Uniform stationary measure of space-inhomogeneous quantum walks in one dimension

TL;DR

This paper investigates stationary measures of space-inhomogeneous quantum walks in one dimension, demonstrating conditions under which the uniform measure remains stationary, and highlighting differences from classical random walks.

Contribution

It introduces a class of space-inhomogeneous quantum walks where the uniform measure is stationary, extending known results from homogeneous cases.

Findings

Uniform measure is stationary for certain space-inhomogeneous QWs.

Differences between quantum walks and classical random walks are discussed.

Results apply to walks on lines and cycles.

Abstract

The discrete-time quantum walk (QW) is a quantum version of the random walk (RW) and has been widely investigated for the last two decades. Some remarkable properties of QW are well known. For example, QW has a ballistic spreading, i.e., QW is quadratically faster than RW. For some cases, localization occurs: a walker stays at the starting position forever. In this paper, we consider stationary measures of two-state QWs on the line. It was shown that for any space-homogeneous model, the uniform measure becomes the stationary measure. However, the corresponding result for space-inhomogeneous model is not known. Here, we present a class of space-inhomogeneous QWs on the line and cycles in which the uniform measure is stationary. Furthermore, we briefly discuss a difference between QWs and RWs.