The discrete-time quaternionic quantum walk on a graph

Konno, Norio; Mitsuhashi, Hideo; Sato, Iwao

doi:10.1007/s11128-015-1205-8

Cited by 16 publications

(17 citation statements)

References 34 publications

(41 reference statements)

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“…Recently the author extended the QW to a new walk called quaternionic quantum walk (QQW) determined by a unitary matrix whose component is quaternion [13]. In general, the behavior of QQW is different from usual QW [15,24], it is interesting to compare our method with a time-series one based on the QQW model. Moreover an extension from the QQW time-series model to the Clifford algebra time-series one would be also attractive.…”

Section: Discussionmentioning

confidence: 99%

A new time-series model based on quantum walk

Konno

2018

Quantum Stud.: Math. Found.

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The quantum walk (QW) was introduced as a quantum counterpart of the classical random walk. A number of non-classical properties of the QW have been shown, e.g., ballistic spreading, anti-bellshaped limit density, localization. Since around 2000, extensive research has been conducted in both theoretical aspects as well as the practical application of QWs. However, the application of a QW to the time-series analysis is not known. On the other hand, it is well known that the ARMA or GARCH models have been widely used in economics and finance. These models are studied under some suitable stationarity conditions. In this paper, we propose a new time-series model based on the QW, which does not assume such a stationarity. Therefore, our method would be applicable to the non-stationary time series.Abbr. title: A new time-series model based on quantum walk

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Section: Discussionmentioning

confidence: 99%

A new time-series model based on quantum walk

Konno

2018

Quantum Stud.: Math. Found.

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“…Hence χ(Ξ n (l, m)) is divided into the Ξ (1) n (l, m) ∈ M(4, C) and Ξ (2) n (l, m) ∈ M(4, C). Each matrix is composed by the combination of corresponding P i and Q i .…”

Section: Casementioning

confidence: 99%

“…3 (1, 2) = P 2 Q 2 2 + Q 2 P 2 Q 2 + Q 2 2 P 2 . Moreover each component of Ξ (1) n (l, m) and Ξ (2) n (l, m) corresponds to the sum of possible paths with the coin operator given as the following U (1) and U (2) , respectively.…”

Section: Casementioning

confidence: 99%

Probability Distributions and Weak Limit Theorems of Quaternionic Quantum Walks in One Dimension

Saito

2018

IIS

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The discrete-time quantum walk (QW) is determined by a unitary matrix whose component is complex number. Konno (2015) extended the QW to a walk whose component is quaternion.We call this model quaternionic quantum walk (QQW). The probability distribution of a class of QQWs is the same as that of the QW. On the other hand, a numerical simulation suggests that the probability distribution of a QQW is different from the QW. In this paper, we clarify the difference between the QQW and the QW by weak limit theorems for a class of QQWs. Definitons and models 2.1 QuaternionThe quaternion was introduced as an extension of the complex number by Hamilton in 1843. Let R, C and H be the set of real, complex numbers and quaternions, respectively. Then x ∈ H is expressed as follows.x = x 0 + x 1 i + x 2 j + x 3 k ∈ H, * saito-kei-nb@ynu.jp Abbr. title: Probability distributions and weak limit theorems of quaternionic quantum walks in one dimension AMS 2000 subject classifications: 60F05, 81P68

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“…We shall review the way to obtain all the right eigenvalues of a quaternionic matrix. Expositions of these contents can be found in [5,21,24,30]. [40] gives an overview of the quaternionic matrix theory.…”

Section: Right Eigenvalues and Root Subspaces Of A Quaternionic Matrixmentioning

confidence: 99%

“…It is known that right eigenvalues of a square quaternionic matrix are given by the following manner. The detail of its proof can be found in, for example [21]. Theorem 2.4 ([21], Theorem 2.8).…”

Section: Right Eigenvalues and Root Subspaces Of A Quaternionic Matrixmentioning

confidence: 99%

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

Konno

Mitsuhashi

Sato

2017

Graphs and Combinatorics

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We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.Abbr. title: Quaternionic quantum walks of Szegedy type AMS 2010 subject classifications: 60F05, 05C50, 15A15, 11R52 quantum walk on a graph in this study. The discrete-time quantum walk in one dimension lattice was intensively studied by Ambainis et al. [4]. One of its most striking properties is the spreading property of the walker. The standard deviation of the position grows linearly in time, quadratically faster than classical random walk. On the other hand, a walker stays at the starting position, i.e., localization occurs. The reviews and books on quantum walks are Kempe [18], Konno [19], Venegas-Andraca [37], Cantero et al. [8], Manouchehri and Wang [27], Portugal [28] for examples.One of the most important discrete-time quantum walks on graphs is the Grover walk on a graph. The Grover walk on a graph was first formulated in [13] for complete graphs. After Grover's approach, several researchers, for example [38,3,39] proposed quantum walks on graphs. Emms et al. [9] and Godsil and Guo [12] applied the spectral property of the Grover walk to graph isomorphism problem. In the present article, we focus on the discrete-time quantum walk on a graph based on the quantum search algorithm proposed by Szegedy [39]. This walk arises naturally from a Markov chain, and is called the Szegedy walk.Zeta functions of graphs originate from the Ihara zeta function for a regular graph introduced by Ihara. In [17], Ihara defined a p-adic analogue of the Selberg zeta function associated to a certain kind of discrete subgroups of the 2 by 2 projective linear group over p-adic fields. After that, the Ihara zeta function has been extensively studied as a zeta function of a graph by many researchers [17,32,35,36,14,6,34,11,23]. Sato [31] defined a new zeta function (the second weighted zeta function) of a graph by modifying a determinant expression of the (first) weighted zeta function which had been proposed by Mizuno and Sato [25]. This new zeta function and its determinant formula play essential roles in the determination of eigenvalues of the quantum walk on a graph in Konno and Sato [22] and in another proof of the Smilansky's formula [33] for the characteristic polynomial of the bond scattering matrix of a graph in Mizuno and Sato [26]. Ren et al. [29] found an interesting relation between the Ihara zeta fun...

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The discrete-time quaternionic quantum walk on a graph

Cited by 16 publications

References 34 publications

A new time-series model based on quantum walk

A new time-series model based on quantum walk

Probability Distributions and Weak Limit Theorems of Quaternionic Quantum Walks in One Dimension

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

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