著者

Norio KONNO Yuki SHIMIZU Masato TAKEI

出版者

東北大学大学院情報科学研究科ジャーナル編集委員会

雑誌

Interdisciplinary Information Sciences (ISSN:13409050)

巻号頁・発行日

vol.23, no.1, pp.1-8, 2017 (Released:2017-03-31)

参考文献数

17

被引用文献数

5

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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.

著者

Shimpei ENDO Takako ENDO Norio KONNO Etsuo SEGAWA Masato TAKEI

出版者

東北大学大学院情報科学研究科ジャーナル編集委員会

雑誌

Interdisciplinary Information Sciences (ISSN:13409050)

巻号頁・発行日

pp.2016.R.01, (Released:2016-03-25)

参考文献数

24

被引用文献数

6

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We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW) with one defect at the origin, which has two different quantum coins in positive and negative parts. We call the QW ``the two-phase QW with one defect'', which we treated concerning localization theorems. The two-phase QW with one defect has been expected to be a mathematical model of topological insulator which is an intense issue both theoretically and experimentally. In this paper, we derive the weak limit theorem describing the ballistic spreading, and as a result, we obtain the mathematical expression of the whole picture of the asymptotic behavior. Our approach is based mainly on the generating function of the weight of the passages. We emphasize that the time-averaged limit measure is symmetric for the origin , however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.