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

Quantum Information and Computation

Limit distributions for different forms of four-state quantum walks on a two-dimensional lattice

MachidaTakuya, M. ChandrashekarC., KonnoNorio, BuschThomas

Published Oct 1, 2015• 0 citations
Publisher

Abstract

Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin ...
DOI
10.5555/2871363.2871373

Related papers

Differences between Quantum Walks and Classical Random Walks in Limit Distributions

Yutaka Shikano

AIP conference proceedings•Jan 1, 2011•4 citations

Limit distributions for different forms of four-state quantum walks on a two-dimensional lattice

Takuya Machida, C. M. Chandrashekar, Norio Konno +1 more

Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin systems. For two-dimensional quantum walks using a four-state coin system, however, the only known limit distribution is for a walk using a parameterized Grover coin operation and analytical complexities have been a major obstacle for obtaining long-time limit distributions for other coins. In this work however, we present two new types of long-time limit distributions for walks using different forms of coin-flip operations in a four-state coin system. This opens the road towards understanding the dynamics and asymptotic behaviour for higher state coin system from a mathematical view point.

Quantum Information and Computation•Oct 1, 2015•5 citations

Differences between Quantum Walks and Classical Random Walks in Limit Distributions

Yutaka Shikano, Andrei Yu. Khrennikov, Gregg S. Jaeger +2 more

AIP conference proceedings•Jan 1, 2011•1 citation

Optimization for the propagation of a multiparticle quantum walk in a one-dimensional lattice

Da’er Feng, Shengshi Pang

The quantum walk is a quantum counterpart of the classical random walk that exhibits nonclassical behaviors and outperforms the classical random walk in various aspects. It has been known that a single particle can be propagated by a discrete-time quantum walk with a quadratic time scaling in the variance of position distribution, beating the linear time scaling in a classical random walk. In this paper, we consider the discrete-time quantum walk for multiple particles in a one-dimensional lattice, and investigate the optimization of the joint coin state to enhance the spatial propagation of the particles in the lattice. We study the asymptotic evolution of position distribution for multiple particles in the long-time limit, and analytically optimize the joint coin state to derive the maximum variance of the position distribution between the particles after the evolution of the quantum walk. An interesting result is that an optimized coin state always possesses specific exchange symmetry which can be characterized by a graph consisting of two disconnected complete subgraphs and the exchange symmetry can significantly influence the position correlations between the particles, showing the critical role of coin symmetry in the propagation of multiple particles by the quantum walk. We further study the entanglement of the optimized coin states to show the relation of the coin correlations to the particle position distribution.

arXiv:2212.05452•Physical Review A•May 1, 2024•1 citation

Limit Distribution for a Time-Inhomogeneous 2-State Quantum Walk

Takuya Machida

Journal of Computational and Theoretical Nanoscience•Jun 20, 2013•4 citations

Two-state quantum walk on two- and three-dimensional lattices

C. M. Chandrashekar

We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.

arXiv:1103.2704•Mar 14, 2011•4 citations
Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin systems. For two-dimensional quantum walks using a four-state coin system, however, the only known limit distribution is for a walk using a parameterized Grover coin operation and analytical complexities have been a major obstacle for obtaining long-time limit distributions for other coins. In this work however, we present two new types of long-time limit distributions for walks using different forms of coin-flip operations in a four-state coin system. This opens the road towards understanding the dynamics and asymptotic behaviour for higher state coin system from a mathematical view point.
The quantum walk is a quantum counterpart of the classical random walk that exhibits nonclassical behaviors and outperforms the classical random walk in various aspects. It has been known that a single particle can be propagated by a discrete-time quantum walk with a quadratic time scaling in the variance of position distribution, beating the linear time scaling in a classical random walk. In this paper, we consider the discrete-time quantum walk for multiple particles in a one-dimensional lattice, and investigate the optimization of the joint coin state to enhance the spatial propagation of the particles in the lattice. We study the asymptotic evolution of position distribution for multiple particles in the long-time limit, and analytically optimize the joint coin state to derive the maximum variance of the position distribution between the particles after the evolution of the quantum walk. An interesting result is that an optimized coin state always possesses specific exchange symmetry which can be characterized by a graph consisting of two disconnected complete subgraphs and the exchange symmetry can significantly influence the position correlations between the particles, showing the critical role of coin symmetry in the propagation of multiple particles by the quantum walk. We further study the entanglement of the optimized coin states to show the relation of the coin correlations to the particle position distribution.
We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.