Grover algorithm and absolute zeta functions

TL;DR

This work investigates the link between Grover's quantum search algorithm and absolute zeta functions by analyzing the Grover time-evolution matrix $U_N$. It shows that $U_N$ has finite period $T(U_N)$ only for $N=2$ and $N=4$ (with $T(U_2)=4$ and $T(U_4)=6$), enabling the construction of the matrix zeta function $\\zeta_{U_N}$, its absolute automorphic form weight, and, when $T(U_N)<\infty$, the explicit absolute zeta function $\\zeta_{\\zeta_{U_N}}$ together with a functional equation via Kurokawa's theory. For the special cases $N=2$ and $N=4$, the authors derive closed-form expressions for $\\zeta_{U_N}$ and $\\zeta_{\\zeta_{U_N}}$, including central values and gamma-factor expressions, while for general $N$ they provide an expansion of $\\zeta_{\\zeta_{U_N}}(s)$ in a cyclotomic-root–based product form. This work forges a novel connection between quantum algorithm dynamics and absolute zeta theory, suggesting that number-theoretic tools can inform the study of quantum information processing.

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.

Theorems & Definitions (13)

• Proposition 2.1
• Proposition 3.1
• Corollary 3.2
• Proposition 3.3
• Proposition 3.4
• Theorem 3.5
• Definition 3.1
• Theorem 4.1
• Definition 5.1
• Proposition 5.1