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arXiv:quant-ph/0501169 · Physical Review A

Decoherence in quantum walks on the hypercube

Gorjan Alagic, Alexander C. Russell

Published Dec 5, 2005• 66 citations
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Abstract

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.

DOI
10.1103/physreva.72.062304

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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.
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.
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.
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.
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.
The effect of decoherence on the continuous-time quantum walk on the hypercube is revisited. Previously, an exact solution was found for a decoherence model that preserved the effective tensor-product form of the dynamics. Here a different model is presented, inspired by perfect quantum state transfer in qubit networks. A perturbative solution is found for the dynamics of this model which is not of a tensor-product form. In contrast to previous results, the hitting probability has a lower bound that is independent of the hypercube dimension.
We present a study of the effects of decoherence in the operation of a discrete quantum walk on a line, cycle, and hypercube. We find high sensitivity to decoherence, increasing with the number of steps in the walk, as the particle is becoming more delocalized with each step. However, the effect of a small amount of decoherence is to enhance the properties of the quantum walk that are desirable for the development of quantum algorithms. Specifically, we observe a highly uniform distribution on the line, a very fast mixing time on the cycle, and more reliable hitting times across the hypercube.