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Physica Scripta

Recurrence in coined quantum walks

T. Kiss, L. Kecskés, M. Štefaňák, Igor Jex

Published Jul 1, 2009• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/2009/t135/014055

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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.
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.
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 propose a definition for the P\'olya number of continuous-time quantum walks to characterize their recurrence properties. The definition involves a series of measurements on the system, each carried out on a different member from an ensemble in order to minimize the disturbance caused by it. We examine various graphs, including the ring, the line, the higher-dimensional integer lattices, and a number of other graphs, and we calculate their P\'olya number. For the timing of the measurements, a Poisson process as well as regular timing are discussed. We find that the speed of decay for the probability at the origin is the key for recurrence.
The quantum mechanical generalisation of random walks (called Quantum Walks) present us with a broader spectrum of possibilities compared to their classical counterparts. The aim of the presented study is to explore a new portion of this area by incorporating a new step in the process of the Quantum Walk unique to quantum mechanics: the measurement. Our focus lies in the characterising number of the recurrence behaviour of the walk (Polya-number). We observe the effect of the standard projective measurement, a yes-no measurement on the origin and the effect of different measurement schemes (periodic and random) on the definition and the numeric value of the Polya-number.
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.