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A Direct Reduction from the Polynomial to the Adversary Method

Author Aleksandrs Belovs

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ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Polynomials
  • Quantum Adversary Bound

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Abstract

The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other.
It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known.
In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.

Cite As Get BibTex

Aleksandrs Belovs. A Direct Reduction from the Polynomial to the Adversary Method. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.TQC.2024.11

Author Details

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

Funding

  • Belovs, Aleksandrs: This research is supported by the Latvian Quantum Initiative under European Union Recovery and Resilience Facility project no. 2.3.1.1.i.0/1/22/I/CFLA/001. Part of this work has been supported by the ERDF project number 1.1.1.5/18/A/020 "Quantum algorithms: from complexity theory to experiment."

Acknowledgements

The author is thankful to Shalev Ben-David for the suggestion to apply this construction to the polynomial method, and to anonymous referees for their helpful comments on the previous versions of the paper.

References

  1. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In Proc. of 48th ACM STOC, pages 863-876, 2016. URL: https://doi.org/10.1145/2897518.2897644.
  2. Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, and Avishay Tal. Degree vs. approximate degree and quantum implications of Huang’s sensitivity theorem. In Proc. of 53rd ACM STOC, pages 1330-1342, 2021. URL: https://doi.org/10.1145/3406325.3451047.
  3. 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.
  4. Andris Ambainis. Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences, 64(4):750-767, 2002. URL: https://doi.org/10.1006/jcss.2002.1826.
  5. Andris Ambainis. Polynomial degree vs. quantum query complexity. In Proc. of 44th IEEE FOCS, pages 230-239, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238197.
  6. Andris Ambainis and Aleksandrs Belovs. An exponential separation between quantum query complexity and the polynomial degree. In Proc. of 38th IEEE CCC, volume 264 of LIPIcs, pages 24:1-24:13. Dagstuhl, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.24.
  7. Howard Barnum and Michael Saks. A lower bound on the quantum query complexity of read-once functions. Journal of Computer and System Sciences, 69(2):244-258, 2004. URL: https://doi.org/10.1016/j.jcss.2004.02.002.
  8. Howard Barnum, Michael Saks, and Mario Szegedy. Quantum decision trees and semi-definite programming. In Proc. of 18th IEEE CCC, pages 179-193, 2003. URL: https://doi.org/10.1109/CCC.2003.1214419.
  9. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  10. Aleksandrs Belovs. Adversary lower bound for element distinctness, 2012. URL: https://arxiv.org/abs/1204.5074.
  11. 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. URL: https://doi.org/10.4230/LIPIcs.TQC.2017.3.
  12. Aleksandrs Belovs and Ansis Rosmanis. On the power of non-adaptive learning graphs. Computational Complexity, 23(2):323-354, 2014. URL: https://doi.org/10.1007/s00037-014-0084-1.
  13. Aleksandrs Belovs and Robert Špalek. Adversary lower bound for the k-sum problem. In Proc. of 4th ACM ITCS, pages 323-328, 2013. URL: https://doi.org/10.1145/2422436.2422474.
  14. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and weaknesses of quantum computing. SIAM Journal on Computing, 26(5):1510-1523, 1997. URL: https://doi.org/10.1137/S0097539796300933.
  15. Harry Buhrman and Robert Špalek. Quantum verification of matrix products. In Proc. of 17th ACM-SIAM SODA, pages 880-889, 2006. URL: https://arxiv.org/abs/quant-ph/0409035.
  16. Mark Bun, Robin Kothari, and Justin Thaler. The polynomial method strikes back: Tight quantum query bounds via dual polynomials. In Proc. of 50th ACM STOC, pages 297-310, 2018. URL: https://doi.org/10.1145/3188745.3188784.
  17. Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. Information and Computation, 243:2-25, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.003.
  18. Mark Bun and Justin Thaler. Dual polynomials for collision and element distinctness. Theory of Computing, 12(16):1-34, 2016. URL: https://doi.org/10.4086/toc.2016.v012a016.
  19. Sebastian Dörn and Thomas Thierauf. The quantum query complexity of algebraic properties. In Proc. of 16th FCT, volume 4639 of LNCS, pages 250-260. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-74240-1_22.
  20. Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. Quantum query complexity of some graph problems. In Proc. of 31st ICALP, volume 3142 of LNCS, pages 481-493. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27836-8_42.
  21. Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proc. of 28th ACM STOC, pages 212-219, 1996. URL: https://doi.org/10.1145/237814.237866.
  22. 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.
  23. 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.
  24. Qipeng Liu and Mark Zhandry. On finding quantum multi-collisions. In Proc. of 38th EUROCRYPT, volume 11478 of LNCS, pages 189-218, 2019. URL: https://doi.org/10.1007/978-3-030-17659-4_7.
  25. Loïck Magnin and Jérémie Roland. Explicit relation between all lower bound techniques for quantum query complexity. In Proc. of 30th STACS, volume 20 of LIPIcs, pages 434-445. Dagstuhl, 2013. URL: https://doi.org/10.4230/LIPIcs.STACS.2013.434.
  26. Nikhil S. Mande, Justin Thaler, and Shuchen Zhu. Improved approximate degree bounds for k-distinctness. In Proc. of 15th TQC, volume 158 of LIPIcs, pages 2:1-2:22, 2020. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.2.
  27. Noam Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. URL: https://doi.org/10.1137/0220062.
  28. Noam Nisan and Mario Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 4(4):301-313, 1994. URL: https://doi.org/10.1007/BF01263419.
  29. 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.
  30. 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.
  31. Ben W. Reichardt and Robert Špalek. Span-program-based quantum algorithm for evaluating formulas. Theory of Computing, 8:291-319, 2012. URL: https://doi.org/10.4086/toc.2012.v008a013.
  32. Alexander A Sherstov. The pattern matrix method. SIAM Journal on Computing, 40(6):1969, 2011. URL: https://doi.org/10.1137/080733644.
  33. Alexander A. Sherstov. Approximating the and-or tree. Theory of Computing, 9(20):653-663, 2013. URL: https://doi.org/10.4086/toc.2013.v009a020.
  34. Mark Zhandry. How to record quantum queries, and applications to quantum indifferentiability. In Proc. of 39th CRYPTO, volume 11693 of LNCS, pages 239-268, 2019. URL: https://doi.org/10.1007/978-3-030-26951-7_9.
  35. Shengyu Zhang. On the power of Ambainis lower bounds. Theoretical Computer Science, 339(2):241-256, 2005. URL: https://doi.org/10.1016/j.tcs.2005.01.019.
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