Periodicity for the Hadamard Walk on Cycles

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Author(s)

    • KONNO Norio
    • Department of Applied Mathematics, Faculty of Engineering, Yokohama National University
    • SHIMIZU Yuki
    • Department of Applied Mathematics, Faculty of Engineering, Yokohama National University
    • TAKEI Masato
    • Department of Applied Mathematics, Faculty of Engineering, Yokohama National University

Abstract

The present paper treats the period <i>T</i><sub><i>N</i></sub> of the Hadamard walk on a cycle <i>C</i><sub><i>N</i></sub> with <i>N</i> vertices. Dukes (2014) considered the periodicity of more general quantum walks on <i>C</i><sub><i>N</i></sub> and showed <i>T</i><sub>2</sub>=2, <i>T</i><sub>4</sub>=8, <i>T</i><sub>8</sub>=24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., <i>N</i> = 2,4,8. Our method is based on a path counting and cyclotomic polynomials which is different from his approach based on the property of eigenvalues for unitary matrix that determines the evolution of the walk.

Journal

  • Interdisciplinary Information Sciences

    Interdisciplinary Information Sciences 23(1), 1-8, 2017

    The Editorial Committee of the Interdisciplinary Information Sciences

Codes

  • NII Article ID (NAID)
    130005519585
  • Text Lang
    ENG
  • ISSN
    1340-9050
  • Data Source
    J-STAGE 
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