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.
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