Absolute zeta functions for zeta functions of quantum walks

TL;DR

This paper explores the connection between quantum walks, specifically Grover walks, and absolute zeta functions, revealing their expression through multiple gamma functions and establishing a novel link with absolute automorphic forms.

Contribution

It introduces the concept of absolute zeta functions derived from Grover walks and demonstrates their expression via multiple gamma functions, bridging quantum walks and absolute mathematics.

Findings

Zeta function of Grover walk is an absolute automorphic form.

Absolute zeta functions are expressed by multiple gamma functions.

Analogues of absolute zeta functions are constructed for different graphs.

Abstract

This paper presents a connection between the quantum walk and the absolute mathematics. The quantum walk is a quantum counterpart of the classical random walk. We especially deal with the Grover walk on a graph. The Grover walk is a typical model of quantum walks. The time evolution of the Grover walk is obtained by a unitary matrix that is called the Grover matrix. We define the zeta function determined by the Grover matrix. First we prove that the zeta function of the Grover walk is the absolute automorphic form that constructs the absolute zeta function. Next we calculate the absolute zeta function defined by Grover walks on some graphs. The absolute zeta functions of the Grover walks are expressed by the multiple gamma function. Other types of absolute zeta functions are obtained as an analogue of the multiple gamma function.