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 requiredb y 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(n3/2) in the discrete-time case, slower than the classical walk, andnonexisten t 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-connectedgraphs. 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.