  The mixing time of a discrete-time quantum walk on the hypercube is
considered. The mean probability distribution of a Markov chain on a hypercube
is known to mix to a uniform distribution in time O(n log n). We show that the
mean probability distribution of a discrete-time quantum walk on a hypercube
mixes to a (generally non-uniform) distribution pi(x) in time O(n) and the
stationary distribution is determined by the initial state of the walk. An
explicit expression for pi(x) is derived for the particular case of a symmetric
walk. These results are consistent with those obtained previously for a
continuous-time quantum walk. The effect of decoherence due to randomly
breaking links between connected sites in the hypercube is also considered. We
find that the probability distribution mixes to the uniform distribution as
expected. However, the mixing time has a minimum at a critical decoherence rate
$p \approx 0.1$. A similar effect was previously reported for the QW on the
N-cycle with decoherence from repeated measurements of position. A controlled
amount of decoherence helps to obtain--and preserve--a uniform distribution
over the $2^n$ sites of the hypercube in the shortest possible time.
