How does Grover walk recognize the shape of crystal lattice?

Abstract

We consider the support of the limit distribution of the Grover walk on crystal lattices with the linear scaling. The orbit of the Grover walk is denoted by the parametric plot of the pseudo-velocity of the Grover walk in the wave space. The region of the orbit is the support of the limit distribution. In this paper, we compute the regions of the orbits for the triangular, hexagonal and kagome lattices. We show every outer frame of the support is described by an ellipse. The shape of the ellipse depends only on the realization of the fundamental lattice of the crystal lattice in \(\mathbb {R}^2\).

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Correspondence to Etsuo Segawa.

Additional information

Norio Konno 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). Etsuo Segawa acknowledges financial supports from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant Nos. 16K17637, 16K03939). The research by Hyun Jae Yoo was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03936006).

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Ko, C.K., Konno, N., Segawa, E. et al. How does Grover walk recognize the shape of crystal lattice?. Quantum Inf Process 17, 167 (2018). https://doi.org/10.1007/s11128-018-1886-x

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Keywords

  • Grover walks
  • Crystal lattice
  • Limit theorem