Almost uniform sampling via quantum walks

Many classical randomized algorithms (e.g. approximation algorithms for #P-complete problems) utilize the following random walk algorithm for almost uniform sampling from a state space S of cardinality N: run a symmetric ergodic Markov chain P on S for long enough to obtain a random state from within total variation distance of the uniform distribution over S. The running time of this algorithm, the so-called mixing time of P, is O(δ−1(logN+log−1)), where δ is the spectral gap of P. We present a natural quantum version of this algorithm based on repeated measurements of the quantum walk Ut = e−iPt. We show that it samples almost uniformly from S with logarithmic dependence on −1 just as the classical walk P does; previously, no such quantum walk algorithm was known. We then outline a framework for analysing its running time and formulate two plausible conjectures which together would imply that it runs in time O(δ−1/2log N log−1) when P is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.