A family of quantum walks on a finite graph corresponding to the generalized weighted zeta function

TL;DR

This paper introduces a family of quantum walks based on graph zeta functions, allowing analysis of their behavior through characteristic polynomials of transition matrices on finite graphs with multi-edges and loops.

Contribution

It establishes a novel connection between quantum walks and generalized weighted zeta functions, providing a method to analyze quantum walk dynamics on complex finite graphs.

Findings

Derived the characteristic polynomial of the quantum walk transition matrix.

Enabled analysis of quantum walk behavior on graphs with multi-edges and loops.

Linked quantum walks to generalized weighted zeta functions.

Abstract

This paper gives the quantum walks determined by graph zeta functions. The result enables us to obtain the characteristic polynomial of the transition matrix of the quantum walk, and it determines the behavior of the quantum walk. We treat finite graphs allowing multi-edges and multi-loops.