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Unitary equivalence classes of split-step quantum walks

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Abstract

In this study, we investigate the unitary equivalence of split-step quantum walks (SSQWs). Introduced by Kitagawa and generalized by Suzuki, SSQWs are two-state discrete-time quantum walks on a one-dimensional lattice. In this work, we define a new class of quantum walks that includes all SSQWs by making use of the rank conditions, which state that (1) the rank of operators representing the walk from vertex x to vertex \(x\pm 1\) is less than or equal to 1 and that (2) the rank from vertex x to x (i.e., to itself) is less than or equal to 2. Initially, we abstractly define quantum walks in this class, then clarify by presenting the explicit form of such quantum walks. We also calculate their unitary equivalence classes and show that four real parameters are required for each vertex to parameterize the unitary equivalence classes. Further, we consider the unitary equivalence of Suzuki’s SSQWs and prove that all parameters of Suzuki’s SSQWs can be real. Finally, we discuss the chiral symmetry of SSQWs.

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Yokohama National University, Hodogaya, Yokohama, 240-8501, Japan

    Akihiro Narimatsu

  2. Department of Mathematics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan

    Hiromichi Ohno

  3. Department of General Science and Education, National Institute of Technology, Hachinohe College, Hachinohe, 039-1192, Japan

    Kazuyuki Wada

Authors
  1. Akihiro Narimatsu
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  2. Hiromichi Ohno
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  3. Kazuyuki Wada
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Correspondence to Hiromichi Ohno.

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Narimatsu, A., Ohno, H. & Wada, K. Unitary equivalence classes of split-step quantum walks. Quantum Inf Process 20, 368 (2021). https://doi.org/10.1007/s11128-021-03323-6

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