Spectral and asymptotic properties of Grover walks on crystal lattices

Higuchi, Yusuke; Konno, Norio; Sato, Iwao; Segawa, Etsuo

doi:10.1016/j.jfa.2014.09.003

Cited by 79 publications

(85 citation statements)

References 16 publications

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“…Summarizing the above statements with the preceding work [2], we obtain the complete characterization of the spectrum of quantum walks on graphs. …”

Section: The Final Resultsmentioning

confidence: 57%

“…The new insight of our study is the presence of the generalized eigenspace of the linear operatorT. We expect that such generalized eigenstructures reflect not only the geometric feature of underlying graphs such as their bipartiteness and underlying random walks such as their reversibility ( [2]), but also performance of quantum search algorithms on graphs [7,10]. We also expect that our result explicitly reveals such hidden structure and will lead to deeper study of spectra and asymptotic behavior of quantum walks from the viewpoint of functional analysis and geometry.…”

Section: Introductionmentioning

confidence: 93%

“…As can be seen in preceding works such as [2], SpecðUÞ consists of eigenvalues inherited from those of a self-adjoint operator T : K 1 ! K 1 via the spectral mapping property and specific ones to U.…”

Section: Invariant Subspaces Of Umentioning

confidence: 99%

“…Ã Remark 3.18. It is shown in [2] that the discriminant operator T has the eigenvalue 1 if and only if the underlying random walk on a finite graph G is reversible. Moreover, the operator T has the eigenvalue À1 if and only if the graph G is bipartite, namely, its set of vertices can be partitioned into two parts V 1 and V 2 such that each edge has one vertex in V 1 and one vertex in V 2 .…”

Section: <mentioning

confidence: 99%

“…In [2], the spectral mapping theorem of the twisted Szegedy walk U is derived with spectral decomposition of the associated self-adjoint operator T. According to [2], the eigenstructure of T induces those of the operator of the form…”

Section: Introductionmentioning

confidence: 99%

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A Note on the Spectral Mapping Theorem of Quantum Walk Models

Matsue

Ogurisu

Segawa

2017

IIS

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We discuss the description of eigenspace of a quantum walk model U with an associating linear operator T in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of U without the spectral decomposition of T. Arguments in this direction reveal the eigenspaces of U characterized by the generalized kernels of linear operators given by T.

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“…Summarizing the above statements with the preceding work [2], we obtain the complete characterization of the spectrum of quantum walks on graphs. …”

Section: The Final Resultsmentioning

confidence: 57%

Section: Introductionmentioning

confidence: 93%

Section: Invariant Subspaces Of Umentioning

confidence: 99%

Section: <mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Note on the Spectral Mapping Theorem of Quantum Walk Models

Matsue

Ogurisu

Segawa

2017

IIS

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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 staggered quantum walk model

Portugal

Santos

Fernandes

et al. 2015

Quantum Inf Process

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There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work we focus on the equivalence of two of them, known as Szegedy's and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy's model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy's framework. Our analysis also works when there are marked vertices. We show that Szegedy's spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs.

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Spectral and asymptotic properties of Grover walks on crystal lattices

Cited by 79 publications

References 16 publications

A Note on the Spectral Mapping Theorem of Quantum Walk Models

A Note on the Spectral Mapping Theorem of Quantum Walk Models

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

The staggered quantum walk model

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