Periodicity for the Hadamard Walk on Cycles

Abstract

The present paper treats the period TN of the Hadamard walk on a cycle CN with N vertices. Dukes (2014) considered the periodicity of more general quantum walks on CN and showed T2 = 2,T4 = 8,T8 = 24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., N = 2,4,8. Our method is based on a path counting and cyclotomic polynomials which is different from his approach based on the property of eigenvalues for unitary matrix that determines the evolution of the walk.