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Partition-based discrete-time quantum walks

Quantum Information Processing volume 17, Article number: 100 (2018) Cite this article

Abstract

We introduce a family of discrete-time quantum walks, called two-partition model, based on two equivalence-class partitions of the computational basis, which establish the notion of local dynamics. This family encompasses most versions of unitary discrete-time quantum walks driven by two local operators studied in literature, such as the coined model, Szegedy’s model, and the 2-tessellable staggered model. We also analyze the connection of those models with the two-step coined model, which is driven by the square of the evolution operator of the standard discrete-time coined walk. We prove formally that the two-step coined model, an extension of Szegedy model for multigraphs, and the two-tessellable staggered model are unitarily equivalent. Then, selecting one specific model among those families is a matter of taste not generality.

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Notes

  1. A clique of G is a set of vertices that induces a complete subgraph of G.

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Author information

Authors and Affiliations

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

    Norio Konno

  2. National Laboratory of Scientific Computing - LNCC, Av. Getúlio Vargas 333, Petrópolis, RJ, 25651-075, Brazil

    Renato Portugal

  3. Oyama National College of Technology, Oyama, Tochigi, 323-0806, Japan

    Iwao Sato

  4. Graduate School of Information Sciences, Tohoku University, Aoba, Sendai, 980-8579, Japan

    Etsuo Segawa

Authors
  1. Norio Konno
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  2. Renato Portugal
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  3. Iwao Sato
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  4. Etsuo Segawa
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Corresponding author

Correspondence to Etsuo Segawa.

Additional information

NK is partially supported by the Grant-in-Aid for Scientific Research (Challenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No. 15K13443). RP acknowledges financial support from Faperj (Grant No. E-26/102.350/2013) and CNPq (Grant No. 303406/2015-1). IS is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 15K04985). ES acknowledges financial support from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant Nos. 16K17637, 16H03939).

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Konno, N., Portugal, R., Sato, I. et al. Partition-based discrete-time quantum walks. Quantum Inf Process 17, 100 (2018). https://doi.org/10.1007/s11128-017-1807-4

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  • DOI: https://doi.org/10.1007/s11128-017-1807-4

Keywords

  • Quantum walk
  • Coined walk
  • Szegedy’s walk
  • Staggered walk
  • Graph tessellation
  • Hypergraph walk
  • Unitary equivalence
  • Intersection graph
  • Bipartite graph