Limit theorems for a localization model of 2-state quantum walks

TL;DR

This paper establishes limit theorems for a 2-state quantum walk model that exhibits localization, providing insights into its stationary distribution and convergence behavior, which are not observed in standard quantum walks.

Contribution

The paper introduces a novel 2-state quantum walk model with a half-time matrix operation that can localize, and proves two limit theorems describing its long-term behavior.

Findings

The quantum walk can localize around the origin.

Two limit theorems are established: stationary distribution and convergence in distribution.

Localization occurs in a model where it typically does not in standard quantum walks.

Abstract

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present two limit theorems, that is, one is a stationary distribution and the other is a convergence theorem in distribution.