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Transport distance between Grover walks on graphs and coarse Ricci curvature

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Abstract

One direction to investigate the relation between quantum walks and their underlying graphs is to define geometric quantity concerning quantum walks. In order to contribute to this direction, we define a transport distance between Grover walks, which can be seen as a quantum analogue of symmetric random walks. We employ signed optimal transport theory to formulate this distance. Also, we define coarse Ricci curvature induced by Grover walks and investigate its property. It has been found that this coarse Ricci curvature has similar properties to those of coarse Ricci curvature induced by random walks.

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Acknowledgements

We would like to express our gratitude to Tomoya Akamatsu for valuable comments. This work was supported by JSPS KAKENHI Grant Number JP22KJ1408.

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Authors and Affiliations

  1. Department of Mathematics, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

    Yasuaki Fujitani

  2. Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka, 560-8531, Japan

    Chusei Kiumi

Authors
  1. Yasuaki Fujitani
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  2. Chusei Kiumi
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Contributions

Y.F. provided the fundamental idea for the manuscript and wrote the manuscript text mainly. C.K. wrote the Python implementation mainly. All authors contributed to the analysis of the main statements and reviewed the manuscript.

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Correspondence to Yasuaki Fujitani.

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Fujitani, Y., Kiumi, C. Transport distance between Grover walks on graphs and coarse Ricci curvature. Quantum Inf Process 23, 180 (2024). https://doi.org/10.1007/s11128-024-04373-2

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