On the relation between quantum walks and zeta functions

TL;DR

This paper explores the connection between quantum walks on graphs and zeta functions, providing explicit formulas for transition matrix spectra using the second weighted zeta function.

Contribution

It introduces a new explicit formula linking quantum walk transition matrices to zeta functions, offering novel proofs for spectral properties.

Findings

Derived explicit formulas for transition matrix spectra

Provided new proofs for spectral properties of quantum walks

Linked quantum walks to zeta function theory

Abstract

We present an explicit formula for the characteristic polynomial of the transition matrix of the discrete-time quantum walk on a graph via the second weighted zeta function. As applications, we obtain new proofs for the results on spectra of the transition matrix and its positive support.