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Odd-periodic Grover Walks

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Abstract

The Grover walk is one of the most well-studied quantum walks on graphs. In this paper, we investigate its periodicity to reveal the relationship between the quantum walk and the underlying graph, focusing particularly on the characterization of graphs exhibiting a periodic Grover walk. Graphs having a periodic Grover walk with periods of 2, 3, 4, and 5 have previously been characterized. It is expected that graphs exhibiting a periodic Grover walk with odd period correspond to cycles with odd length. We address that problem and are able to perfectly characterize the class of graphs exhibiting an odd-periodic Grover walk by using a combinatorial method.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We would like to express our gratitude to professors Nobuaki Obata, Etsuo Segawa, Yusuke Higuchi, and Osamu Ogurisu for their continued support and fruitful comments. This study was supported by JSPS KAKENHI Grant Number 22K13952.

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  1. General Education, Ishikawa College, National Institute of Technology, Ishikawa, 929-0392, Japan

    Yusuke Yoshie

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  1. Yusuke Yoshie
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Yoshie, Y. Odd-periodic Grover Walks. Quantum Inf Process 22, 316 (2023). https://doi.org/10.1007/s11128-023-04078-y

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