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arXiv:1011.2066 · Physica Scripta

Full revivals in 2D quantum walks

M. Štefaňák, Bálint Kollár, T. Kiss, Igor Jex

Published Sep 1, 2010• 24 citations
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Abstract

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.

DOI
10.1088/0031-8949/2010/t140/014035

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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.
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.
Abstract Full state revivals in a quantum walk can be viewed as returning of the walker to the initial quantum state in a periodic fashion during the propagation of the walk. In this paper we show that for any given number of spatial dimensions, a coin operator can be constructed to generate a quantum walk having full revivals with any desired period. From the point of view of quantum computation and simulations, these coin operators can be useful in implementing quantum walks, which oscillate between any two states with a finite periodicity.
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.
We analyze the role of dimensionality in the time evolution of discrete-time quantum walks through the example of the three-state walk on a two-dimensional triangular lattice. We show that the three-state Grover walk does not lead to trapping (localization) or recurrence to the origin, in sharp contrast to the Grover walk on the two-dimensional square lattice. We determine the power-law scaling of the probability at the origin with the method of stationary phase. We prove that only a special subclass of coin operators can lead to recurrence, and there are no coins that lead to localization. The propagation for the recurrent subclass of coins is quasi-one dimensional.
We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.