Quantum walks defined by digraphs and generalized Hermitian adjacency matrices

Kubota, Sumihisa; Segawa, Etsuo; Taniguchi, Tetsuji

doi:10.1007/s11128-021-03033-z

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Introduction3

Grover Walks2

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2021

2024

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Cited by 18 publications

(14 citation statements)

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“…We omit a proof. See Section 3 in [23] for more general claim and its proof. Relationship between U-spectra and T -spectra has been studied not only in the Grover walks but also in more general models [12,19,21].…”

Section: Grover Walksmentioning

confidence: 99%

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

Kubota¹

2021

Preprint

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Section: Grover Walksmentioning

confidence: 99%

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

Kubota¹

2021

Preprint

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“…The spectral mapping theorems reduce eigenvalue analysis of time evolution operators to eigenvalue analysis of other self-adjoint operators. They bring quantum walks into close connection with functional analysis [34] and spectral graph theory [20]. In the study of periodicity of discrete-time quantum walk, field theory and algebraic number theory have also been leveraged [18,35].…”

Section: Introductionmentioning

confidence: 99%

Periodicity of quantum walks defined by mixed paths and mixed cycles

Kubota,

Sekido,

Yata

2021

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In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of η-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of η-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define n + 1 types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss periodicity. We show that the mixed paths are periodic for any η.In addition, we provide a necessary and sufficient condition for a mixed cycle to be periodic and determine their periods.

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“…The notion of the Hermitian adjacency matrix (with α = i) was introduced by Liu and Li [10] and independently by Guo and Mohar [6]. Recently Mohar [11] introduced another type of Hermitian adjacency matrices, which coincides with ours (see also [9]) in the important case α = 1+i √ 3 2 , i.e., the primitive sixth root of unity. Since H is a Hermitian matrix, the eigenvalues of H are all real, and their algebraic and geometric multiplicities coincide.…”

Section: Introductionmentioning

confidence: 82%