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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., Comment: 17 pages

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., Comment: 16 pages

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., Comment: 14 pages

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., Comment: 24 pages. arXiv admin note: text overlap with arXiv:2205.00457; text overlap with arXiv:1905.13182 by other authors

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., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2202.05966

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., Comment: 16 pages

The Ihara expression of a weighted zeta function for a general finite digraph is given. It unifies all the Ihara expressions obtained for known zeta functions for finite digraphs. Any digraph in this paper permits multi-edges and multi-loops.

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., Comment: 27 pages. arXiv admin note: text overlap with arXiv:2109.07664, arXiv:2104.10287

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., Comment: 10 pages. arXiv admin note: text overlap with arXiv:2011.14162

We consider a spectral analysis on the quantum walks on graph $G=(V,E)$ with the local coin operators $\{C_u\}_{u\in V}$ and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues $\kappa,\kappa'$ and $p=\dim(\ker(\kappa-C_u))$ for any $u\in V$ with $1\leq p\leq \delta(G)$, where $\delta(G)$ is the minimum degrees of $G$. We show that this quantum walk can be decomposed into a cellular automaton on $\ell^2(V;\mathbb{C}^p)$ whose time evolution is described by a self adjoint operator $T$ and its remainder. We obtain how the eigenvalues and its eigenspace of $T$ are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on $\mathbb{Z}^d$ with the moving shift in the Fourier space., Comment: 29 pages, 1 figure

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., Comment: 9 pages, minor corrections, Quantum Studies: Mathematics and Foundations, Volume 9, pp.165-173 (2022)

We present the characteristic polynomial for the transition matrix of a vertex-face walk on a graph, and obtain its spectra. Furthermore, we express the characteristic polynomial for the transition matrix of a vertex-face walk on the 2-dimensional torus by using its adjacency matrix, and obtain its spectra. As an application, we define a new walk-type zeta function with respect to the transition matrix of a vertex-face walk on the 2-dimensional torus, and present its explicit formula., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2103.12971, arXiv:2011.14162

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., Comment: 19 pages

We define a zeta function woth respect to the twisted Grover matrix of a mixed digraph, and present an exponential expression and a determinant expression of this zeta function. As an application, we give a trace formula with respect to the twisted Grover matrix of a mixed digraph., Comment: 17 pages

Recently, Gnutzmann and Smilansky presented a formula for the bond scattering matrix of a graph with respect to a Hermitian matrix. We present another proof for this Gnutzmann and Smilansky's formula by a technique used in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilansky's formula to a regular covering of a graph. Finally, we define an $L$-fuction of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilansky's formula to a regular covering of a graph by using its $L$-functions., Comment: 21 pages. arXiv admin note: substantial text overlap with arXiv:1211.4719

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., Comment: 31 pages

Recently the Ihara zeta function for the finite graph was extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno-Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and the zeta function based on the Konno-Sato theorem "Grover/Zeta Correspondence" here., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2011.14162

We define a zeta function of a finite graph derived from time evolution matrix of quantum walk, and give its determinant expression. Furthermore, we generalize the above result to a periodic graph., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1910.12782

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As applications, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph., Comment: 16 pages. arXiv admin note: text overlap with arXiv:2011.14162

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., Comment: 8 pages, Yokohama Mathematical Journal (in press)

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., Comment: The draft is improved: Typos are fixed, Corollaries 4.3 and 4.5 are improved, and Proposition 4.6 and Remark 4.7 are added

We present the structure theorem for the positive support of the cube of the Grover transition matrix of the discrete-time quantum walk (the Grover walk) on a general graph $G$ under same condition. Thus, we introduce a zeta function on the positive support of the cube of the Grover transition matrix of $G$, and present its Euler product and its determinant expression. As a corollary, we give the characteristic polynomial for the positive support of the cube of the Grover transition matrix of a regular graph, and so obtain its spectra. Finally, we present the poles and the radius of the convergence of this zeta function., Comment: arXiv admin note: text overlap with arXiv:1103.0079

