Probability measure for the Szegedy walk on path graph

Ide Yusuke, Norio Konno, Tomoyuki, Terada

Published Jan 1, 2022• 0 citations

Abstract

In this paper, we consider a spectral analysis of the Szegedy walk on path graph. Here, we have an explicit formula for the probability measure of the walk by using eigenvalues and eigenvectors of the Jacobi matrix related to the birth and death chain.

DOI: 10.18880/0002000033

Related papers

A spectral analysis of the correlated random walk

Yusuke Ide, Akihiro Narimatsu

In this paper, we consider a spectral analysis of the Correlated Random Walk (CRW) on the path. We apply an analytical method for the Quantum Walk to CRW. For the isospectral coin cases, we obtain all of the eigenvalues and the corresponding eigenvectors of the time evolution operator of CRW. We show that most of them are described by the eigenvalues and the eigenvectors of the Jacobi matrix related to the random walks on the path (the birth and death chains). The remaining eigenvector corresponding to an eigenvalue 1, is characterized by the stationary measure of the same birth and death chain. As a consequence, we derive the limiting distribution.

arXiv:2310.20220Linear and Multilinear AlgebraMar 28, 20260 citations

Scaling limit of the time averaged distribution for continuous time quantum walk and Szegedy's walk on the path

Ide, Yusuke

In this paper, we consider Szegedy's walk, a type of discrete time quantum walk, and corresponding continuous time quantum walk related to the birth and death chain. We show that the scaling limit of time averaged distribution for the continuous time quantum walk induces that of Szegedy's walk if there exists the spectral gap on so-called the corresponding Jacobi matrix .

arXiv:2301.00283Dec 31, 20220 citations

Spectral analysis of quantum Markov chains on the line and their statistics.

Manuel D. de la Iglesia, Carlos F. Lardizabal, Newton Loebens

Inspired by the classical spectral analysis of birth-death chains using orthogonal polynomials, we study an analogous set of constructions in the context of open quantum dynamics and related walks. In such setting, block tridiagonal matrices and matrix-valued orthogonal polynomials are the natural objects to be considered. We recall the problems of the existence of a matrix of measures or weight matrix together with concrete calculations of basic statistics of the walk, such as site recurrence and first passage time probabilities, with these notions being defined in terms of a quantum trajectories formalism. The discussion concentrates on the models of quantum Markov chains, due to S. Gudder, and on the particular class of open quantum walks, due to S. Attal et al. The folding trick for birth-death chains on the integers is revisited in this setting together with applications of the matrix-valued Stieltjes transform associated with the measures, thus extending recent results on the subject. Finally, we consider the case of non-symmetric weight matrices and explore some examples.

Jul 21, 20210 citations

Multi-dimensional continuous time quantum walks related to the birth and death chains

Yusuke Ide, Norio Konno, Akihiro Narimatsu

In this paper, we consider multi-dimensional birth and death chains and continuous time quantum walks (CTQWs) related to them. For continuous time quantum walk (CTQW) related to our forms of multi-dimensional birth and death chains, we obtain the time- scaled independence between multiple dimensions about the transition probability of CTQW. By using this feature, we analyze CTQW on the path graph, which is related to 1-dimensional Ehrenfest model. We also have a random variable which is related to our models and converges to the standard Gaussian distribution.

arXiv:2408.10574International Journal of Quantum InformationDec 11, 20241 citation

A weighted graph zeta function involved in the Szegedy walk

Ayaka Ishikawa, Norio Konno

We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.

arXiv:2110.05068Quantum Information and ComputationDec 28, 20211 citation

Periodicity of the Discrete-time Quantum Walk on a Finite Graph

Yusuke Higuchi, Norio Konno, Iwao Sato +1 more

In this paper we discuss the periodicity of the evolution matrix of Szegedy walk, which is a special type of quantum walk induced by the classical simple random walk, on a finite graph. We completely characterize the periods of Szegedy walks for complete graphs, compete bipartite graphs and strongly regular graphs. In addition, we discuss the periods of Szegedy walk induced by a non-reversible random walk on a cycle.

Interdisciplinary Information SciencesJan 1, 201726 citations