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Recently, a new model of quantum walk, utilizing recycled coins, wasintroduced; however little is yet known about its properties. In this paper, westudy its behavior on the cycle graph. In particular, we will consider its timeaveraged distribution and how it is affected by the walk's "memory parameter" -a real parameter, between zero and eight, which affects the walk's coin flipoperator. Despite an infinite number of different parameters, our analysisprovides evidence that only a few produce non-uniform behavior. Our analysisalso shows that the initial state, and cycle size modulo four all affect thebehavior of this walk. We also prove an interesting relationship between therecycled coin model and a different memory-based quantum walk recentlyproposed.

History Dependent Quantum Walk on the Cycle with an Unbalanced Coin