Periodicity of the Discrete-time Quantum Walk on a Finite Graph

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

doi:10.4036/iis.2017.a.10

Cited by 24 publications

(32 citation statements)

References 22 publications

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“…Trivial (or [26]) Complete graphs, complete bipartite graphs, SRGs [13] Generalized Bethe trees [24] Hamming graphs, Johnson graphs [35] Cycles (3-state) [17] Complete graphs with self loops [15] Table 1: Previous works on periodicity of Grover walks on undirected graphs sections for terminologies and definitions. The first main result states that if a bipartite regular graph with four or five distinct adjacency eigenvalues is periodic, then the second largest eigenvalue can only take three different values.…”

Section: Paths and Cyclesmentioning

confidence: 99%

“…In this section, we derive a general fact on bipartite regular periodic graphs with four or five distinct adjacency eigenvalues. Note that the bipartite regular graph with two distinct adjacency eigenvalues is the complete graph K 2 , which is known to be periodic [13]. Bipartite regular graphs with three distinct adjacency eigenvalues have the A-spectra of the form…”

Section: Bipartite Regular Graphs With At Most Five Distinct Adjacenc...mentioning

confidence: 99%

“…Disconnected strongly regular graphs are disjoint unions of the complete graphs with the same size (See Lemma 10.1.1 in [9]). This implies that the graphs are the complete bipartite graphs K k,k , which are periodic shown in [13]. In the end, we are concerned with bipartite regular graphs with four or five distinct adjacency eigenvalues.…”

Section: Bipartite Regular Graphs With At Most Five Distinct Adjacenc...mentioning

confidence: 99%

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Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

Kubota¹

2021

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Section: Paths and Cyclesmentioning

confidence: 99%

Section: Bipartite Regular Graphs With At Most Five Distinct Adjacenc...mentioning

confidence: 99%

Section: Bipartite Regular Graphs With At Most Five Distinct Adjacenc...mentioning

confidence: 99%

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Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

Kubota¹

2021

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“…In the proof of Theorem 1.2 of [25], Yoshie claimed that if the discriminant has a rational eigenvalue, then it must be one of the five listed above. We also find approaches that rational eigenvalues restrict candidates of periodic graphs in [9]. Considerations of quantum walks with algebraic integers were also given in [11,13,22].…”

Section: Lemma 23 ([4]mentioning

confidence: 90%

“…We restrict times to consider, but this does not diminish worth of the theorem. This is because both Γ 2,m and Γ 3,m are periodic graphs [9], i.e., U(Γ 2,m ) 4 = I and U(Γ 3,m ) 12 = I, where U(Γ) is the time evolution matrix of a graph Γ. Thus, it suffices to study only times less than their periods.…”

Section: Introductionmentioning

confidence: 99%

Perfect state transfer in Grover walks between states associated to vertices of a graph

Kubota¹,

Segawa²

2021

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We study perfect state transfer in Grover walks, which are typical discretetime quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, together with the time.

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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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Periodicity of the Discrete-time Quantum Walk on a Finite Graph

Cited by 24 publications

References 22 publications

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

Perfect state transfer in Grover walks between states associated to vertices of a graph

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

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