Recurrence and P\'olya number of quantum walks

TL;DR

This paper investigates the recurrence properties of d-dimensional unbiased quantum walks, deriving conditions for recurrence and localization, and highlighting how quantum walk recurrence depends on topology, coin choice, and initial state.

Contribution

It introduces a criterion for recurrence and localization in quantum walks, showing how initial states and topology influence recurrence behavior, unlike classical walks.

Findings

Recurrence probability depends on topology, coin, and initial state.

A sufficient condition for quantum walk recurrence is established.

Quantum walks can switch between recurrent and transient by initial state alteration.

Abstract

We analyze the recurrence probability (P\'olya 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 localisation of quantum walks. In contrast to classical walks, where the P\'olya 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 to change the character of the quantum walk from recurrent to transient by altering the initial state.