著者

Takako ENDO Hikari KAWAI Norio KONNO

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.57-64, 2017 (Released:2017-03-31)

参考文献数

22

被引用文献数

2

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This study is motivated by the previous work [14]. We treat 3 types of the one-dimensional quantum walks (QWs), whose time evolutions are described by diagonal unitary matrices except at one defected point. In this paper, we call the QW defined by diagonal unitary matrices, ``the diagonal QW'', and we consider the stationary distributions of general 2-state diagonal QW with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW with one defect. One of the purposes of our study is to characterize the QWs by the stationary measure, which may lead to answer the basic and natural question, ``What are stationary measures for one-dimensional QWs?''. In order to analyze the stationary distribution, we focus on the corresponding eigenvalue problems and the definition of the stationary measure.

著者

Yusuke HIGUCHI Norio KONNO Iwao SATO Etsuo SEGAWA

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.75-86, 2017 (Released:2017-03-31)

参考文献数

23

被引用文献数

1

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In this paper we discuss the periodicity of the evolution matrix of Szegedy walk, which is a special type of quantum walk induced by the classical simple random walk, on a finite graph. We completely characterize the periods of Szegedy walks for complete graphs, compete bipartite graphs and strongly regular graphs. In addition, we discuss the periods of Szegedy walk induced by a non-reversible random walk on a cycle.

著者

Norio KONNO Hideo MITSUHASHI Iwao SATO

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.9-17, 2017 (Released:2017-03-31)

参考文献数

14

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We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function.

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.