Document Open Access Logo

Approximate Degree Lower Bounds for Oracle Identification Problems

Authors Mark Bun, Nadezhda Voronova

PDF
Thumbnail PDF

File

LIPIcs.TQC.2023.1.pdf
  • Filesize: 0.83 MB
  • 24 pages

Document Identifiers

Author Details

Mark Bun
  • Department of Computer Science, Boston University, MA, USA
Nadezhda Voronova
  • Department of Computer Science, Boston University, MA, USA

Funding

Supported in part by NSF awards CCF-1947889 and CNS-2046425.
  • Bun, Mark: Supported in part by a Sloan Research Fellowship.

Acknowledgements

We thank Arkadev Chattopadhyay for suggesting the problem of determining the approximate degree of ordered search, and Arkadev and Justin Thaler for many helpful conversations about it. We also thank the anonymous TQC 2023 reviewers for helpful suggestions on the presentation.

Cite As Get BibTex

Mark Bun and Nadezhda Voronova. Approximate Degree Lower Bounds for Oracle Identification Problems. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.TQC.2023.1

Abstract

The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function.
We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string x ∈ {0, 1}ⁿ given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of x. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of Ω(n/log² n) for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Communication complexity
  • Theory of computation → Quantum complexity theory
Keywords
  • Approximate degree
  • quantum query complexity
  • communication complexity
  • ordered search
  • polynomial approximations
  • polynomial method

