A zeta function related to the transition matrix of the discrete-time quantum walk on a graph

TL;DR

This paper introduces a zeta function related to the positive support of the cube of the Grover transition matrix in discrete-time quantum walks on graphs, providing new spectral and structural insights.

Contribution

It establishes a structure theorem, defines a novel zeta function, and derives its Euler product, determinant expression, and spectral properties for quantum walk transition matrices.

Findings

Derived the characteristic polynomial for regular graphs.

Provided the spectra of the positive support of the cube of the Grover matrix.

Analyzed the poles and convergence radius of the zeta function.

Abstract

We present the structure theorem for the positive support of the cube of the Grover transition matrix of the discrete-time quantum walk (the Grover walk) on a general graph $G$ under same condition. Thus, we introduce a zeta function on the positive support of the cube of the Grover transition matrix of $G$, and present its Euler product and its determinant expression. As a corollary, we give the characteristic polynomial for the positive support of the cube of the Grover transition matrix of a regular graph, and so obtain its spectra. Finally, we present the poles and the radius of the convergence of this zeta function.