Spectral analysis for a multi-dimensional split-step quantum walk with a defect

Fuda, Toru; Narimatsu, Akihiro; Saito, Kei; Suzuki, Akito

doi:10.48550/arxiv.2008.08846

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…To clarify various properties of quantum walks, analysis of eigenvalues of the time evolution is significant. The spectral mapping theorem of quantum walks [6,7,8,11,16,19] is a useful tool for analyzing eigenvalues and is a fundamental theorem for connecting quantum and classical systems. For the finite dimension case, the spectral mapping theorem associates the eigenvalues of two matrices, the time evolution U and a self-adjoint matrix T , by lifting the eigenvalues of T on to the unit circle on the complex plane.…”

Section: Introductionmentioning

confidence: 99%

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A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

Kubota¹,

Saito²,

Yoshie³

2021

Preprint

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The conventional spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. This theorem gives most of the eigenvalues of the time evolution U by lifting the eigenvalues of an induced self-adjoint matrix T onto the unit circle on the complex plane. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs. Moreover, graphs we can consider for a quantum walk with such a shift operator is characterized by a triangulation. We call these graphs triangulable graphs in this paper. One of the differences between our spectral mapping theorem and the conventional one is that lifting the eigenvalues of T − 1/2 onto the unit circle gives most of the eigenvalues of U .

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Section: Introductionmentioning

confidence: 99%

“…Thus, one of the excellent features of the spectral mapping theorem is that it allows us to apply well-known results for classical models to quantum walks. For examples of the applications of the spectral mapping theorem, there are some results of the periodicity [12,14,15,21,22] and the limit averaged measure [8,13,18].…”

Section: Introductionmentioning

confidence: 99%

A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

Kubota¹,

Saito²,

Yoshie³

2021

Preprint

Self Cite

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show abstract

“…This theorem enables us to analyze a corresponding self-adjoint operator instead of a unitary operator U . In this aspects, there are several results on the spectral analysis by the spectral mapping theorem [10,11,12].…”

Section: Introductionmentioning

confidence: 99%

The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition

Matsuzawa¹,

Suzuki²,

Tanaka³

et al. 2021

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Split-step quantum walks possess chiral symmetry. From a supercharge, that is the imaginary part of the time evolution operator, the Witten index can be naturally defined and it gives the lower bound of the dimension of eigenspaces of 1 or −1. The first three authors studied the Witten index as a Fredholm index under the condition that the supercharge satisfies the Fredholm condition. In this paper, we establish the Witten index formula under the non-Fredholm condition. To derive it, we employ the spectral shift function induced by the fourth-order difference operator with a rank one perturbation on the half-line. Under two-phase limit conditions and trace conditions, the Witten index only depends on parameters of two-side limits. Especially, half-integer indices appear.

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Spectral analysis for a multi-dimensional split-step quantum walk with a defect

Cited by 2 publications

References 14 publications

A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition

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