸฉโŸจ0|โŠ—๐‘†ฬ‚ 0+|1โŸฉโŸจ1|โŠ—๐‘†ฬ‚ 1S^i=|0โŸฉโŸจ0|โŠ—S^0+|1โŸฉโŸจ1|โŠ—S^1, where ๐‘†ฬ‚ 0S^0 and ๐‘†ฬ‚ 1S^1 are symmetric unitary scattering matrices of the TLI in a state |0โŸฉ|0โŸฉ or |1โŸฉ|1โŸฉ. This scattering problem is modeled by the evolution of the qubit register ๐‘ˆฬ‚ ๐‘›bit|๐‘ž๐‘–โŸฉโŠ—(|๐‘ž1โŸฉโŠ—โ‹ฏโŠ—|๐‘ž๐‘›โŸฉ)U^nbit|qiโŸฉโŠ—(|q1โŸฉโŠ—โ‹ฏโŠ—|qnโŸฉ), where |๐‘ž๐‘–โŸฉ|qiโŸฉ qubit describes the state of the TLI and the remaining qubits describe the state of scattered particles. The basis states |0๐‘–โŸฉ|0iโŸฉ and |1๐‘–โŸฉ|1iโŸฉ, ๐‘–=1,โ€ฆ๐‘›i=1,โ€ฆncorrespond 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 ๐‘ก=๐œt=ฯ„ and ๐‘ก=2๐œt=2ฯ„. The corresponding evolutions are described by unitary rotations ๐‘†ฬ‚ (1)iโ‹…๐‘ˆฬ‚ i(๐œ)S^i(1)โ‹…U^i(ฯ„) and ๐‘†ฬ‚ (2)iโ‹…๐‘ˆฬ‚ i(๐œ)โ‹…๐‘†ฬ‚ (1)iโ‹…๐‘ˆฬ‚ i(๐œ)S^i(2)โ‹…U^i(ฯ„)โ‹…S^i(1)โ‹…U^i(ฯ„) for one- and two-particle situation, where ๐‘ˆฬ‚ i(๐œ)=exp(โˆ’๐‘–๐ปฬ‚ i๐œ/โ„)U^i(ฯ„)=expโก(โˆ’iH^iฯ„/โ„) describes the free evolution of the TLI between the particles arrivals and ๐‘†ฬ‚ (๐‘–)iS^i(i), ๐‘–=1,2i=1,2 is the scattering operator for the i-th particle. Next, in both cases we let the system freely evolve during a time ๐œฯ„ that makes the resulting 2-qubit and 3-qubit evolution operators: ๐‘ˆฬ‚ 2bit=๐‘ˆฬ‚ i(๐œ)โ‹…๐‘†ฬ‚ (1)iโ‹…๐‘ˆฬ‚ i(๐œ)U^2bit=U^i(ฯ„)โ‹…S^i(1)โ‹…U^i(ฯ„) and ๐‘ˆฬ‚ 3bit=๐‘ˆฬ‚ i(๐œ)โ‹…๐‘†ฬ‚ (2)iโ‹…๐‘ˆฬ‚ i(๐œ)โ‹…๐‘†ฬ‚ (1)iโ‹…๐‘ˆฬ‚ i(๐œ)U^3bit=U^i(ฯ„)โ‹…S^i(2)โ‹…U^i(ฯ„)โ‹…S^i(1)โ‹…U^i(ฯ„) more symmetric, that simplifies the form of ๐‘ˆฬ‚ ๐‘…U^R unitary rotation entering in the time-reversal procedure. The 2-qubit scattering model is endowed with the symmetric evolution operator ๐‘ˆฬ‚ 2bitU^2bit and, therefore, its time reversal requires only the complex conjugation operation ๐’ฏฬ‚ =๐’ฆฬ‚ T^=K^. At variance, the evolution operator ๐‘ˆฬ‚ 3bitU^3bit of the 3-qubit model is non symmetric and its time reversal requires an additional unitary rotation ๐’ฏฬ‚ =๐‘ˆฬ‚ ๐‘…๐’ฆฬ‚ T^=U^RK^. It follows from the relation SWAP12โ‹…๐‘ˆฬ‚ 3bitโ‹…SWAP12=๐‘ˆฬ‚ ๐‘ก3bitSWAP12โ‹…U^3bitโ‹…SWAP12=U^3bitt, where SWAP12|๐‘ž1โŸฉโŠ—|๐‘ž2โŸฉ=|๐‘ž2โŸฉโŠ—|๐‘ž1โŸฉSWAP12|q1โŸฉโŠ—|q2โŸฉ=|q2โŸฉโŠ—|q1โŸฉ is the swap operation, that the required unitary operation ๐‘ˆฬ‚ ๐‘…=SWAP12U^R=SWAP12. The corresponding quantum circuits realizing ๐‘ˆฬ‚ 2bitU^2bit and ๐‘ˆฬ‚ 3bitU^3bit are shown on Fig. 3A and C, see the detail in SI.

