  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. A rigorous proof of this result involves
an intricate argument about analytic properties of Bessel functions.
