Periodicity for the Hadamard walk on cycles

Norio Konno; Yuki Shimizu; Masato Takei

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arXiv · 1504.06396

Periodicity for the Hadamard walk on cycles

Norio Konno·Yuki Shimizu·Masato Takei

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Abstract

The present paper treats the period T_N of the Hadamard walk on a cycle C_N with N vertices. Dukes (2014) considered the periodicity of more general quantum walks on C_N and showed T_2 =2, T_4=8, T_8=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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Norio Konno, Yuki Shimizu, Masato Takei. 2015-04-24. Periodicity for the Hadamard walk on cycles. https://arxiv.org/abs/1504.06396

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