Figure 3
Figure 3

According to the results of the Section 2, the unitary implementation of the complex conjugation for a 2- or 3-qubit 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 1.5โ€“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 n-bit AND Boolean function34,๐‘0โˆง๐‘1โˆงโ‹ฏโˆง๐‘๐‘›โˆ’1=12๐‘›โˆ’1(โˆ‘๐‘–1๐‘๐‘–1โˆ’โˆ‘๐‘–1<๐‘–2๐‘๐‘–1โŠ•๐‘๐‘–2+โˆ‘๐‘–1<๐‘–2<๐‘–3๐‘๐‘–1โŠ•๐‘๐‘–2โŠ•๐‘๐‘–3+โ‹ฏ+(โˆ’1)๐‘›โˆ’1๐‘0โŠ•โ‹ฏโŠ•๐‘๐‘›โˆ’1),b0โˆงb1โˆงโ‹ฏโˆงbnโˆ’1=12nโˆ’1(โˆ‘i1bi1โˆ’โˆ‘i1<i2bi1โŠ•bi2+โˆ‘i1<i2<i3bi1โŠ•bi2โŠ•bi3+โ‹ฏ+(โˆ’1)nโˆ’1b0โŠ•โ‹ฏโŠ•bnโˆ’1),(5)

where ๐‘0โˆง๐‘1โˆงโ€ฆb0โˆงb1โˆงโ€ฆ is 1 if and only if all ๐‘0=๐‘1=โ‹ฏ=1b0=b1=โ‹ฏ=1 and ๐‘0โŠ•๐‘1โŠ•โ€ฆb0โŠ•b1โŠ•โ€ฆ is modulo 2 addition. Consider, for instance, a 2-qubit situation. The complex conjugation ๐‘ˆฬ‚ ๐œ“=โˆ‘3๐‘˜=0ฮฆฬ‚ ๐‘˜(๐œ™๐‘˜)U^ฯˆ=โˆ‘k=03ฮฆ^k(ฯ•k) can be alternatively represented as๐‘ˆฬ‚ ๐œ“=exp[โˆ’2๐‘–๐œ™00๐‘ยฏ0โˆง๐‘ยฏ1โˆ’2๐‘–๐œ™01๐‘ยฏ0โˆง๐‘1โˆ’2๐‘–๐œ™10๐‘0โˆง๐‘ยฏ1โˆ’2๐‘–๐œ™11๐‘0โˆง๐‘1],U^ฯˆ=exp[โˆ’2iฯ•00bยฏ0โˆงbยฏ1โˆ’2iฯ•01bยฏ0โˆงb1โˆ’2iฯ•10b0โˆงbยฏ1โˆ’2iฯ•11b0โˆงb1],(6)

where ๐‘ยฏ๐‘–bยฏi denotes a negation of the bit value. Summing up all four components in the exponent according to the Eq. (5) one decomposes ๐‘ˆฬ‚ ๐œ“U^ฯˆ operator into a product of 1-qubit and 2-qubit phase shifts operations:๐‘ˆฬ‚ ๐œ“=exp[โˆ’๐‘–๐›ผ0๐‘0โˆ’๐‘–๐›ฝ0๐‘ยฏ0]exp[โˆ’๐‘–๐›ผ1๐‘1โˆ’๐‘–๐›ฝ1๐‘ยฏ1]exp[โˆ’๐‘–๐›ผ01๐‘0โŠ•๐‘1โˆ’๐‘–๐›ฝ01๐‘0โŠ•๐‘1โŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏโŽฏ],U^ฯˆ=expโก[โˆ’iฮฑ0b0โˆ’iฮฒ0bยฏ0]exp[โˆ’iฮฑ1b1โˆ’iฮฒ1bยฏ1]exp[โˆ’iฮฑ01b0โŠ•b1โˆ’iฮฒ01b0โŠ•b1ยฏ],(7)

where aiฮฒj are linear combinations of the phase shifts ๐œ™๐‘˜ฯ•k. The modulo 2 addition ๐‘0โŠ•๐‘1b0โŠ•b1 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 (๐‘›โˆ’1)2๐‘›โˆ’1(nโˆ’1)2nโˆ’1 CNOT gates for the complex conjugation of an arbitrary n-qubit state that wins over the approach discussed in Section 2 for ๐‘›โ‰ค48nโ‰ค48. In particular, at ๐‘›=2n=2 and 3 one needs only two or eight CNOT gates, respectively. The corresponding 2- and 3-qubit quantum circuits are shown on Fig. 3B and 3D.

