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arXiv:2201.03307 · Physical Review A

Induced on-demand revival in coined quantum walks on infinite d -dimensional lattices

Mahesh N. Jayakody, Ismael L. Paiva, Asiri Nanayakkara, Eliahu Cohen

Published Mar 7, 2022• 4 citations
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Abstract

The study of recurrences and revivals in quantum systems has attracted a great deal of interest because of its importance in the control of quantum systems and its potential use in developing new technologies. In this paper, we introduce a protocol to induce full-state revivals in a huge class of quantum walks on a $d$-dimensional lattice governed by a $c$-dimensional coin system. The protocol requires two repeated interventions in the coin degree of freedom. We also present a characterization of the walks that admit such a protocol. Moreover, we modify the quantity known as the P\'olya number, typically used in the study of recurrences in classical random walks and quantum walks, to create a witness of the first revival of the walk.

DOI
10.1103/physreva.105.032413

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The study of recurrences and revivals in quantum systems has attracted a great deal of interest because of its importance in the control of quantum systems and its potential use in developing new technologies. In this paper, we introduce a protocol to induce full-state revivals in a huge class of quantum walks on a ddd-dimensional lattice governed by a ccc-dimensional coin system. The protocol requires two repeated interventions in the coin degree of freedom. We also present a characterization of the walks that admit such a protocol. Moreover, we modify the quantity known as the P'olya number, typically used in the study of recurrences in classical random walks and quantum walks, to create a witness of the first revival of the walk.
The P\'olya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. \ifmmode \check{S}\else \v{S}\fi{}tefa\ifmmode \check{n}\else \v{n}\fi{}\'ak et al., Phys. Rev. Lett. 100, 020501 (2008)] which is based on a specific measurement scheme. The P\'olya number of a quantum walk depends, in general, on the choice of the coin and the initial coin state, in contrast to classical random walks where the lattice dimension uniquely determines it. We analyze several examples to depict the variety of possible recurrence properties. First, we show that for the class of quantum walks driven by Hadamard tensor-product coins, the P\'olya number is independent of the initial conditions and the actual coin operators, thus resembling the property of the classical walks. We provide an estimation of the P\'olya number for this class of quantum walks. Second, we examine the two-dimensional Grover walk, which exhibits localization and thus is recurrent, except for a particular initial state for which the walk is transient. We generalize the Grover walk to show that one can construct in arbitrary dimensions a quantum walk which is recurrent. This is in great contrast with classical walks which are recurrent only for the dimensions d=1,2d=1,2d=1,2. Finally, we analyze the recurrence of the 2D Fourier walk. This quantum walk is recurrent except for a two-dimensional subspace of the initial states. We provide an estimation of the P\'olya number in its dependence on the initial state.
Recurrence in the classical random walk is well known and described by the Pólya number. For quantum walks, recurrence is similarly understood in terms of the probability of a localized quantum walker to return to its origin. Under certain circumstances the quantum walker may also return to an arbitrary initial quantum state in a finite number of steps. Quantum state revivals in quantum walks on cycles using coin operators which are constant in time and uniform across the path have been described before but only incompletely. In this paper we find the general conditions for which full-quantum state revival will occur.
Recurrence of quantum walks on lattices can be characterized by the generalized Pólya number. Its value reflects the difference between a classical and a quantum system. The dimension of the lattice is not a unique parameter in the quantum case; both the coin operator and the initial quantum state of the coin influence the recurrence in a nontrivial way. In addition, the definition of the Pólya number involves measurement of the system. Depending on how measurement is included in the definition, the recurrence properties vary. We show that in the limiting case of frequent, strong measurements, one can approach the classical dynamics. Comparing various cases, we have found numerical indication that our previous definition of the Pólya number provides an upper limit.
The physical process in the macroscopic world unfolds along a single time direction, while the evolution of a quantum system is reversible in principle. How to recover a quantum system to its past state is a complex issue of both fundamental and practical interests. In this article, we experimentally demonstrate a novel method for recovering the state in quantum walks (QWs), also known as full-state revival. Moreover, we observe two other important phenomena in QWs, recurrence and periodicity, via simplifying and repeatedly implementing the scheme, respectively. Our experiments show that obtaining these phenomena requires neither any information of the initial state nor full information of the coin operations. Our work sheds new light on quantum state engineering and recovery, and the initialization of quantum devices based on QWs.
In this study we show a way of achieving the reverse evolution of n-dimensional quantum walks by introducing interventions on the coin degree of freedom during the forward progression of the coin-walker system. Only a single intervention is required to reverse a quantum walker on a line to its initial positon and the number of interventions increases with the dimensionality of the walk. We present an analytical treatment to prove these results. This reversion scheme can be used to generate periodic bounded quantum walks and to control the locations where particle can be found with highest probability. From the point of view of quantum computations and simulations, this scheme could be useful in resetting quantum operations and implementing certain quantum gates.
Recurrence of a random walk is described by the Pólya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full revival of its quantum state. Localization for two-dimensional quantum walks is known to exist in the sense of non-vanishing probability distribution in the asymptotic limit. We show, on the example of the 2D Grover walk, that one can exploit the effect of localization to construct stationary solutions. Moreover, we find full revivals of a quantum state with a period of two steps. We prove that there cannot be longer cycles for a four-state quantum walk. Stationary states and revivals result from interference, which has no counterpart in classical random walks.