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Equivalence of Szegedy’s and coined quantum walks

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Abstract

Szegedy’s quantum walk is a quantization of a classical random walk or Markov chain, where the walk occurs on the edges of the bipartite double cover of the original graph. To search, one can simply quantize a Markov chain with absorbing vertices. Recently, Santos proposed two alternative search algorithms that instead utilize the sign-flip oracle in Grover’s algorithm rather than absorbing vertices. In this paper, we show that these two algorithms are exactly equivalent to two algorithms involving coined quantum walks, which are walks on the vertices of the original graph with an internal degree of freedom. The first scheme is equivalent to a coined quantum walk with one walk step per query of Grover’s oracle, and the second is equivalent to a coined quantum walk with two walk steps per query of Grover’s oracle. These equivalences lie outside the previously known equivalence of Szegedy’s quantum walk with absorbing vertices and the coined quantum walk with the negative identity operator as the coin for marked vertices, whose precise relationships we also investigate.

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Acknowledgements

Thanks to Peter Høyer for pointing out Fact 3.2 of [18]. This work was supported by the U.S. Department of Defense Vannevar Bush Faculty Fellowship of Scott Aaronson.

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  1. Thomas G. Wong

    Present address: Department of Physics, Creighton University, 2500 California Plaza, Omaha, NE, 68102, USA

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  1. Department of Computer Science, University of Texas at Austin, 2317 Speedway, Austin, TX, 78712, USA

    Thomas G. Wong

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  1. Thomas G. Wong
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Correspondence to Thomas G. Wong.

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Wong, T.G. Equivalence of Szegedy’s and coined quantum walks. Quantum Inf Process 16, 215 (2017). https://doi.org/10.1007/s11128-017-1667-y

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