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

Complete characterization of mixing time for the continuous quantum walk on the hypercube with markovian decoherence model

Milosh Drezgich, Andrew P. Hines, Mohan Sarovar, Shankar Sastry

Published Sep 1, 2009• 6 citations
Publisher

Abstract

The n-dimensional hypercube quantum random walk (QRW) is a particularilyappealing example of a quantum walk because it has a natural implementationon a register on n qubits. However, any real implementation will encounterdecoherence effects due to interactions with uncontrollable degrees of freedom.We present a complete characterization of the mixing properties of the hypercubeQRW under a physically relevant Markovian decoherence model. In the localdecoherence model considered the non-unitary dynamics are modeled as a sum ofprojections on individual qubits to an arbitrary direction on the Bloch sphere. Weprove that there is always classical (asymptotic) mixing in this model and specifythe conditions under which instantaneous mixing always exists. And we show thatthe latter mixing property, as well as the classical mixing time, depend heavilyon the exact environmental interaction and its strength. Therefore, algorithmicapplications of the QRW on the hypercube, if they intend to employ mixingproperties, need to consider both the walk dynamics and the precise decoherencemodel.

DOI
10.5555/2011804.2011812

Related papers

Complete characterization of mixing time for the continuous quantum walk on the hypercube with Markovian decoherence model

Milosh Drezgich, Andrew P. Hines, Mohan Sarovar +1 more

The n-dimensional hypercube quantum random walk (QRW) is a particularily appealing example of a quantum walk because it has a natural implementation on a register on n qubits. However, any real implementation will encounter decoherence effects due to interactions with uncontrollable degrees of freedom. We present a complete characterization of the mixing properties of the hypercube QRW under a physically relevant Markovian decoherence model. In the local decoherence model considered the non-unitary dynamics are modeled as a sum of projections on individual qubits to an arbitrary direction on the Bloch sphere. We prove that there is always classical (asymptotic) mixing in this model and specify the conditions under which instantaneous mixing always exists. And we show that the latter mixing property, as well as the classical mixing time, depend heavily on the exact environmental interaction and its strength. Therefore, algorithmic applications of the QRW on the hypercube, if they intend to employ mixing properties, need to consider both the walk dynamics and the precise decoherence model.

arXiv:0811.4472•Quantum Information and Computation•Sep 1, 2009•2 citations

Quantum Walks on the Hypercube

Cristopher Moore, Alexander C. Russell

Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the n-dimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the quantum walk mixes in (π/4)n steps, faster than the O(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is {\em exactly} uniform at this time. More importantly, these walks expose several subtleties in the definition of mixing time for quantum walks. Even though the continuous-time walk has an O(n) instantaneous mixing time at which it is precisely uniform, it never approaches the uniform distribution when the stopping time is chosen randomly as in [AharonovAKV2001]. Our analysis treats interference between terms of different phase more carefully than is necessary for the walk on the cycle; previous general bounds predict an exponential, rather than linear, mixing time for the hypercube.

arXiv:quant-ph/0104137•Apr 29, 2001•9 citations

Decoherence in quantum walks on the hypercube

Gorjan Alagic, Alexander C. Russell

We study a natural notion of decoherence on quantum random walks over the hypercube. We prove that this model possesses a decoherence threshold beneath which the essential properties of the hypercubic quantum walk, such as linear mixing times, are preserved. Beyond the threshold, we prove that the walks behave like their classical counterparts.

arXiv:quant-ph/0501169•Physical Review A•Dec 5, 2005•66 citations

On mixing in continuous-time quantum walks on some circulant graphs

A. Ahmadi, Ryan Belk, Christino Tamon +1 more

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

Quantum Information and Computation•Nov 1, 2003•54 citations

Mixing times in quantum walks on the hypercube

Franklin de Lima Marquezino, Renato Portugal, Gonzalo Abal +1 more

The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time $O(n\text{ }\text{log}\text{ }n)$. We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution $\ensuremath{\pi}(x)$ in time $O(n)$, and the stationary distribution is determined by the initial state of the walk. An explicit expression for $\ensuremath{\pi}(x)$ is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p\ensuremath{\approx}0.1$. A similar effect was previously reported for a quantum walk on an $N$-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps in obtaining---and preserving---a uniform distribution over the ${2}^{n}$ sites of the hypercube in the shortest possible time.

