Maneuvering periods of 2D quantum walks with the coin operator

Recurrence in classical random walks is well known and the idea has been investigated in quantum walks in many aspects. The recurrence in quantum walks is termed when the walker returns to the origin with a nonzero probability and if the final coin state is also the same as the initial coin state then the quantum walk is said to have a full revival. So far, full revival 2D quantum walks with a period larger than two steps have not been found and it has been argued that four-state quantum walks cannot have periods longer than two steps. In this paper, with the aid of simple 2D non-local coins we show that some four-state quantum walks can have full revivals with any even period and the periodicity can be controlled with a slight change of a single parameter within the coin operator.