Simulating quantum backflow on a quantum computer

Quantum backflow is a counterintuitive effect in which the probability density of a free particle moves in the direction opposite to the particle’s momentum. If the particle is electrically charged, then the effect can be viewed as the contrast between the direction of electric current and that of the momentum. To date, there has been no direct experimental observation of quantum backflow. However, the effect has been simulated numerically (using classical computers) and optically (using classical light). In this study, we present the first simulation of quantum backflow using a real quantum computer.


I. INTRODUCTION
Imagine a bead of mass M and electric charge Q constrained to move without friction along a rigid ring of radius R (Fig. 1).Suppose the bead moves freely (i.e., FIG. 1.A particular manifestation of the quantum backflow effect. in the absence of any external forces other than the constraint force) in the counterclockwise direction, so that the time derivative of the azimuthal angle θ is nonnegative, dθ/dt ≥ 0. The laws of classical physics guarantee that, at every point of the ring, momentum density and electric current density are both nonnegative, i.e., both are either zero or point tangentially to the ring in the direction of increasing θ.Interestingly, the situation can be drastically different if the motion of the bead is governed by the laws of quantum mechanics: There are quantum states of the bead for which its instantaneous momentum and electric current point in the opposite directions (Fig. 1).This counterintuitive scenario is one particular manifestation of a broad class of phenomena concerned with the classically forbidden flow of probability commonly referred to as quantum backflow (QB).
QB was first mentioned in the context of the arrival time problem in quantum mechanics [1,2].The first indepth analysis of QB for a free particle on a line was carried out by Bracken and Melloy [3].In particular, they showed that the effect is weak: Only a small amount of probability -less than 4% [4] -can be transported in the direction opposite to the particle's momentum.Interestingly, QB can be significantly more pronounced for rotational motion.Thus, the classically forbidden probability transfer can reach values up to approximately 0.116816 in the case of a particle rotating on a ring (Fig. 1) [5] and can be arbitrarily high in two-dimensional systems [6][7][8].Recently, the problem of QB on a ring has also been considered for the case of a massless Dirac fermion [9].The literature on QB is substantial, and reviewing it goes beyond the scope of the present paper.The reader is referred to Ref. [10] for an elementary introduction and to Refs.[9,11] for an extensive list of references to more recent results in the area.
As of today, QB has not been observed experimentally.A promising experimental scheme that could lead to the observation of QB in Bose-Einstein condensates was proposed in Ref. [12], but, to our knowledge, has never been realized in practice.
While a direct experimental observation of QB remains an open challenge, there has been exciting progress in simulating the QB effect using classical light [13][14][15].The simulations utilize the analogy between the dynamics of quantum particles and the transverse spreading of light beams.In this analogy, the momentum and probability current of a quantum particle are represented by the transverse wave vector and Poynting vector of the light beam, respectively.Thus, the optical experiments reported in Refs.[13,14] can be regarded as simulating QB for a particle on a line, whereas the experiment in Ref. [15] is the optics counterpart of QB for a twodimensional rotational motion.
In this paper, we demonstrate how QB for a particle on a ring (Fig. 1) can be simulated using a quantum computer.The demonstration involves the following two steps: (i) we use N qubits to encode a particle state |ψ⟩ comprised of 2 N eigenstates with non-negative angular momentum, and (ii) pass |ψ⟩ through a quantum circuit designed to compute the probability (or electric) current at a given point on the ring.A negative readout is a manifestation of QB.Theoretical results derived in this paper are applicable to the case of arbitrary N , whereas the actual experimental demonstration, utilizing the IBM-Q quantum computer [16], is performed for the cases of N = 1 and N = 2.
The paper is organized as follows.In Sec.II, we specify the system and formulate the QB effect.In Sec.III, we introduce a finite-dimensional probability current operator and derive its decomposition in terms of sums of tensor products of one-qubit operators.This decomposition is what makes the following quantum-computer simulation of QB possible.In Sec.IV, we construct a concrete example, valid for arbitrary N , of a quantum state exhibiting QB.We later use this state in our quantum simulation.Section V presents an experimental simulation of QB performed on the IBM-Q quantum computer.In Sec.VI we summarize our work and make concluding remarks.Throughout the paper, we set the particle mass and electric charge, the ring radius, and the Planck constant equal to unity, i.e.M = Q = R = ℏ = 1.

