  In this paper, we present a quantum Bernoulli noises approach to quantum
walks on hypercubes. We first obtain an alternative description of a general
hypercube and then, based on the alternative description, we find that the
operators $\partial_k^* + \partial_k$ behave actually as the shift operators,
where $\partial_k$ and $\partial_k^*$ are the annihilation and creation
operators acting on Bernoulli functionals, respectively. With the above
operators as the shift operators on the position space, we introduce a
discrete-time quantum walk model on a general hypercube and obtain an explicit
formula for calculating its probability distribution at any time. We also
establish two limit theorems showing that the averaged probability distribution
of the walk even converges to the uniform probability distribution. Finally, we
show that the walk produces the uniform measure as its stationary measure on
the hypercube provided its initial state satisfies some mild conditions. Some
other results are also proven.