We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for the zeta function of an (infinite) periodic graph., Comment: 14 pages

Max-plus algebra is a kind of idempotent semiring over $\mathbb{R}_{\max}:=\mathbb{R}\cup\{-\infty\}$ with two operations $\oplus := \max$ and $\otimes := +$.In this paper, we introduce a new model of a walk on one dimensional lattice on $\mathbb{Z}$, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the $\ell^2$-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk.Moreover, spectral analysis on the total time evolution operator is also given., Comment: 17 pages, 1 figures

Staggered quantum walks on graphs are based on the concept of graph tessellation and generalize some well-known discrete-time quantum walk models. In this work, we address the class of 2-tessellable quantum walks with the goal of obtaining an eigenbasis of the evolution operator. By interpreting the evolution operator as a quantum Markov chain on an underlying multigraph, we define the concept of quantum detailed balance, which helps to obtain the eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of the double discriminant matrix of the quantum Markov chain. To obtain the remaining eigenvectors, we have to use the quantum detailed balance conditions. If the quantum Markov chain has a quantum detailed balance, there is an eigenvector for each fundamental cycle of the underlying multigraph. If the quantum Markov chain does not have a quantum detailed balance, we have to use two fundamental cycles linked by a path in order to find the remaining eigenvectors. We exemplify the process of obtaining the eigenbasis of the evolution operator using the kagome lattice (the line graph of the hexagonal lattice), which has symmetry properties that help in the calculation process., Comment: 21 pages, 3 figures

We obtain a structure theorem of the positive support of the $n$-th power of the Grover walk on $k$-regular graph whose girth is greater than $2(n-1)$. This structure theorem is provided by the parity of the amplitude of another quantum walk on the line which depends only on $k$. The phase pattern of this quantum walk has a curious regularity. We also exactly show how the spectrum of the $n$-th power of the Grover walk is obtained by lifting up that of the adjacency matrix to the complex plain., Comment: 22 pages, 21 figures. The statement of Theorem 4.2 is revised

We introduce a family of discrete-time quantum walks, called two-partition model, based on two equivalence-class partitions of the computational basis, which establish the notion of local dynamics. This family encompasses most versions of unitary discrete-time quantum walks driven by two local operators studied in literature, such as the coined model, Szegedy's model, and the 2-tessellable staggered model. We also analyze the connection of those models with the two-step coined model, which is driven by the square of the evolution operator of the standard discrete-time coined walk. We prove formally that the two-step coined model, an extension of Szegedy model for multigraphs, and the two-tessellable staggered model are unitarily equivalent. Then, selecting one specific model among those families is a matter of taste not generality., Comment: 27 pages, 7 Figures

In the previous report [Phys. Rev. B {\bf{62}} 13812 (2000)], by proposing the mechanism under which electric conductivity is caused by the activational hopping conduction with the Wigner surmise of the level statistics, the temperature-dependent of electronic conductivity of a highly disordered carbon system was evaluated including apparent metal-insulator transition. Since the system consists of small pieces of graphite, it was assumed that the reason why the level statistics appears is due to the behavior of the quantum chaos in each granular graphite. In this article, we revise the assumption and show another origin of the Wigner surmise, which is more natural for the carbon system based on a recent investigation of graph zeta function in graph theory., Comment: 4 pages

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1505.00683

Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra., Comment: 32 pages

We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function., Comment: 15 pages

We establish a generalization of the second weighted zeta function of a graph to the case of quaternions. For an arc-weighted graph whose weights are quaternions, we define the second weighted zeta function by using the Study determinant that is a quaternionic determinant for quaternionic matrices defined by Study. This definition is regarded as a quaternionic analogue of the determinant expression of Hashimoto type for the Ihara zeta function of a graph. We derive the Study determinant expression of Bass type and the Euler product for the quaternionic second weighted zeta function., Comment: arXiv admin note: text overlap with arXiv:1507.06761