Quantum random walks on the N-cycle subject to decoherence on the coin degree of freedom

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For a discrete time quantum walk (QW) on the N-cycle, allowing for decoherence on the coin, we derive a number of new results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long-run, to a uniform distribution, regardless of the initial state of the system and irrespective of the parity of the number of nodes N. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time M(ε) for the time-everaged probability distribution is of order no greater than O(N²/ε).