Absolute zeta functions and periodicity of quantum walks on cycles

TL;DR

This paper explores the relationship between quantum walks on cycle graphs and absolute zeta functions, deriving explicit forms and showing their automorphic properties, thus linking quantum computation and number theory.

Contribution

It establishes a novel connection between quantum walks and absolute zeta functions, providing explicit formulas and demonstrating their automorphic nature.

Findings

Derived explicit forms of absolute zeta functions for quantum walks.

Showed that these zeta functions are absolute automorphic forms.

Analyzed the periods of Hadamard and Grover walks on cycles.

Abstract

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over $\mathbb{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.