Recurrence properties of unbiased coined quantum walks on infinite $d$-dimensional lattices

TL;DR

This paper investigates the recurrence properties of unbiased quantum walks on infinite lattices, revealing how initial states and coin choices influence recurrence, contrasting with classical walk behaviors, and providing analytical results for various quantum walk models.

Contribution

It analyzes the recurrence of quantum walks, showing dependence on initial states and coins, and constructs examples of recurrent walks in higher dimensions, unlike classical counterparts.

Findings

Quantum walks driven by independent coins have Pólya number independent of initial conditions.

The 2-D Grover walk is recurrent except for a specific initial state.

Recurrent quantum walks can be constructed in arbitrary dimensions.

Abstract

The P\'olya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. \v{S}tefa\v{n}\'ak, I. Jex and T. Kiss, Phys. Rev. Lett. \textbf{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 independent coins for all spatial dimensions, 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 analytical estimation of the P\'olya number for this class of quantum walks. Second, we examine the 2-D Grover walk, which exhibits localisation 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 the classical walks which are recurrent only for the dimensions $d=1,2$. Finally, we analyze the recurrence of the 2-D Fourier walk. This quantum walk is recurrent except for a two-dimensional subspace of the initial states. We provide an analytical formula of the P\'olya number in its dependence on the initial state.