Periodicity for space-inhomogeneous quantum walks on the cycle

TL;DR

This paper investigates the periodicity of space-inhomogeneous quantum walks on a cycle, analyzing spectral properties for different coin configurations and extending known results to new cases.

Contribution

It introduces a spectral analysis method for isospectral coins and links the periodicity of the entire system to the coin at the origin in non-isospectral cases.

Findings

Spectral analysis for isospectral coin cases

Extension of periodicity results to some isospectral coins

Periodicity determined by the coin at the origin in non-isospectral cases

Abstract

In this paper, we consider periodicity for space-inhomogeneous quantum walks on the cycle. For isospectral coin cases, we propose a spectral analysis. Based on the analysis, we extend the result for periodicity for Hadamard walk to some isospectral coin cases. For non-isospectral coin cases, we consider the the system that uses only one general coin at the origin and the identity coin at the other sites. In this case, we show that the periodicity of the general coin at the origin determines the periodicity for the whole system.