Quantum Algorithms for Classical Probability Distributions

Author Aleksandrs Belovs

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Aleksandrs Belovs
  • Faculty of Computing, University of Latvia, Riga, Latvia


Most of all I would like to thank Anrás Gilyén for the suggestion to work on this problem. I am also grateful to Frédéric Magniez, Shalev Ben-David, and Anurag Anshu for useful discussions, as well as to anonymous reviewers for their comments. Part of this research was performed while at the Institute for Quantum Computing in Waterloo, Canada. I would like to thank Ashwin Nayak for hospitality.

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Aleksandrs Belovs. Quantum Algorithms for Classical Probability Distributions. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 16:1-16:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships. Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions. The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • quantum query complexity
  • quantum adversary method
  • distinguishing probability distributions
  • Hellinger distance


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