On the relation between quantum walks and absolute zeta functions

TL;DR

This paper explores the mathematical connection between quantum walks, specifically the Grover walk, and absolute zeta functions, revealing their automorphic properties and explicit examples like the cycle graph.

Contribution

It establishes a novel link between quantum walks and absolute zeta functions, showing their automorphic nature and providing explicit computations for specific graphs.

Findings

Zeta function from quantum walk is an absolute automorphic form.

Explicit computation of the absolute zeta function for cycle graphs.

Connection between quantum walk matrices and multiple gamma functions.

Abstract

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, the absolute zeta function can be considered as a zeta function over F_1. This paper presents a connection between the quantum walk and the absolute zeta function. First we deal with a zeta function determined by a time evolution matrix of the Grover walk on a graph. The Grover walk is a typical model of the quantum walk. Then we prove that the zeta function given by the quantum walk is an absolute automorphic form of weight depending on the number of edges of the graph. Furthermore we consider an absolute zeta function for the zeta function based on a quantum walk. As an example, we compute an absolute zeta function for the cycle graph and show that it is expressed as the multiple gamma function of order 2.