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

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

2009-08-11

arxiv.org/abs/0908.1524v3

Phys. Rev. E 81, 031113 (2010)

Physics

Quantum Physics

Scientific paper

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 $\bar{M(\epsilon)}$ for the time-everaged probability distribution is of order no greater than $O(N^2/\epsilon)$.

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