Quaternionic quantum walks of Szegedy type and zeta functions of graphs

TL;DR

This paper introduces a quaternionic extension of Szegedy quantum walks on graphs, providing spectral analysis and explicit eigenvalue characterization using zeta functions, advancing quantum walk theory.

Contribution

It presents the first quaternionic extension of Szegedy walks, deriving spectral mapping theorems and eigenvalue formulas via zeta functions of graphs.

Findings

Explicit eigenvalues of quaternionic Szegedy walk obtained

Spectral mapping theorem extended to quaternionic setting

Eigenvectors linked to doubly weighted matrix eigenvalues

Abstract

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.