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Iwao Sato

Publications and source records attributed to Iwao Sato.

At least 19 recordsLinked to original sources
arXiv2025

The graph zeta functions with respect to the group matrix of a finite group

In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group $Γ$. Furthermore, we give another proof of Dedekind Theorem for the group determinant of $Γ$ by the decomposition formula for a matrix of a group covering of a digraph. Finally, we treat the weighted complexity of the complete graph with entries of the group matrix of $Γ$ as arc weights.

math.CO↗
arXiv2025

Grover algorithm and absolute zeta functions

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.

quant-ph↗
arXiv2025

Periodicity and absolute zeta functions of multi-state Grover walks on cycles

Quantum walks, the quantum counterpart of classical random walks, are extensively studied for their applications in mathematics, quantum physics, and quantum information science. This study explores the periods and absolute zeta functions of Grover walks on cycle graphs. Specifically, we investigate Grover walks with an odd number of states and determine their periods for cycles with any number of vertices greater than or equal to two. In addition, we compute the absolute zeta functions of M-type Grover walks with finite periods. These results advance the understanding of the properties of Grover walks and their connection to absolute zeta functions.

quant-ph↗
arXiv2024

Absolute zeta functions for zeta functions of quantum walks

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 a graph. The Grover walk is a typical model of quantum walks. The time evolution of the Grover walk is obtained by a unitary matrix that is called the Grover matrix. We define the zeta function determined by the Grover matrix. First we prove that the zeta function of the Grover walk is the absolute automorphic form that constructs the absolute zeta function. Next we calculate the absolute zeta function defined by Grover walks on some graphs. The absolute zeta functions of the Grover walks are expressed by the multiple gamma function. Other types of absolute zeta functions are obtained as an analogue of the multiple gamma function.

quant-ph↗
arXiv2024

Absolute zeta functions and periodicity of quantum walks on cycles

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.

quant-ph↗
arXiv2024

A quantization of interacting particle systems

Interacting particle systems studied in this paper are probabilistic cellular automata with nearest-neighbor interaction including the Domany-Kinzel model. A special case of the Domany-Kinzel model is directed percolation. We regard the interacting particle system as a Markov chain on a graph. Then we present a new quantization of the interacting particle system. After that, we introduce a zeta function of the quantized model and give its determinant expression. Moreover, we calculate the absolute zeta function of the quantized model for the Domany-Kinzel model.

quant-ph↗
arXiv2024

Absolute zeta functions for zeta functions of quantum cellular automata

Our previous work dealt with the zeta function for the interacting particle system (IPS) including quantum cellular automaton (QCA) as a typical model in the study of ``IPS/Zeta Correspondence". On the other hand, the absolute zeta function is a zeta function over F_1 defined by a function satisfying an absolute automorphy. This paper proves that a new zeta function given by QCA is an absolute automorphic form of weight depending on the size of the configuration space. As an example, we calculate an absolute zeta function for a tensor-type QCA, and show that it is expressed as the multiple gamma function. In addition, we obtain its functional equation by the multiple sine function.

quant-ph↗
arXiv2023

Alternating Walk/Zeta Correspondence

We consider the alternating zeta function and the alternating $L$-function of a graph $G$, and express them by using the Ihara zeta function of $G$. Next, we define a generalized alternating zeta function of a graph, and express the generalized alternating zeta function of a vertex-transitive regular graph by spectra of the transition probability matrix of the symmetric simple random walk on it and its Laplacian. Furthermore, we present an integral expression for the limit of the generalized alternating zeta functions of a series of vertex-transitive regular graphs. As an example, we treat the generalized alternating zeta functions of a finite torus. Finally, we treat the relation between the Mahler measure and the alternating zeta function of a graph.

math.CO↗
arXiv2023

Ronkin/Zeta Correspondence

The Ronkin function was defined by Ronkin in the consideration of the zeros of almost periodic function. Recently, this function has been used in various research fields in mathematics, physics and so on. Especially in mathematics, it has a closed connections with tropical geometry, amoebas, Newton polytopes and dimer models. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on Zeta Correspondence. The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Ronkin function and our zeta function for random walks and quantum walks. Firstly we consider this relation in the case of one-dimensional random walks. Afterwards we deal with higher-dimensional random walks. For comparison with the case of the quantum walk, we also treat the case of one-dimensional quantum walks. Our results bridge between the Ronkin function and the zeta function via quantum walks for the first time.

math-ph↗
arXiv2022

Walk/Zeta Correspondence

Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno-Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.

quant-ph↗
arXiv2022

The limit theorem with respect to the matrices on non-backtracking paths of a graph

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

math.CO↗
arXiv2022

Metzler/Zeta Correspondence

We present an explicit formula for the determinant on the Metzler matrix of a digraph $D$. Furthermore, we introduce a walk-type zeta function with respect to this Metzler matrix of the symmetric digraph of a finite torus, and express its limit formula by using the integral expression.

math.CO↗
arXiv2022

CTM/Zeta Correspondence

In our previous work, we investigated the relation between zeta functions and discrete-time models including random and quantum walks. In this paper, we introduce a zeta function for the continuous-time model (CTM) and consider CTMs including the corresponding random and quantum walks on the d-dimensional torus.

quant-ph↗
arXiv2022

Mahler/Zeta Correspondence

The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on "Zeta Correspondence". The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Mahler measure and our zeta function for quantum walks. Firstly we consider this relation in the case of one-dimensional quantum walks. Afterwards we deal with higher-dimensional quantum walks. For comparison with the case of the quantum walk, we also treat the case of higher-dimensional random walks. Our results bridge between the Mahler measure and the zeta function via quantum walks for the first time.

quant-ph↗
arXiv2022

IPS/Zeta Correspondence

Our previous works presented zeta functions by the Konno-Sato theorem or the Fourier analysis for one-particle models including random walks, correlated random walks, quantum walks, and open quantum random walks. This paper introduces a new zeta function for multi-particle models with probabilistic or quantum interactions, called the interacting particle system (IPS). We compute the zeta function for some tensor-type IPSs.

quant-ph↗
arXiv2022

A Generalized Grover/Zeta Correspondence

We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function and a generalized zeta function of a graph $G$ with respect to its generalized Grover matrix as an analog of the Ihara zeta function and present explicit formulas for their zeta functions for a vertex-transitive graph. As applications, we express the limit on the generalized zeta functions of a family of finite vertex-transitive regular graphs by an integral. Furthermore, we give the limit on the generalized zeta functions of a family of finite tori as an integral expression.

math.CO↗
arXiv2021

A note on the Grover walk and the generalized Ihara zeta function of the one-dimensional integer lattice

Chinta, Jorgenson and Karlsson introduced a generalized version of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. On the other hand, Konno and Sato obtained a formula of the characteristic polynomial of the Grover matrix by using the determinant expression for the second weighted zeta function of a finite graph. In this paper, we focus on a relationship between the Grover walk and the generalized Ihara zeta function. That is to say, we treat the generalized Ihara zeta function of the one-dimensional integer lattice as a limit of the Ihara zeta function of the cycle graph.

math.CO↗