Quantum Walks on the Hypercube

Abstract

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 instantaneous mixing time is (π/4)n steps, faster than the Θ(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is exactly uniform at this time. On the other hand, we show that the average mixing time as defined by Aharonov et al. [AAKV01] is Ω(n 3/2) in the discrete-time case, slower than the classical walk, and nonexistent in the continuous-time case. This suggests that the instantaneous mixing time is a more relevant notion than the average mixing time for quantum walks on large, well-connected graphs. 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 average mixing time when applied to the hypercube.