II. QUANTUM BACKFLOW FOR CIRCULAR MOTION
The Hamiltonian of the particle-on-a-ring system (Fig. 1) is Ĥ = The states are orthonormal, and form a complete basis.Now suppose that the particle is in a state |ψ⟩ given by a superposition of 2 N stationary states of the lowest possible energy and non-negative angular momentum: The expansion coefficients a m satisfy the normalization condition, By construction, any angular momentum measurement performed on |ψ⟩ is guaranteed to return a non-negative result.However, the probability (or electric) current at some fixed point θ = θ 0 on the ring can be negative.This is the essence of the QB effect.

III. CURRENT OPERATOR
We now introduce a probability current operator and derive its representation in terms of tensor products of one-qubit gates.Without any loss of generality, and in order to simplify the calculations to follow, we set θ 0 = 0.Then, substituting Eq. (2) into Eq.( 4) and making use of and Eq. ( 1), we obtain Alternatively, J can be written as where the operator represents the (scaled) probability current at θ 0 = 0 for quantum states in the 2 N -dimensional subspace of the Hilbert space spanned by |0⟩, |1⟩, . .., |2 N − 1⟩.
In the rest of this section, we show how the operator ĴN , for any N = 1, 2, 3, . .., can be decomposed into a sum of tensor products of the following two-dimensional operators: the identity operator Î and the Pauli gates X and Ẑ.The main result of this section is given by Eq. ( 17).This decomposition is essential for one's ability to simulate QB on a quantum computer.

A. N = 1 case
We begin by considering the N = 1 case.In the matrix representation defined by Clearly, Ĵ1 can be decomposed as where We now turn to the N = 2 case.Writing The last matrix can be rewritten as follows: where Ĵ1 is given by Eqs. ( 7) and ( 8), and Noticing that we find Finally, using we arrive at the following decomposition:

C. General case
We now generalize the method of Sec.III B to construct an explicit decomposition of ĴN , for arbitrary N .To this end, we first write ĴN , defined by Eq. ( 6), in the matrix form.Following the convention adopted in Secs.III A and III B, we take |m⟩ to be represented by the 2 N -dimensional column vector with the n th element equal to δ mn .Then, Since the last matrix can be written as , it is easy to see that [cf.Eq. ( 9)] where ĈN−1 is the 2 N −1 ×2 N −1 matrix of ones.It follows from Eq. ( 15) that 1 into Eq.( 16), we get the following recurrence relation [cf.Eq. ( 10)]: It is straightforward to verify that the solution to this recurrence relation is [cf.Eq. ( 10)] Substituting Eq. ( 11) into the last expression, and using Finally, using Eq. ( 13), we arrive at the following explicit decomposition of ĴN [cf.Eq. ( 14)]: Equation ( 17) constitutes the main result of this section.

IV. EXAMPLE OF A BACKFLOWING STATE
Decomposition (17) allows us to devise a quantum computing circuit for measuring the probability current at a fixed point on the ring.What we also need for a quantum simulation of QB is a quantum state that would give rise to a substantially negative probability current.In this section, we present an explicit example of such a state.
Consider the state |ψ⟩ defined by Eq. ( 2) with The state is normalized.Indeed, using identities it is straightforward to verify that the normalization condition (3) is fulfilled.Now, substituting Eq. ( 18) into Eq.( 5) and performing a straightforward calculation, taking into account that Clearly J < 0, for all N ≥ 1, meaning that |ψ⟩, defined by Eq. ( 18) exhibits QB.In particular, and For the purpose of clarity, we would like to point out that the state defined by Eq. ( 18) is not the only state exhibiting negative probability current (see Ref. [5] for other examples of backflowing states.)Nor is it the state maximizing the backflow probability transfer [5].The main reasons we use the state given by Eq. ( 18) in our study are its simplicity -specifically, the fact that the expansion coefficients have a simple linear dependence on the quantum number m -and its generality, as the state gives rise to negative probability current for any N .
It is also interesting to note that, for the state defined by Eq. ( 18), J → −∞ as N → ∞.This example shows that, just as in the particle-on-a-line case [3], the instantaneous probability current for non-negative angular momentum states in a ring is unbounded from below.

