Localization of discrete-time quantum walks on a half line via the CGMV method

TL;DR

This paper analyzes discrete-time quantum walks on a half line using spectral methods, specifically the CGMV approach, to derive spectral measures and demonstrate localization phenomena.

Contribution

It applies the CGMV method to quantum walks on a half line, providing new spectral analysis and an alternative proof of localization on homogeneous trees.

Findings

Spectral measure for the quantum walk obtained

Alternative proof of localization on homogeneous trees

Spectral analysis methodology demonstrated

Abstract

We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. [1] showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle [2], expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al. [3].