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 863-876. ACM, 2016. Google Scholar
  2. A. Ambainis. A better lower bound for quantum algorithms searching an ordered list. In 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039), pages 352-357, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814606.
  3. Andris Ambainis. Polynomial degree vs. quantum query complexity. J. Comput. Syst. Sci., 72(2):220-238, 2006. URL: https://doi.org/10.1016/j.jcss.2005.06.006.
  4. Andris Ambainis, Kazuo Iwama, Akinori Kawachi, Hiroyuki Masuda, Raymond H. Putra, and Shigeru Yamashita. Quantum identification of boolean oracles. In Volker Diekert and Michel Habib, editors, STACS 2004, 21st Annual Symposium on Theoretical Aspects of Computer Science, Montpellier, France, March 25-27, 2004, Proceedings, volume 2996 of Lecture Notes in Computer Science, pages 105-116. Springer, 2004. Google Scholar
  5. Andris Ambainis, Kazuo Iwama, Akinori Kawachi, Rudy Raymond, and Shigeru Yamashita. Improved algorithms for quantum identification of boolean oracles. Theor. Comput. Sci., 378(1):41-53, 2007. Google Scholar
  6. Andris Ambainis and Ashley Montanaro. Quantum algorithms for search with wildcards and combinatorial group testing. Quantum Inf. Comput., 14(5-6):439-453, 2014. URL: https://doi.org/10.26421/QIC14.5-6-4.
  7. James Aspnes, Richard Beigel, Merrick Furst, and Steven Rudich. The expressive power of voting polynomials. In Proceedings of the twenty-third annual ACM symposium on Theory of Computing, pages 402-409, 1991. Google Scholar
  8. Howard Barnum, Michael Saks, and Mario Szegedy. Quantum decision trees and semidefinite programming. Technical report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 2001. Google Scholar
  9. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald De Wolf. Quantum lower bounds by polynomials. Journal of the ACM (JACM), 48(4):778-797, 2001. Google Scholar
  10. Aleksandrs Belovs. Quantum algorithms for learning symmetric juntas via the adversary bound. Comput. Complex., 24(2):255-293, 2015. URL: https://doi.org/10.1007/s00037-015-0099-2.
  11. Shalev Ben-David, Adam Bouland, Ankit Garg, and Robin Kothari. Classical lower bounds from quantum upper bounds. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 339-349. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00040.
  12. Michael Ben-Or and Avinatan Hassidim. The bayesian learner is optimal for noisy binary search (and pretty good for quantum as well). In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 221-230, 2008. URL: https://doi.org/10.1109/FOCS.2008.58.
  13. Ethan Bernstein and Umesh V. Vazirani. Quantum complexity theory. In S. Rao Kosaraju, David S. Johnson, and Alok Aggarwal, editors, Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 11-20. ACM, 1993. URL: https://doi.org/10.1145/167088.167097.
  14. Harry Buhrman, Richard Cleve, Ronald de Wolf, and Christof Zalka. Bounds for small-error and zero-error quantum algorithms. In 40th Annual Symposium on Foundations of Computer Science, FOCS '99, 17-18 October, 1999, New York, NY, USA, pages 358-368. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814607.
  15. Harry Buhrman and Ronald de Wolf. A lower bound for quantum search of an ordered list. Information Processing Letters, 70(5):205-209, 1999. URL: https://doi.org/10.1016/S0020-0190(99)00069-1.
  16. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci., 288(1):21-43, 2002. Google Scholar
  17. Harry Buhrman, Ilan Newman, Hein Röhrig, and Ronald de Wolf. Robust polynomials and quantum algorithms. Theory Comput. Syst., 40(4):379-395, 2007. Google Scholar
  18. Mark Bun, Robin Kothari, and Justin Thaler. The polynomial method strikes back: Tight quantum query bounds via dual polynomials. Theory Comput., 16:1-71, 2020. URL: https://doi.org/10.4086/toc.2020.v016a010.
  19. Mark Bun and Justin Thaler. Approximate degree in classical and quantum computing. Found. Trends Theor. Comput. Sci., 15(3-4):229-423, 2022. Google Scholar
  20. Mark Bun and Nadezhda Voronova. Approximate degree lower bounds for oracle identification problems, 2023. URL: https://arxiv.org/abs/2303.03921.
  21. Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-to-communication lifting using low-discrepancy gadgets. SIAM J. Comput., 50(1):171-210, 2021. URL: https://doi.org/10.1137/19M1310153.
  22. Arkadev Chattopadhyay, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Composition and simulation theorems via pseudo-random properties. Electron. Colloquium Comput. Complex., page 14, 2017. URL: https://eccc.weizmann.ac.il/report/2017/014, URL: https://arxiv.org/abs/TR17-014.
  23. Andrew M. Childs, Andrew J. Landahl, and Pablo A. Parrilo. Quantum algorithms for the ordered search problem via semidefinite programming. Phys. Rev. A, 75:032335, March 2007. URL: https://doi.org/10.1103/PhysRevA.75.032335.
  24. Andrew M. Childs and Troy Lee. Optimal quantum adversary lower bounds for ordered search. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I: Tack A: Algorithms, Automata, Complexity, and Games, volume 5125 of Lecture Notes in Computer Science, pages 869-880. Springer, 2008. Google Scholar
  25. Richard Cleve, Kazuo Iwama, François Le Gall, Harumichi Nishimura, Seiichiro Tani, Junichi Teruyama, and Shigeru Yamashita. Reconstructing strings from substrings with quantum queries. In Scandinavian Workshop on Algorithm Theory, pages 388-397. Springer, 2012. Google Scholar
  26. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. A limit on the speed of quantum computation for insertion into an ordered list, 1998. URL: https://doi.org/10.48550/ARXIV.QUANT-PH/9812057.
  27. Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Invariant quantum algorithms for insertion into an ordered list, 1999. URL: https://doi.org/10.48550/ARXIV.QUANT-PH/9901059.
  28. Peter Høyer, Troy Lee, and Robert Spalek. Negative weights make adversaries stronger. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 526-535. ACM, 2007. Google Scholar
  29. Peter Hoyer, Troy Lee, and Robert Spalek. Negative weights make adversaries stronger. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 526-535, 2007. Google Scholar
  30. Peter Høyer, Jan Neerbek, and Yaoyun Shi. Quantum complexities of ordered searching, sorting, and element distinctness. Algorithmica, 34(4):429-448, 2002. Google Scholar
  31. William M. Hoza. Quantum communication-query tradeoffs. CoRR, abs/1703.07768, 2017. URL: https://arxiv.org/abs/1703.07768.
  32. Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Junichi Teruyama. Quantum counterfeit coin problems. Theor. Comput. Sci., 456:51-64, 2012. URL: https://doi.org/10.1016/j.tcs.2012.05.039.
  33. Hartmut Klauck, Robert Spalek, and Ronald de Wolf. Quantum and classical strong direct product theorems and optimal time-space tradeoffs. SIAM J. Comput., 36(5):1472-1493, 2007. URL: https://doi.org/10.1137/05063235X.
  34. Robin Kothari. An optimal quantum algorithm for the oracle identification problem. In Ernst W. Mayr and Natacha Portier, editors, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), STACS 2014, March 5-8, 2014, Lyon, France, volume 25 of LIPIcs, pages 482-493. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2014. Google Scholar
  35. Ashley Montanaro and Changpeng Shao. Quantum algorithms for learning graphs and beyond. CoRR, abs/2011.08611, 2020. URL: https://arxiv.org/abs/2011.08611.
  36. Ilan Newman. Private vs. common random bits in communication complexity. Inf. Process. Lett., 39(2):67-71, 1991. URL: https://doi.org/10.1016/0020-0190(91)90157-D.
  37. Noam Nisan. The communication complexity of threshold gates. In Combinatorics, Paul Erdös is Eighty, number 1 in Bolyai Society Mathematical Studies, pages 301-315, 1993. Google Scholar
  38. Ben Reichardt. Reflections for quantum query algorithms. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 560-569. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.44.
  39. Alexander A. Sherstov. The pattern matrix method. SIAM J. Comput., 40(6):1969-2000, 2011. URL: https://doi.org/10.1137/080733644.
  40. Alexander A Sherstov. Making polynomials robust to noise. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 747-758, 2012. Google Scholar
  41. Alexander A. Sherstov. Strong direct product theorems for quantum communication and query complexity. SIAM J. Comput., 41(5):1122-1165, 2012. Google Scholar
  42. Alexander A. Sherstov. Algorithmic polynomials. SIAM J. Comput., 49(6):1173-1231, 2020. URL: https://doi.org/10.1137/19M1278831.
  43. Steven Skiena and Gopalakrishnan Sundaram. Reconstructing strings from substrings. J. Comput. Biol., 2(2):333-353, 1995. URL: https://doi.org/10.1089/cmb.1995.2.333.
  44. Wim van Dam. Quantum oracle interrogation: Getting all information for almost half the price. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 362-367. IEEE Computer Society, 1998. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact