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

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Abstract

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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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama, 240-8501, Japan

    Sho Kubota

  2. Department of Information Systems Creation, Faculty of Engineering, Kanagawa University, Kanagawa, Yokohama, 221-8686, Japan

    Kei Saito

  3. General Education, Ishikawa College, National Institute of Technology, Ishikawa, 929-0392, Japan

    Yusuke Yoshie

Authors
  1. Sho Kubota
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  2. Kei Saito
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  3. Yusuke Yoshie
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Correspondence to Kei Saito.

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S.K. is supported by JSPS KAKENHI (Grant No. 20J01175)

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Kubota, S., Saito, K. & Yoshie, Y. A new type of spectral mapping theorem for quantum walks with a moving shift on graphs. Quantum Inf Process 21, 159 (2022). https://doi.org/10.1007/s11128-022-03493-x

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