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Quantum Algorithms for Classical Probability Distributions

Author Aleksandrs Belovs

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

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Abstract

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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ESA.2019.16

Author Details

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

Funding

This research is partly supported by the ERDF grant number 1.1.1.2/VIAA/1/16/113.

Acknowledgements

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.

References

  1. Scott Aaronson and Yaoyun Shi. Quantum Lower Bounds for the Collision and the Element Distinctness Problems. Journal of the ACM, 51(4):595-605, 2004. URL: https://doi.org/10.1145/1008731.1008735.
  2. Dorit Aharonov and Amnon Ta-Shma. Adiabatic quantum state generation. SIAM Journal on Computing, 37(1):47-82, 2007. URL: https://doi.org/doi.org/10.1137/060648829.
  3. Andris Ambainis. Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range. Theory of Computing, 1:37-46, 2005. Google Scholar
  4. Andris Ambainis, Aleksandrs Belovs, Oded Regev, and Ronald de Wolf. Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing. In Proc. of 27th ACM-SIAM SODA, pages 903-922, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch65.
  5. Andris Ambainis, Loïck Magnin, Martin Rötteler, and Jérémie Roland. Symmetry-assisted adversaries for quantum state generation. In Proc. of 26th IEEE CCC, pages 167-177, 2011. URL: https://doi.org/10.1109/CCC.2011.24.
  6. Alp Atıcı and Rocco A. Servedio. Improved bounds on quantum learning algorithms. Quantum Information Processing, 4(5):355-386, 2005. URL: https://doi.org/10.1007/s11128-005-0001-2.
  7. Ziv Bar-Yossef. The Complexity of Massive Data Set Computations. PhD thesis, UC Berkeley, 2002. Google Scholar
  8. Aleksandrs Belovs. Learning-graph-based Quantum Algorithm for k-distinctness. In Proc. of 53rd IEEE FOCS, pages 207-216, 2012. URL: https://doi.org/10.1109/FOCS.2012.18.
  9. Aleksandrs Belovs. Span programs for functions with constant-sized 1-certificates. In Proc. of 44th ACM STOC, pages 77-84, 2012. URL: https://doi.org/10.1145/2213977.2213985.
  10. Aleksandrs Belovs. Quantum Algorithms for Learning Symmetric Juntas via the Adversary Bound. Computational Complexity, 24(2):255-293, 2015. URL: https://doi.org/10.1007/s00037-015-0099-2.
  11. Aleksandrs Belovs. Variations on Quantum Adversary, 2015. URL: http://arxiv.org/abs/1504.06943.
  12. Aleksandrs Belovs, Gilles Brassard, Peter Høyer, Marc Kaplan, Sophie Laplante, and Louis Salvail. Provably secure key establishment against quantum adversaries. In Proc. of 12th TQC, volume 73 of LIPIcs, pages 3:1-3:17. Dagstuhl, 2018. Google Scholar
  13. Aleksandrs Belovs and Ben W. Reichardt. Span programs and quantum algorithms for st-connectivity and claw detection. In Proc. of 20th ESA, volume 7501 of LNCS, pages 193-204. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-33090-2_18.
  14. Rajendra Bhatia. Positive definite matrices. Princeton University Press, 2009. Google Scholar
  15. Gilles Brassard, Peter Høyer, and Alain Tapp. Quantum cryptanalysis of hash and claw-free functions. In Proc. of 3rd LATIN, volume 1380 of LNCS, pages 163-169. Springer, 1998. URL: https://doi.org/10.1007/BFb0054319.
  16. Sergey Bravyi, Aram W. Harrow, and Avinatan Hassidim. Quantum algorithms for testing properties of distributions. IEEE Transactions on Information Theory, 57:3971-3981, 2011. URL: https://doi.org/10.1109/TIT.2011.2134250.
  17. Nader H. Bshouty and Jeffrey C. Jackson. Learning DNF over the uniform distribution using a quantum example oracle. SIAM Journal on Computing, 28(3):1136-1153, 1998. URL: https://doi.org/10.1137/S0097539795293123.
  18. Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Ronald de Wolf. New Results on Quantum Property Testing. In Proc. of 30th FSTTCS, volume 8 of LIPIcs, pages 145-156. Dagstuhl, 2010. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2010.145.
  19. András Gilyén and Tongyang Li. Distributional property testing in a quantum world, 2019. URL: http://arxiv.org/abs/1902.00814.
  20. Yassine Hamoudi and Frédéric Magniez. Quantum Chebyshev’s Inequality and Applications, 2018. URL: http://arxiv.org/abs/1807.06456.
  21. Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proc. of 39th ACM STOC, pages 526-535, 2007. URL: https://doi.org/10.1145/1250790.1250867.
  22. Michael Jarret, Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita. Quantum Algorithms for Connectivity and Related Problems. In Proc. of 26th ESA, volume 112 of LIPIcs, pages 49:1-49:13. Dagstuhl, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.49.
  23. Troy Lee, Frédéric Magniez, and Miklos Santha. A learning graph based quantum query algorithm for finding constant-size subgraphs. Chicago Journal of Theoretical Computer Science, 2012(10), 2012. Google Scholar
  24. Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In Proc. of 52nd IEEE FOCS, pages 344-353, 2011. URL: https://doi.org/10.1109/FOCS.2011.75.
  25. Tongyang Li and Xiaodi Wu. Quantum query complexity of entropy estimation. IEEE Transactions on Information Theory, page 1–1, 2018. URL: https://doi.org/10.1109/TIT.2018.2883306.
  26. Ashley Montanaro. Quantum speedup of Monte Carlo methods. Proceedings of the Royal Society A, 471(2181), 2015. URL: https://doi.org/10.1098/rspa.2015.0301.
  27. Ashley Montanaro. The quantum complexity of approximating the frequency moments. Quantum Information & Computation, 16:1169-1190, 2016. Google Scholar
  28. Ben W. Reichardt. Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function. In Proc. of 50th IEEE FOCS, pages 544-551, 2009. URL: https://doi.org/10.1109/FOCS.2009.55.
  29. Ben W. Reichardt. Reflections for quantum query algorithms. In Proc. of 22nd ACM-SIAM SODA, pages 560-569, 2011. URL: https://doi.org/10.1137/1.9781611973082.44.
  30. Mark Zhandry. A Note on the Quantum Collision and Set Equality Problems. Quantum Information & Computation, 15(7&8):557-567, 2015. Google Scholar
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