Improved Approximate Degree Bounds for k-Distinctness

Authors Nikhil S. Mande, Justin Thaler, Shuchen Zhu

Thumbnail PDF


  • Filesize: 0.61 MB
  • 22 pages

Document Identifiers

Author Details

Nikhil S. Mande
  • Georgetown University, Washington DC, USA
Justin Thaler
  • Georgetown University, Washington DC, USA
Shuchen Zhu
  • Georgetown University, Washington DC, USA


JT is grateful to Robin Kothari for extremely useful suggestions and discussions surrounding Theorem 2, and to Mark Bun for essential discussions regarding Theorem 19. SZ would like to thank Yao Ji for several helpful conversations.

Cite AsGet BibTex

Nikhil S. Mande, Justin Thaler, and Shuchen Zhu. Improved Approximate Degree Bounds for k-Distinctness. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element Distinctness) is well-understood, there is a polynomial gap between the known upper and lower bounds for all constants k>2. Specifically, the best known upper bound is O (N^{(3/4)-1/(2^{k+2}-4)}) (Belovs, FOCS 2012), while the best known lower bound for k≥ 2 is Ω̃(N^{2/3} + N^{(3/4)-1/(2k)}) (Aaronson and Shi, J. ACM 2004; Bun, Kothari, and Thaler, STOC 2018). For any constant k ≥ 4, we improve the lower bound to Ω̃(N^{(3/4)-1/(4k)}). This yields, for example, the first proof that 4-distinctness is strictly harder than Element Distinctness. Our lower bound applies more generally to approximate degree. As a secondary result, we give a simple construction of an approximating polynomial of degree Õ(N^{3/4}) that applies whenever k ≤ polylog(N).

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Quantum Query Complexity
  • Approximate Degree
  • Dual Polynomials
  • k-distinctness


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


  1. Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595-605, 2004. URL:
  2. Andris Ambainis. Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory of Computing, 1(1):37-46, 2005. URL:
  3. Andris Ambainis. Polynomial degree vs. quantum query complexity. Journal of Computer and System Sciences, 72(2):220-238, 2006. Google Scholar
  4. Andris Ambainis. Quantum walk algorithm for element distinctness. SIAM J. Comput., 37(1):210-239, 2007. URL:
  5. Srinivasan Arunachalam, Jop Briët, and Carlos Palazuelos. Quantum query algorithms are completely bounded forms. SIAM Journal on Computing, 48(3):903-925, 2019. Google Scholar
  6. 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
  7. Paul Beame and Widad Machmouchi. The quantum query complexity of AC^0. Quantum Information & Computation, 12(7-8):670-676, 2012. Google Scholar
  8. Aleksandrs Belovs. Learning-graph-based quantum algorithm for k-distinctness. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 207-216, 2012. URL:
  9. 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
  10. Mark Bun, Robin Kothari, and Justin Thaler. The polynomial method strikes back: Tight quantum query bounds via dual polynomials. CoRR, abs/1710.09079, version 3, 2017. Google Scholar
  11. Mark Bun, Robin Kothari, and Justin Thaler. The polynomial method strikes back: tight quantum query bounds via dual polynomials. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 297-310, 2018. Google Scholar
  12. Mark Bun and Justin Thaler. Hardness amplification and the approximate degree of constant-depth circuits. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 268-280, 2015. Google Scholar
  13. Mark Bun and Justin Thaler. A nearly optimal lower bound on the approximate degree of AC^0. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 1-12, 2017. Google Scholar
  14. Mark Bun and Justin Thaler. The large-error approximate degree of ac^0. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  15. Samuel Kutin. Quantum lower bound for the collision problem with small range. Theory of Computing, 1(1):29-36, 2005. URL:
  16. Troy Lee. A note on the sign degree of formulas. CoRR, abs/0909.4607, 2009. Google Scholar
  17. Tongyang Li and Xiaodi Wu. Quantum query complexity of entropy estimation. IEEE Trans. Inf. Theory, 65(5):2899-2921, 2019. Google Scholar
  18. Qipeng Liu and Mark Zhandry. On finding quantum multi-collisions. In Advances in Cryptology - EUROCRYPT 2019 - 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Darmstadt, Germany, May 19-23, 2019, Proceedings, Part III, pages 189-218, 2019. URL:
  19. Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. Computational Complexity, 4:301-313, 1994. Google Scholar
  20. Alexander A. Razborov and Alexander A. Sherstov. The sign-rank of AC^0. SIAM J. Comput., 39(5):1833-1855, 2010. URL:
  21. Alexander A. Sherstov. The pattern matrix method. SIAM J. Comput., 40(6):1969-2000, 2011. Google Scholar
  22. Alexander A. Sherstov. Strong direct product theorems for quantum communication and query complexity. SIAM J. Comput., 41(5):1122-1165, 2012. Google Scholar
  23. Alexander A Sherstov. The intersection of two halfspaces has high threshold degree. SIAM Journal on Computing, 42(6):2329-2374, 2013. Google Scholar
  24. Alexander A Sherstov. Algorithmic polynomials. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 311-324, 2018. Google Scholar
  25. Alexander A. Sherstov. The power of asymmetry in constant-depth circuits. SIAM J. Comput., 47(6):2362-2434, 2018. Google Scholar
  26. Alexander A Sherstov and Justin Thaler. Vanishing-error approximate degree and QMA complexity. arXiv preprint arXiv:1909.07498, 2019. Google Scholar
  27. Yaoyun Shi and Yufan Zhu. Quantum communication complexity of block-composed functions. Quantum Information & Computation, 9(5):444-460, 2009. Google Scholar
  28. Robert Špalek. A dual polynomial for OR. CoRR, abs/0803.4516, 2008. Google Scholar