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:1506.01940

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

Forrest Ingram-Johnson, Chaobin Liu, Nelson Petulante

Published Jun 5, 2015• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1506.01940

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

Quantum Walks

Takuya Machida

Quantum walks are quantum counterparts of random walks. While probability distributions of random walks diffusively spread out as the walkers are updating, quantum walks have ballistic behavior. Some of the ballistic behaviors have been revealed in long-time limit theorems and their probability distributions are all far away from the Gaussian distributions, which are known as limit distributions of random walks. In this chapter, we are going to be seeing limit distributions for a standard quantum walk on the line and two kinds of time-dependent quantum walk on the line.

InTech eBooks•Aug 24, 2016•1 citation

Coin-decohered quantum random walks on the NNN-cycle

Chaobin Liu, Nelson Petulante

For a discrete time quantum walk (QW) on the $N$-cycle, allowing for decoherence on the coin, we derive a number of new results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long-run, to a uniform distribution, regardless of the initial state of the system and irrespective of the parity of the number of nodes $N$. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time $\bar{M(\epsilon)}$ for the time-everaged probability distribution is of order no greater than $O(N^2/\epsilon)$.

•Aug 11, 2009•0 citations

Infinite reduction in absorbing time in quantum walks over classical ones

Shuva Mondal, Amrita Mandal, Ujjwal Sen

We study the absorption time and spreading rate of the discrete-time quantum walk propagating on a line in the presence or absence of an absorber. We analytically establish that in the presence of an absorber, the average absorption time of the quantum walker is finite, contrary to the behavior of a classical random walker, indicating an infinite resource reduction on moving over to a quantum version of a walker. Furthermore, numerical simulations indicate a reversal of this behavior due to the insertion of disorder in the walker's step lengths. Additionally, we demonstrate that in the presence of an absorber, there is a speed-up in the spreading rate, and that a disordered quantum walk that is sub-ballistic regains the ballistic spreading of a clean quantum walk.

quant-pharXiv:2602.14880•Feb 17, 2026•0 citations

Decoherence versus entanglement in coined quantum walks

Olivier Maloyer, Viv M. Kendon

Quantum versions of random walks on the line and cycle show a quadratic improvement in their spreading rate and mixing times respectively. The addition of decoherence to the quantum walk produces a more uniform distribution on the line, and even faster mixing on the cycle by removing the need for time-averaging to obtain a uniform distribution. We calculate numerically the entanglement between the coin and the position of the quantum walker and show that the optimal decoherence rates are such that all the entanglement is just removed by the time the final measurement is made.

arXiv:quant-ph/0612229•New Journal of Physics•Apr 11, 2007•67 citations

Quantum random walks with decoherent coins

Todd A. Brun, Hilary A. Carteret, Andris Ambainis

The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.

arXiv:quant-ph/0210180•Physical Review A•Mar 14, 2003•155 citations
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.
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.
Quantum walks are quantum counterparts of random walks. While probability distributions of random walks diffusively spread out as the walkers are updating, quantum walks have ballistic behavior. Some of the ballistic behaviors have been revealed in long-time limit theorems and their probability distributions are all far away from the Gaussian distributions, which are known as limit distributions of random walks. In this chapter, we are going to be seeing limit distributions for a standard quantum walk on the line and two kinds of time-dependent quantum walk on the line.
For a discrete time quantum walk (QW) on the NNN-cycle, allowing for decoherence on the coin, we derive a number of new results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long-run, to a uniform distribution, regardless of the initial state of the system and irrespective of the parity of the number of nodes NNN. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time M(ϵ)ˉ\bar{M(\epsilon)}M(ϵ)ˉ​ for the time-everaged probability distribution is of order no greater than O(N2/ϵ)O(N^2/\epsilon)O(N2/ϵ).
We study the absorption time and spreading rate of the discrete-time quantum walk propagating on a line in the presence or absence of an absorber. We analytically establish that in the presence of an absorber, the average absorption time of the quantum walker is finite, contrary to the behavior of a classical random walker, indicating an infinite resource reduction on moving over to a quantum version of a walker. Furthermore, numerical simulations indicate a reversal of this behavior due to the insertion of disorder in the walker's step lengths. Additionally, we demonstrate that in the presence of an absorber, there is a speed-up in the spreading rate, and that a disordered quantum walk that is sub-ballistic regains the ballistic spreading of a clean quantum walk.
Quantum versions of random walks on the line and cycle show a quadratic improvement in their spreading rate and mixing times respectively. The addition of decoherence to the quantum walk produces a more uniform distribution on the line, and even faster mixing on the cycle by removing the need for time-averaging to obtain a uniform distribution. We calculate numerically the entanglement between the coin and the position of the quantum walker and show that the optimal decoherence rates are such that all the entanglement is just removed by the time the final measurement is made.
The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.