  The n-dimensional hypercube quantum random walk (QRW) is a particularily
appealing example of a quantum walk because it has a natural implementation on
a register on $n$ qubits. However, any real implementation will encounter
decoherence effects due to interactions with uncontrollable degrees of freedom.
We present a complete characterization of the mixing properties of the
hypercube QRW under a physically relevant Markovian decoherence model. In the
local decoherence model considered the non-unitary dynamics are modeled as a
sum of projections on individual qubits to an arbitrary direction on the Bloch
sphere. We prove that there is always classical (asymptotic) mixing in this
model and specify the conditions under which instantaneous mixing
\textit{always} exists. And we show that the latter mixing property, as well as
the classical mixing time, depend heavily on the exact environmental
interaction and its strength. Therefore, algorithmic applications of the QRW on
the hypercube, if they intend to employ mixing properties, need to consider
both the walk dynamics and the precise decoherence model.
