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:quant-ph/0310171 · Physica A Statistical Mechanics and its Applications

Quantum random walk on the line as a Markovian process

A. Romanelli, A. C. Sicardi Schifino, R. Siri, Gonzalo Abal, A. Auyuanet, R. Donangelo

Published Mar 18, 2004• 73 citations
arXivHTMLPDF
DOI
10.1016/j.physa.2004.02.061

Related papers

Non-Markovian decoherent quantum walks

Peng Xue, Yongsheng Zhang

Quantum walks act in obviously different ways from their classical counterparts, but decoherence will lessen and close this gap between them. To understand this process, it is necessary to investigate the evolution of quantum walks under different decoherence situations. In this article, we study a non-Markovian decoherent quantum walk on a line. In a short time regime, the behavior of the walk deviates from both ideal quantum walks and classical random walks. The position variance as a measure of the quantum walk collapses and revives for a short time, and tends to have a linear relation with time. That is, the walker's behavior shows a diffusive spread over a long time limit, which is caused by non-Markovian dephasing affecting the quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of the quantum correlations, and observe both collapse and revival in the short time regime, and the tendency to be zero in the long time limit. Therefore, quantum walks with non-Markovian decoherence tend to have diffusive spreading behavior over long time limits, while in the short time regime they oscillate between ballistic and diffusive spreading behavior, and the quantum correlation collapses and revives due to the memory effect.

arXiv:1203.5630•Chinese Physics B•Jul 1, 2013•13 citations

The quantum random walk within the Schrödinger-Langevin approach

Manuel O. Cáceres, Ana Karina Chattah

Journal of Molecular Liquids•Apr 1, 1997•7 citations

Mapping a Hadamard Quantum Walk to a Unique Case of a Birth and Death Process

Arie Bar-Haim

A new model maps a quantum random walk described by a Hadamard operator to a particular case of a birth and death process. The model is represented by a 2D Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries (this is possible by increasing the dimensionality of the system). The probability distribution of the walker population is preserved using the Markovian property. By applying a proper transformation to the population distribution of the random walk, the probability distributions of the quantum states |0>, 1> are revealed. Thus, the new model has two unique properties: it reveals the probability distribution of the quantum states as a unitary system and preserves the population distribution of the random walker as a Markovian system.

arXiv:2104.04286•Apr 9, 2021•0 citations

Quantum Walks: A Markovian Perspective

Diego de Falco, Dario Tamascelli

arXiv:0801.4515•Lecture notes in computer science•Jan 1, 2008•0 citations

On the behavior of quantum walks confined to a cycle coupled with a half line

Forrest Ingram-Johnson, Chaobin Liu, Nelson Petulante

When confined to a topological environment consisting of a cycle coupled with a half-line, quantum walks exhibit long-term statistical tendencies which differ dramatically from the tendencies of classical random walks in the same environment. In particular, as suggested by numerical simulations, the probability distribution of the walker's position resolves, in part, into a non-vanishing distribution on the cycle and, in part, into a ballistic distribution on the half-line. By contrast, for a classical random walk, the probability distribution of the walker's position tends always to vanish on the cycle and to migrate completely to the half-line as a purely diffusive process.

arXiv:1506.01940•Jun 5, 2015•0 citations

Quantum walks : a Markovian perspective

Diego de Falco, Dario Tamascelli, Centro Interdipartimentale +1 more

•Jan 1, 2008•2 citations
Quantum walks act in obviously different ways from their classical counterparts, but decoherence will lessen and close this gap between them. To understand this process, it is necessary to investigate the evolution of quantum walks under different decoherence situations. In this article, we study a non-Markovian decoherent quantum walk on a line. In a short time regime, the behavior of the walk deviates from both ideal quantum walks and classical random walks. The position variance as a measure of the quantum walk collapses and revives for a short time, and tends to have a linear relation with time. That is, the walker's behavior shows a diffusive spread over a long time limit, which is caused by non-Markovian dephasing affecting the quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of the quantum correlations, and observe both collapse and revival in the short time regime, and the tendency to be zero in the long time limit. Therefore, quantum walks with non-Markovian decoherence tend to have diffusive spreading behavior over long time limits, while in the short time regime they oscillate between ballistic and diffusive spreading behavior, and the quantum correlation collapses and revives due to the memory effect.
A new model maps a quantum random walk described by a Hadamard operator to a particular case of a birth and death process. The model is represented by a 2D Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries (this is possible by increasing the dimensionality of the system). The probability distribution of the walker population is preserved using the Markovian property. By applying a proper transformation to the population distribution of the random walk, the probability distributions of the quantum states |0>, 1> are revealed. Thus, the new model has two unique properties: it reveals the probability distribution of the quantum states as a unitary system and preserves the population distribution of the random walker as a Markovian system.
When confined to a topological environment consisting of a cycle coupled with a half-line, quantum walks exhibit long-term statistical tendencies which differ dramatically from the tendencies of classical random walks in the same environment. In particular, as suggested by numerical simulations, the probability distribution of the walker's position resolves, in part, into a non-vanishing distribution on the cycle and, in part, into a ballistic distribution on the half-line. By contrast, for a classical random walk, the probability distribution of the walker's position tends always to vanish on the cycle and to migrate completely to the half-line as a purely diffusive process.