PERIODICITY FOR THE 3-STATE QUANTUM WALK ON CYCLES

Dukes (2014) and Konno, Shimizu, and Takei (2017) studied the periodicity for 2-state quantum walks whose coin operator is the Hadamard matrix on cycle graph C-N with N vertices. The present paper treats the periodicity for 3-state quantum walks on C-N. Our results follow from a new method based on the cyclotomic field. This method gives a necessary condition for the coin operator for quantum walks to be periodic. Moreover, we reveal the period T-N of typical two kinds of quantum walks, the Grover and Fourier walks. We prove that both walks do not have any finite period except for N = 3, in which case T-3 = 6 (Grover), = 12 (Fourier).