For a discrete time quantum walk (QW) on the $N$-cycle, allowing for decoherence on the coin, we derive a number of 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 $\overline{M\left(ϵ\right)}$ for the time-averaged probability distribution is of order no greater than $O\left(N^2/ϵ\right)$.

• Received 29 November 2009

DOI:https://doi.org/10.1103/PhysRevE.81.031113