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The n-dimensional hypercube quantum random walk (QRW) is a particularilyappealing example of a quantum walk because it has a natural implementationon a register on n qubits. However, any real implementation will encounterdecoherence effects due to interactions with uncontrollable degrees of freedom.We present a complete characterization of the mixing properties of the hypercubeQRW under a physically relevant Markovian decoherence model. In the localdecoherence model considered the non-unitary dynamics are modeled as a sum ofprojections on individual qubits to an arbitrary direction on the Bloch sphere. Weprove that there is always classical (asymptotic) mixing in this model and specifythe conditions under which instantaneous mixing always exists. And we show thatthe latter mixing property, as well as the classical mixing time, depend heavilyon the exact environmental interaction and its strength. Therefore, algorithmicapplications of the QRW on the hypercube, if they intend to employ mixingproperties, need to consider both the walk dynamics and the precise decoherencemodel.

Complete characterization of mixing time for the continuous quantum walk on the hypercube with Markovian decoherence model