Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2408.10574 · International Journal of Quantum Information

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

Yusuke Ide, Norio Konno, Akihiro Narimatsu

Published Dec 11, 2024• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0219749924500527

Related papers

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:2301.00283•Dec 31, 2022•0 citations

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:2310.20220•Linear and Multilinear Algebra•Mar 28, 2026•0 citations

Spacetime structures of continuous-time quantum walks

Oliver Mülken, A. Blumen

The propagation by continuous-time quantum walks (CTQWs) on one-dimensional lattices shows structures in the transition probabilities between different sites reminiscent of quantum carpets. For a system with periodic boundary conditions, we calculate the transition probabilities for a CTQW by diagonalizing the transfer matrix and by a Bloch function ansatz. Remarkably, the results obtained for the Bloch function ansatz can be related to results from (discrete) generalized coined quantum walks. Furthermore, we show that here the first revival time turns out to be larger than for quantum carpets.

arXiv:quant-ph/0502004•Physical Review E•Mar 22, 2005•68 citations

Birth and death processes and quantum spin chains

F. Alberto Grünbaum, Luc Vinet, Alexei S. Zhedanov

This paper underscores the intimate connection between the quantum walks generated by certain semi-infinite spin chain Hamiltonians and classical birth and death processes. It is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the Karlin-McGregor representation formula of the transition probability functions for classes of birth and death processes. As an application, we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1.

arXiv:1205.4689•Journal of Mathematical Physics•Jun 1, 2013•13 citations

Probability measure for the Szegedy walk on path graph

Ide Yusuke, Norio Konno, Tomoyuki, Terada

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.

•Jan 1, 2022•0 citations

Spectral analysis of hierarchical continuous-time quantum walks

Jirô Akahori, Yusuke Ide, Tomoki Kato +3 more

In this paper, we introduce hierarchical random walks at first. In this model, we use two types of random walkers, global and local walkers. The global walker chooses a local walker at every step, then the chosen local walker moves a single step. After that we construct the corresponding continuous-time quantum walks and discuss their spectral structures. Then we define multi-dimensional continuous-time quantum walks by taking a marginal distribution with respect to the global walker.

arXiv:2510.12043•International Journal of Quantum Information•Mar 30, 2026•0 citations
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.
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.
The propagation by continuous-time quantum walks (CTQWs) on one-dimensional lattices shows structures in the transition probabilities between different sites reminiscent of quantum carpets. For a system with periodic boundary conditions, we calculate the transition probabilities for a CTQW by diagonalizing the transfer matrix and by a Bloch function ansatz. Remarkably, the results obtained for the Bloch function ansatz can be related to results from (discrete) generalized coined quantum walks. Furthermore, we show that here the first revival time turns out to be larger than for quantum carpets.
This paper underscores the intimate connection between the quantum walks generated by certain semi-infinite spin chain Hamiltonians and classical birth and death processes. It is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the Karlin-McGregor representation formula of the transition probability functions for classes of birth and death processes. As an application, we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1.