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

TL;DR

This paper derives explicit formulas for quantum walks on an N-cycle with coin decoherence, showing they tend to a uniform distribution over time with mixing times proportional to N^2, confirming previous numerical results.

Contribution

It provides an explicit formula for the position distribution and analyzes the mixing behavior of quantum walks with coin decoherence on the N-cycle.

Findings

Quantum walks with coin decoherence tend to a uniform distribution.

Mixing time is of order O(N^2/ε).

Results confirm previous numerical simulations.

Abstract

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 $O(N^2/\epsilon)$.