arXiv:0712.0625•Physical Review A•Apr 16, 2008•45 citations

Bounds for mixing time of quantum walks on finite graphs

Vladislav Kargin

Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov et al (2002 Proc. 33rd STOC ( 2001 ) (New York: ACM) pp 50–9 arXiv:quant-ph/0012090v2) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speedup in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e. walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.

arXiv:1004.0188•Journal of Physics A Mathematical and Theoretical•Jul 15, 2010•8 citations
The n-dimensional hypercube quantum random walk (QRW) is a particularilyappealing example of a quantum walk because it has a natural implementationon a register on n qubits. However, any real implementation will encounterdecoherence effects due to interactions with uncontrollable degrees of freedom.We present a complete characterization of the mixing properties of the hypercubeQRW under a physically relevant Markovian decoherence model. In the localdecoherence model considered the non-unitary dynamics are modeled as a sum ofprojections on individual qubits to an arbitrary direction on the Bloch sphere. Weprove that there is always classical (asymptotic) mixing in this model and specifythe conditions under which instantaneous mixing always exists. And we show thatthe latter mixing property, as well as the classical mixing time, depend heavilyon the exact environmental interaction and its strength. Therefore, algorithmicapplications of the QRW on the hypercube, if they intend to employ mixingproperties, need to consider both the walk dynamics and the precise decoherencemodel.
The n-dimensional hypercube quantum random walk (QRW) is a particularily appealing example of a quantum walk because it has a natural implementation on a register on n qubits. However, any real implementation will encounter decoherence effects due to interactions with uncontrollable degrees of freedom. We present a complete characterization of the mixing properties of the hypercube QRW under a physically relevant Markovian decoherence model. In the local decoherence model considered the non-unitary dynamics are modeled as a sum of projections on individual qubits to an arbitrary direction on the Bloch sphere. We prove that there is always classical (asymptotic) mixing in this model and specify the conditions under which instantaneous mixing always exists. And we show that the latter mixing property, as well as the classical mixing time, depend heavily on the exact environmental interaction and its strength. Therefore, algorithmic applications of the QRW on the hypercube, if they intend to employ mixing properties, need to consider both the walk dynamics and the precise decoherence model.
Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the n-dimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the quantum walk mixes in (π/4)n steps, faster than the O(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is {\em exactly} uniform at this time. More importantly, these walks expose several subtleties in the definition of mixing time for quantum walks. Even though the continuous-time walk has an O(n) instantaneous mixing time at which it is precisely uniform, it never approaches the uniform distribution when the stopping time is chosen randomly as in [AharonovAKV2001]. Our analysis treats interference between terms of different phase more carefully than is necessary for the walk on the cycle; previous general bounds predict an exponential, rather than linear, mixing time for the hypercube.
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.
The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n)O(n\text{ }\text{log}\text{ }n)O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution $\ensuremath{\pi}(x)$ in time O(n)O(n)O(n), and the stationary distribution is determined by the initial state of the walk. An explicit expression for $\ensuremath{\pi}(x)$ is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p\ensuremath{\approx}0.1$. A similar effect was previously reported for a quantum walk on an NNN-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps in obtaining---and preserving---a uniform distribution over the 2n{2}^{n}2n sites of the hypercube in the shortest possible time.
Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov et al (2002 Proc. 33rd STOC ( 2001 ) (New York: ACM) pp 50–9 arXiv:quant-ph/0012090v2) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speedup in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e. walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.