The discrete-time quaternionic quantum walk on a graph

TL;DR

This paper analyzes the spectral properties of quaternionic quantum walks on graphs by solving the right eigenvalue problem of quaternionic matrices, providing a complete characterization of eigenvalues and illustrating with examples.

Contribution

It introduces a method to determine all right eigenvalues of quaternionic matrices and applies it to analyze quaternionic quantum walks, extending spectral analysis techniques.

Findings

All right eigenvalues of quaternionic quantum walks can be explicitly determined.

Properties of the transition matrix influence the spectral characteristics of the walk.

Examples illustrate the application of the eigenvalue analysis to specific quaternionic quantum walks.

Abstract

Recently, the quaternionic quantum walk was formulated by the first author as a generalization of discrete-time quantum walks. We treat the right eigenvalue problem of quaternionic matrices to analysis the spectra of its transition matrix. The way to obtain all the right eigenvalues of a quaternionic matrix is given. From the unitary condition on the transition matrix of the quaternionic quantum walk, we deduce some properties about it. Our main results, Theorem 5.3, determine all the right eigenvalues of a quaternionic quantum walk by use of those of the corresponding weighted matrix. In addition, we give some examples of quaternionic quantum walks and their right eigenvalues.