Strong dispersion property for the quantum walk on the hypercube

Abstract

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.