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<p>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 the graph, which is a typical model of quantum walks. The time evolution of the Grover walk is obtained by a unitary matrix called the Grover matrix. We define a new type of the zeta function determined by the Grover matrix. Then, we consider to construct the absolute zeta function. In the previous paper [<CitationRef CitationID="CR10">10</CitationRef>], it is pointed out that there is a relationship between quantum walk and absolute zeta function. In the first half of this paper, we briefly describe the results of the previous research. In the latter part, we compute the absolute zeta function of the quantum walk by two different methods. One is calculated by the cyclotomic polynomial. The other is based on the series expansion. Its explicit derivation can be extended to a broader class of quantum walks on graphs.</p>

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Absolute zeta functions for zeta functions of quantum walks

  • 期刊论文

  • 发表日期：

    2025年12月26日

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  • Jirô Akahori,
  • Norio Konno,
  • Rikuki Okamoto,
  • Iwao Sato

摘要

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 the graph, which is a typical model of quantum walks. The time evolution of the Grover walk is obtained by a unitary matrix called the Grover matrix. We define a new type of the zeta function determined by the Grover matrix. Then, we consider to construct the absolute zeta function. In the previous paper [10], it is pointed out that there is a relationship between quantum walk and absolute zeta function. In the first half of this paper, we briefly describe the results of the previous research. In the latter part, we compute the absolute zeta function of the quantum walk by two different methods. One is calculated by the cyclotomic polynomial. The other is based on the series expansion. Its explicit derivation can be extended to a broader class of quantum walks on graphs.