Eigenvalues of two-state quantum walks induced by the Hadamard walk

TL;DR

This paper investigates the eigenvalues of two-state quantum walks, particularly Hadamard walks with defects and phase variations, revealing their distribution patterns and parameter dependencies through analytical and numerical methods.

Contribution

It introduces a detailed analysis of eigenvalue distributions for specific quantum walk models using the splitted generating function method, expanding understanding of localization phenomena.

Findings

Eigenvalue distributions depend on characteristic parameters.

Numerical simulations confirm analytical results.

Localization properties are linked to eigenvalue distributions.

Abstract

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization of the walks. We revealed the distributions of the eigenvalues given by the splitted generating function method (the SGF method) of the quantum walks we had treated in our previous studies. In particular, we focused on two kinds of the Hadamard walk with one defect models and the two-phase QWs that have phases at the non-diagonal elements of the unitary transition operators. As a result, we clarified the characteristic parameter dependence for the distributions of the eigenvalues with the aid of numerical simulation.