Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s10801-022-01122-5

Other Link： https://link.springer.com/article/10.1007/s10801-022-01122-5/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s10955-022-02901-x

Other Link： https://link.springer.com/article/10.1007/s10955-022-02901-x/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Rinton Press  

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.

DOI： 10.26421/qic22.3-4-4

Language：English   Publishing type：(MISC) Institution technical report and pre-print, etc.  

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.

arXiv

Other Link： http://arxiv.org/pdf/2201.03973v1

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.disc.2021.112674

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

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.

DOI： 10.1007/s40509-021-00263-9

arXiv

Other Link： https://link.springer.com/article/10.1007/s40509-021-00263-9/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s00023-021-01131-3

Other Link： https://link.springer.com/article/10.1007/s00023-021-01131-3/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：The Electronic Journal of Combinatorics  

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 an application, 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.

DOI： 10.37236/10108

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s11128-021-03214-w

Other Link： https://link.springer.com/article/10.1007/s11128-021-03214-w/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

DOI： 10.1142/s0129055x21500197

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.laa.2021.03.020

Language：English   Publishing type：Research paper (scientific journal)   Publisher：MDPI AG  

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

DOI： 10.3390/sym13071134

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Informa UK Limited  

DOI： 10.1080/03081087.2021.1910122

Language：English   Publishing type：(MISC) Institution technical report and pre-print, etc.  

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)   Publisher：The Electronic Journal of Combinatorics  

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 (inifinite) periodic graph.

DOI： 10.37236/9104

Language：English   Publishing type：Research paper (scientific journal)   Publisher：MDPI AG  

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization of the walks. We revealed, for the first time, the distributions of the eigenvalues given by the splitted generating function method (the SGF method) of the space-inhomogeneous quantum walks in one dimension we had treated in our previous studies. Especially, we clarified the characteristic parameter dependence for the distributions of the eigenvalues with the aid of numerical simulation.

DOI： 10.3390/e22010127

Language：English   Publishing type：(MISC) Institution technical report and pre-print, etc.  

In this paper, we consider the scattering theory for a one-dimensional quantum walk with impurities which make reflections and transmissions. We focus on an explicit expression of the scattering operator. Our construction of the formula is based on the counting paths of quantum walkers. The Fourier transform of the scattering operator gives an explicit formula of the scattering matrix which is deeply related with the resonant-tunneling for quantum walks.

Language：English   Publishing type：Research paper (scientific journal)   Publisher：RINTON PRESS, INC  

We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.

Web of Science

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Birkhauser Boston  

A necessary and sufficient conditions for a certain class of periodic unitary transition operators to have eigenvalues are given. Applying this, it is shown that Grover walks in any dimension has both of (Formula presented.) as eigenvalues and it has no other eigenvalues. It is also shown that the lazy Grover walks in any dimension has 1 as an eigenvalue, and it has no other eigenvalues. As a result, a localization phenomenon occurs for these quantum walks. A general conditions for the existence of eigenvalues can be applied also to certain quantum walks of Fourier type. It is shown that the two-dimensional Fourier walk does not have eigenvalues and hence it is not localized at any point. Some other topics, such as Grover walks on the triangular lattice, products and deformations of Grover walks, are also discussed.

DOI： 10.1007/s00041-018-9630-6

Scopus

Other Link： http://link.springer.com/article/10.1007/s00041-018-9630-6/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer New York LLC  

Stationary measures of quantum walks on the one-dimensional integer lattice are well studied. However, the stationary measure for the higher-dimensional case has not been clarified. In this paper, we give the stationary amplitude for quantum walks on the d-dimensional integer lattice with a finite support by solving the corresponding eigenvalue problem. As a corollary, we can obtain the stationary measures of the Grover walks. In fact, the amplitude for the stationary measure is an eigenfunction with eigenvalue 1.

DOI： 10.1007/s11128-017-1737-1

Scopus

Other Link： http://link.springer.com/content/pdf/10.1007/s11128-017-1737-1.pdf

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Yokohama National University  

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