V. IMPLEMENTATION OF QUANTUM BACKFLOW ON A QUANTUM COMPUTER
Equipped with the decomposition ( 17) and the explicit backflowing state example (18), we proceed to simulating QB on the IBM-Q quantum computer [16].The quantum The simulation consists of the following four stages.First, we prepare the system qubit(s) in state |0⟩ S for N = 1, and in states |0⟩ S1 and |0⟩ S2 for N = 2. Second, we apply a set of gates to the system qubit(s) to create the target input state |ψ⟩, Eq. ( 2), with the expansion coefficients given by Eq. (18).Third, we act with singlequbit gates Ûk for N = 1, and with Ûk1 and Ûk2 for N = 2, on |ψ⟩ to obtain a new state |ψ 2 ⟩.The gates Ûk , Ûk1 , and Ûk2 are to be defined below.Finally, the qubits of |ψ 2 ⟩ are measured in the Ẑ basis and the measurement outcome probabilities are used to compute the expectation value of operators Vk .As we explain in detail below, the operators Vk are elementary N -qubit gates allowing one to represent the current operator as with K 1 = 2, K 2 = 7, and λ k 's being some real numbers.We now provide further details of the outlined simulation procedure and present the experimental results, separately in the N = 1 and N = 2 case.
We first initialize the system qubit, S, in state |0⟩ S .We then transform |0⟩ S into the desired backflowing state whose expansion coefficients are obtained from Eq. ( 18) by setting N = 1 and m = 0, 1.The transformation is achieved by acting on |0⟩ S with the Y -rotation operator Ry (α) ≡ exp −i α 2 Ŷ : with α ≃ 5.35589.Next, we apply gate Ûk to the system qubit to obtain Here, Û1 is the Hadamard gate, and Û2 is the identity operator, Û2 = Î .
This choice of the transformations Û1 and Û2 allows us to experimentally determine the sought expectation values ⟨ V1 ⟩ and ⟨ V2 ⟩ by measuring |ψ 2 ⟩ in the Ẑ basis.This works as follows.
The probabilities that the measurement will collapse |ψ 2 ⟩ onto |0⟩ S and |1⟩ S are given by respectively, and so For k = 1, we have which means that where we have used the identity X = Ĥd Ẑ Ĥd .For k = 2, we have This is how, by measuring the difference P 0 − P 1 in the quantum circuit shown in Fig. 2(a), we can evaluate the expectation values ⟨ Vk ⟩ experimentally.
Once the backflowing state |ψ⟩ has been prepared, the rest of the simulation follows the same steps as in the N = 1 case.The transformation from |ψ⟩ to |ψ 2 ⟩ [see Fig.

VI. CONCLUSION
In this paper, we report a quantum simulation of QB within a circular geometry.The system under consideration -a particle moving freely in a circular ring -is well-suited for the simulation, as its discrete angular momentum eigenstates can be effectively modeled using qubits.More specifically, N -qubit circuits can be employed to simulate QB for quantum states that are superpositions of 2 N angular momentum eigenstates.We explicitly design such circuits for the cases when N = 1 and N = 2, and subsequently implement them on the IBM-Q quantum computer.Our quantum simulations demonstrate negative probability current for quantum states comprised solely of non-negative angular momentum states, thereby confirming the presence of QB.The simulated probability current values are reasonably close to the corresponding theoretical values.
The quantum simulations presented in this paper have thus far been limited to one and two qubits, i.e., to the cases of N = 1 and N = 2, due to us having access only to noisy quantum devices.Nonetheless, we have established a comprehensive theoretical framework for simulating QB with an arbitrary number of qubits (i.e., for any value of N ).To elaborate, a quantum simulation of QB involves two main stages: (i) the preparation of a backflowing state, and (ii) the subsequent measurement of probability current.Concerning (i), we have devised a specific example of a backflowing state valid for arbitrary N ; this example is detailed by Eqs. ( 2) and (18).As for (ii), we have derived a universally applicable decomposition of the probability current operator, given by Eq. ( 17), which can be readily employed to construct a current-measuring circuit for arbitrary N .In light of these findings, our theoretical results provide a clear pathway for simulating QB using any number of qubits.
The exploration of novel applications for quantum computing has been an active area of research [20][21][22].The study reported in the present paper expands the domain of potential applications of quantum computing and uncovers new connections between different areas of quantum physics.