An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree

Authors Andris Ambainis, Aleksandrs Belovs

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


We thank Scott Aaronson for writing the open problem survey [Aaronson, 2021] which attracted our attention to this problem. We also thank the anonymous reviewers at the CCC conference for their numerous valuable suggestions on the presentation of the paper.

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Andris Ambainis and Aleksandrs Belovs. An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions. In this paper, we demonstrate an exponential separation between exact polynomial degree and approximate quantum query complexity for a partial Boolean function. For an unbounded alphabet size, we have a constant versus polynomial separation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Polynomials
  • Quantum Adversary Bound
  • Separations in Query Complexity


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  1. Scott Aaronson. Quantum lower bound for the collision problem. In Proc. of 34th ACM STOC, pages 635-642, 2002. URL:
  2. Scott Aaronson. Open problems related to quantum query complexity. ACM Transactions on Quantum Computing, 2(4):1-9, 2021. URL:
  3. 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:
  4. 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:
  5. 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:
  6. Andris Ambainis. Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences, 64(4):750-767, 2002. URL:
  7. Andris Ambainis. Polynomial degree vs. quantum query complexity. In Proc. of 44th IEEE FOCS, pages 230-239, 2003. URL:
  8. Anurag Anshu, Shalev Ben-David, and Srijita Kundu. On query-to-communication lifting for adversary bounds. In Proc. of 36th IEEE CCC, volume 200 of LIPIcs, pages 30:1-30:39, 2021. URL:
  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:
  10. Aleksandrs Belovs and Ansis Rosmanis. On the power of non-adaptive learning graphs. Computational Complexity, 23(2):323-354, 2014. URL:
  11. Aleksandrs Belovs and Ansis Rosmanis. Quantum lower bounds for tripartite versions of the hidden shift and the set equality problems. In Proc. of 13th TQC, volume 111 of LIPIcs, pages 3:1-3:15. Dagstuhl, 2018. URL:
  12. Aleksandrs Belovs and Robert Špalek. Adversary lower bound for the k-sum problem. In Proc. of 4th ACM ITCS, pages 323-328, 2013. URL:
  13. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288:21-43, 2002. URL:
  14. 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:
  15. Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proc. of 39th ACM STOC, pages 526-535, 2007. URL:
  16. 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:
  17. 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:
  18. Gatis Midrijānis. Exact quantum query complexity for total Boolean functions. quant-ph/0403168, 2004. Google Scholar
  19. Noam Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. URL:
  20. Noam Nisan and Mario Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 4(4):301-313, 1994. URL:
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