On the relation between quantum walks and absolute zeta 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.
Keywords
Cite
@article{arxiv.2306.14625,
title = {On the relation between quantum walks and absolute zeta functions},
author = {Norio Konno},
journal= {arXiv preprint arXiv:2306.14625},
year = {2024}
}
Comments
11 pages, minor corrections. arXiv admin note: text overlap with arXiv:2204.00765