The time-reversal experiment runs in several steps: (i) The qubit register that is initially set into the state |๐œ“(0)โŸฉ=|0โ€ฆ0โŸฉ|ฯˆ(0)โŸฉ=|0โ€ฆ0โŸฉ accomplishes the forward time unitary evolution |๐œ“0โŸฉโ†’|๐œ“1โŸฉ=๐‘ˆฬ‚ ๐‘›bit|๐œ“0โŸฉ|ฯˆ0โŸฉโ†’|ฯˆ1โŸฉ=U^nbit|ฯˆ0โŸฉ. Next, (iiโ€ฒ) the unitary complex conjugation operation ๐’ฆฬ‚ =๐‘ˆฬ‚ ๐œ“K^=U^ฯˆ is applied |๐œ“1โŸฉโ†’|๐œ“โˆ—1โŸฉ=๐‘ˆฬ‚ ๐œ“|๐œ“1โŸฉ|ฯˆ1โŸฉโ†’|ฯˆ1โˆ—โŸฉ=U^ฯˆ|ฯˆ1โŸฉ followed by (iiโ€ณ) the unitary transformation ๐‘ˆฬ‚ ๐‘…U^R, |๐œ“โˆ—1โŸฉโ†’|๐’ฏฬ‚ ๐œ“1โŸฉ=๐‘ˆฬ‚ ๐‘…|๐œ“โˆ—1โŸฉ|ฯˆ1โˆ—โŸฉโ†’|T^ฯˆ1โŸฉ=U^R|ฯˆ1โˆ—โŸฉ. As a result, the time-reversed state |๐’ฏฬ‚ ๐œ“1โŸฉ|T^ฯˆ1โŸฉ is generated. Finally, at step (iii) one applies the same forward time unitary evolution |๐’ฏฬ‚ ๐œ“1โŸฉโ†’๐‘ˆฬ‚ ๐‘›bit|๐’ฏฬ‚ ๐œ“1โŸฉ|T^ฯˆ1โŸฉโ†’U^nbit|T^ฯˆ1โŸฉ and measures the resulting state of the register in the computational basis. In practice, the step 2โ€ณ is only needed for the 3-qubit model where ๐‘ˆฬ‚ ๐‘…=SWAP12U^R=SWAP12 requires three additional CNOT gates. In order to save this number of CNOTs we replace the forward evolution operator ๐‘ˆฬ‚ 3bitU^3bit at step (iii) by the new evolution operation obtained from ๐‘ˆฬ‚ 3bitU^3bit via the physical interchange of two particle qubits, rather than to implement the SWAP12 operation at step (iiโ€ณ). Generally, to arrive to the same initial state one has to apply the inverse time-reversal operation ๐’ฏโˆ’1=๐’ฆฬ‚ ๐‘ˆฬ‚ โ€ ๐‘…Tโˆ’1=K^U^Rโ€  to the final state ๐‘ˆฬ‚ ๐‘›bit|๐’ฏฬ‚ ๐œ“1โŸฉU^nbit|T^ฯˆ1โŸฉ. However, if the initial state |๐œ“(0)โŸฉ|ฯˆ(0)โŸฉ was a product state |0โ€ฆ0โŸฉ|0โ€ฆ0โŸฉ this operation is in fact not needed. Indeed, the complex conjugation just changes the overall phase of the qubit register while ๐‘ˆฬ‚ โ€ ๐‘…U^Rโ€  swaps the same qubit states in 2-particle scattering experiment.

