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An improved algorithm for computing hitting probabilities of quantum walks

Author

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  • Zhang, Yanbing
  • Song, Tingting
  • Wu, Zhihao

Abstract

Quantum walks are the quantum corresponding version of classical random walks, providing polynomial and even exponential acceleration for some classical difficult problems. In recent years, there have been numerous researches on quantum walks, among which the hitting probabilities problem is another crucial problem. In 2021, Guan et al. have proposed an HHL-based algorithm to calculate the hitting probabilities of quantum walks, which can effectively transform the original problem into an inverse matrix problem, then achieve an acceleration effect on the corresponding classical algorithm under some specific conditions. However, we propose a new algorithm consisting of an improved quantum matrix inversion algorithm with complexity Oκ2log1.5(κ/ε)polylog(n), where κ is the condition number of the matrix, n+1 is the number of positions of the one-dimensional quantum walks, and ε is the expected accuracy of the output state. Compared to the HHL-based algorithm, our improved algorithm achieves an exponential improvement in the dependence on accuracy.

Suggested Citation

  • Zhang, Yanbing & Song, Tingting & Wu, Zhihao, 2022. "An improved algorithm for computing hitting probabilities of quantum walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 594(C).
    • Handle: RePEc:eee:phsmap:v:594:y:2022:i:c:s0378437122000875
    DOI: 10.1016/j.physa.2022.127009
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    References listed on IDEAS

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    1. Kerstin Beer & Dmytro Bondarenko & Terry Farrelly & Tobias J. Osborne & Robert Salzmann & Daniel Scheiermann & Ramona Wolf, 2020. "Training deep quantum neural networks," Nature Communications, Nature, vol. 11(1), pages 1-6, December.
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