Localization in quantum walks with periodically arranged coin matrices

TL;DR

This paper extends the eigenvalue analysis method for quantum walk localization to models with periodically arranged coin matrices, enabling better understanding of localization phenomena.

Contribution

It generalizes previous eigenvalue analysis techniques to include models with periodic coin matrices, advancing the study of localization in quantum walks.

Findings

Eigenvalue analysis applied to periodic coin matrix models.

Derived time-averaged limit distribution for localized states.

Demonstrated localization in extended quantum walk models.

Abstract

There is a property called localization, which is essential for applications of quantum walks. From a mathematical point of view, the occurrence of localization is known to be equivalent to the existence of eigenvalues of the time evolution operators, which are defined by coin matrices. A previous study proposed an approach to the eigenvalue problem for space-inhomogeneous models using transfer matrices. However, the approach was restricted to models whose coin matrices are the same in positions sufficiently far to the left and right, respectively. This study shows that the method can be applied to extended models with periodically arranged coin matrices. Moreover, we investigate localization by performing the eigenvalue analysis and deriving their time-averaged limit distribution.