The discrete-time quaternionic quantum walk and the second weighted zeta function on a graph

TL;DR

This paper introduces a quaternionic quantum walk on finite graphs, extending the Grover walk, and links its eigenvalues to the second weighted zeta function, revealing new spectral properties and relationships.

Contribution

It develops a quaternionic extension of quantum walks, provides a method to compute eigenvalues using complex eigenvalues, and connects these to the second weighted zeta function.

Findings

All right eigenvalues of the quaternionic quantum walk are determined.

The eigenvalues relate to complex eigenvalues of a derived quaternionic weighted matrix.

The work generalizes the determinant expression of the second weighted zeta function.

Abstract

We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function.