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Supersymmetry for chiral symmetric quantum walks

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Abstract

Quantum walks have attracted attention as a promising platform realizing topological phenomena, and many physicists have introduced various types of indices to characterize topologically protected bound states that are robust against perturbations. In this paper, we introduce an index from a supersymmetric point of view. This allows us to define indices for all chiral symmetric quantum walks such as multi-dimensional split-step quantum walks and quantum walks on graphs, for which there has been no index theory. Moreover, the index gives a lower bound on the number of bound states robust against compact perturbations. We also calculate the index for several concrete examples including the unitary transformation that appears in Grover’s search algorithm.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant No. JP18K03327 and by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University. The author thanks Y. Matsuzawa and Y. Tanaka for helpful comments on the index formula for finite-dimensional cases and for one-dimensional split-step quantum walks.

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  1. Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato, Nagano, 380-8553, Japan

    Akito Suzuki

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Correspondence to Akito Suzuki.

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Suzuki, A. Supersymmetry for chiral symmetric quantum walks. Quantum Inf Process 18, 363 (2019). https://doi.org/10.1007/s11128-019-2474-4

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