Space-homogeneous quantum walks on Z from the viewpoint of complex analysis

TL;DR

This paper introduces a framework for analyzing quantum walks on Z using complex analysis, revealing their structure and limit distributions, and expanding the theoretical understanding of these quantum dynamical systems.

Contribution

It defines analyticity for quantum walks on Z and shows that all such walks are composed of shift operators and continuous-time walks, broadening the theory.

Findings

All analytic space-homogeneous quantum walks on Z are decomposable into shifts and continuous-time walks.

Existence of weak limit distributions for these quantum walks is established.

The theory of quantum walks is extended beyond previously studied cases.

Abstract

The subject of this paper is quantum walks, which are expected to simulate several kinds of quantum dynamical systems. In this paper, we define analyticity for quantum walks on Z. Almost all the quantum walks on $\mathbb{Z}$ which have been already studied are analytic. In the framework of analytic quantum walks, we can enlarge the theory of quantum walks. We obtain not only several generalizations of known results, but also new types of theorems. It is proved that every analytic space-homogeneous quantum walk on Z is essentially a composite of shift operators and continuous-time analytic space-homogeneous quantum walks. We also prove existence of the weak limit distribution for analytic space-homogeneous quantum walks on Z.