Grover algorithm and absolute zeta functions

TL;DR

This paper explores the relationship between the Grover quantum algorithm and absolute zeta functions, focusing on the period of the algorithm to explicitly derive associated zeta functions using Kurokawa's theorem.

Contribution

It establishes a novel connection between quantum algorithms and number theory by relating the Grover algorithm's period to absolute zeta functions.

Findings

Finite periods allow explicit zeta function derivation via Kurokawa's theorem

Zeta function expansions can be computed regardless of the period's finiteness

The study links quantum algorithm properties with algebraic zeta functions

Abstract

The Grover algorithm is one of the most famous quantum algorithms. On the other hand, the absolute zeta function can be regarded as a zeta function over $\mathbb{F}_{1}$ defined by a function satisfying the absolute automorphy. In this study, we show the property of the Grover algorithm and present a relation between the Grover algorithm and the absolute zeta function. We focus on the period of the Grover algorithm, because if the period is finite, then we are able to get an absolute zeta function explicitly by Kurokawa's theorem. In addition, whenever the period is finite or not, an expansion of the absolute zeta function can be obtained by a direct computation.