Localization of space-inhomogeneous three-state quantum walks

TL;DR

This paper develops a transfer matrix method to analyze eigenvalues in space-inhomogeneous three-state quantum walks, providing conditions for localization phenomena crucial for quantum information processing.

Contribution

It extends previous techniques to construct a transfer matrix approach for eigenvalue problems in complex inhomogeneous quantum walks.

Findings

Derived necessary and sufficient conditions for eigenvalues in two-phase quantum walks.

Extended the transfer matrix method to n-state quantum walks with self-loops.

Provided insights into localization properties in inhomogeneous quantum systems.

Abstract

Mathematical analysis on the existence of eigenvalues is essential because it is equivalent to the occurrence of localization, which is an exceptionally crucial property of quantum walks. We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous $n$-state quantum walks in one dimension with $n-2$ self-loops, which is an extension of the technique in a previous study (Quantum Inf. Process 20(5), 2021). This method reveals the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state quantum walk with one defect whose time evolution varies in the negative part, positive part, and at the origin.