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arXiv:0712.0625 · Physical Review A

Mixing times in quantum walks on the hypercube

Franklin de Lima Marquezino, Renato Portugal, Gonzalo Abal, R. Donangelo

Published Apr 16, 2008• 45 citations
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Abstract

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.

DOI
10.1103/physreva.77.042312

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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 \ensuremathπ(x)\ensuremath{\pi}(x)\ensuremathπ(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 \ensuremathπ(x)\ensuremath{\pi}(x)\ensuremathπ(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≈0.1p\ensuremath{\approx}0.1p\ensuremath≈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.
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.
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For a discrete time quantum walk (QW) on the N-cycle, allowing for decoherence on the coin, we derive a number of 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 M(epsilon) for the time-averaged probability distribution is of order no greater than O(N2/).
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.
In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube, and then, based on the alternative description, we find that the operators ∂k*+∂k behave actually as the shift operators, where ∂k and ∂k* are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above-mentioned operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.
Abstract We show that the discrete time quantum walk on the Boolean hypercube of dimension n has a strong dispersion property: if the walk is started in one vertex, then the probability of the walker being at any particular vertex after O ( n ) steps is of an order O ( 1.4818 − n ) . This improves over the known mixing results for this quantum walk which show that the probability distribution after O ( n ) steps is close to uniform but do not show that the probability is small for every vertex. Our result shows that quantum walk on hypercube is interesting for algorithmic applications which require fast dispersion over the state space.