The above time reversal experiment sets the qubit register again into the initial state |0โ€ฆ0โŸฉ|0โ€ฆ0โŸฉ with the probability unity, provided all quantum gates are prefect and no decoherence and relaxation processes are present. The exemplary outcome probabilities ๐‘ƒ๐‘–๐‘—=|โŸจ๐‘๐‘–๐‘๐‘—|๐œ“ฬƒ 0โŸฉ|2Pij=|โŸจbibj|ฯˆ~0โŸฉ|2 and ๐‘ƒ๐‘–๐‘—๐‘˜=|โŸจ๐‘๐‘–๐‘๐‘—๐‘๐‘˜|๐œ“ฬƒ 0โŸฉ|2Pijk=|โŸจbibjbk|ฯˆ~0โŸฉ|2, ๐‘–,๐‘—,๐‘˜=0,1i,j,k=0,1 obtained in a real experiment for the 2- and 3-qubit models are shown on the Fig. 3E. One can see that the probability for observing the correct final state |0โ€ฆ0โŸฉ|0โ€ฆ0โŸฉ is less than 100% and for 2- and 3-qubit experiment are given by 85.3ยฑ0.4%85.3ยฑ0.4% and 49.1ยฑ0.6%49.1ยฑ0.6% correspondingly. This considerable distinction from the perfect scenario comes from the three main sources: (i) The finite coherence time T2 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 โ€˜ibmqx4โ€™ 5-qubit quantum processor, see details in SI. For the 2-qubit experiment two processorโ€™s qubit lines q1 and q2 with the coherence times 41.0โ€‰ฮผs and 43.5โ€‰ฮผs and readout errors ๐œ–๐‘Ÿ1=3.3%ฯตr1=3.3% and ๐œ–๐‘Ÿ2=2.9%ฯตr2=2.9% were involved. For the 3-qubit experiment, the additional q0 qubit line with ๐‘‡2=39.4๐œ‡sT2=39.4ฮผs and the readout error ๐œ–๐‘Ÿ0=4.8%ฯตr0=4.8% was used. The 2-qubit experiment requires six CNOTq2,q1 gates with the gate error ๐œ–๐‘”21=2.786%ฯตg21=2.786%, while the 3-qubit experiment acquires, in addition, six CNOT๐‘ž2,๐‘ž0CNOTq2,q0 and four CNOT๐‘ž1,๐‘ž0CNOTq1,q0gates with the corresponding gate errors ๐œ–๐‘”20=2.460%ฯตg20=2.460% and ๐œ–๐‘”10=1.683%ฯตg10=1.683%. This numbers give us a rough estimate of the net error rates: ๐œ–2bit=1โˆ’(1โˆ’๐œ–๐‘”21)6ฯต2bit=1โˆ’(1โˆ’ฯตg21)6(1โˆ’๐œ–๐‘Ÿ1)(1โˆ’๐œ–๐‘Ÿ2)โ‰ˆ15.6%(1โˆ’ฯตr1)(1โˆ’ฯตr2)โ‰ˆ15.6% and ๐œ–3bit=1โˆ’(1โˆ’๐œ–๐‘”21)6(1โˆ’๐œ–๐‘”20)6(1โˆ’๐œ–๐‘”10)4ฯต3bit=1โˆ’(1โˆ’ฯตg21)6(1โˆ’ฯตg20)6(1โˆ’ฯตg10)4(1โˆ’๐œ–๐‘Ÿ0)(1โˆ’๐œ–๐‘Ÿ1)(1โˆ’๐œ–๐‘Ÿ2)(1โˆ’ฯตr0)(1โˆ’ฯตr1)(1โˆ’ฯตr2)โ‰ˆ34.4%โ‰ˆ34.4%. One can see, that while this estimate agrees with an observed error of a 2-qubit experiment, the error probability for the 3-qubit experiment is underestimated. We argue that a time duration of a single 3-qubit experiment is about 7.5โ€‰ฮผs is comparable with T2 times, while a single 2-qubit experiment takes less time about 3โ€‰ฮผs. Hence, the decoherence effects are more prominent in a 3-qubit 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 paradigm35, where much more accurate two-qubit 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 ๐’ฉN of an evolving quantum state. In our work, we have shown that an isolated d-dimensional quantum particle with quadratic spectrum exhibits a polynomial complexity growth ๐’ฉ(๐œ)=๐œ๐‘‘N(ฯ„)=ฯ„d. Uncovering the ๐’ฉ(๐œ)N(ฯ„) dependence for realistic situations, accounting for the interactions will establish a mechanism and the corresponding time-scale on which time-reversed states can spontaneously emerge. Another fundamental question is whether it is possible at all to design a quantum algorithm that would perform time-reversal more efficiently than using ๐’ช(๐’ฉ)O(N) elementary gates. So far, our time-reversal 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 time-reversal procedure might be helpful for the quantum program testing. Having in hands a multi-qubit 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 time-reversal 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 non-entangled 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 itsย Supplementary Information file.

Acknowledgements
Te 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. 17-02-00396A (G.B.L.) and 18-02-00642A (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 Teoretical Physics โ€œBASISโ€, the Pauli Center for Teoretical Physics. All data are available in the online
supplementary materials.
Author Contributions
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.
Additional Information
Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-019-40765-6.
Competing Interests: Te authors declare no competing interests.

Author: Jesus Padilla

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