Interdisciplinary Information Sciences
Online ISSN : 1347-6157
Print ISSN : 1340-9050
ISSN-L : 1340-9050
Quantum Simulation and Quantum Walks
Periodicity for the Hadamard Walk on Cycles
Norio KONNOYuki SHIMIZUMasato TAKEI
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2017 Volume 23 Issue 1 Pages 1-8

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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.

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© 2017 by the Graduate School of Information Sciences (GSIS), Tohoku University
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