Document Open Access Logo

A Note on the Quantum Query Complexity of Permutation Symmetric Functions

Author André Chailloux

Thumbnail PDF


  • Filesize: 394 kB
  • 7 pages

Document Identifiers

Author Details

André Chailloux
  • Inria de Paris, EPI SECRET, Paris, France

Cite AsGet BibTex

André Chailloux. A Note on the Quantum Query Complexity of Permutation Symmetric Functions. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 19:1-19:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


It is known since the work of [Aaronson and Ambainis, 2014] that for any permutation symmetric function f, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity, more precisely that R(f) = O~(Q^7(f)). In this paper, we improve this result and show that R(f) = O(Q^3(f)) for a more general class of symmetric functions. Our proof is constructive and relies largely on the quantum hardness of distinguishing a random permutation from a random function with small range from Zhandry [Zhandry, 2015].

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • quantum query complexity
  • permutation symmetric functions


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


  1. Scott Aaronson and Andris Ambainis. The Need for Structure in Quantum Speedups. Theory of Computing, 10(6):133-166, 2014. URL:
  2. Scott Aaronson and Yaoyun Shi. Quantum Lower Bounds for the Collision and the Element Distinctness Problems. J. ACM, 51(4):595-605, July 2004. URL:
  3. A. Ambainis. Understanding Quantum Algorithms via Query Complexity. ArXiv e-prints, December 2017. URL:
  4. Andris Ambainis. Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range. Theory of Computing, 1(3):37-46, 2005. URL:
  5. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum Lower Bounds by Polynomials. J. ACM, 48(4):778-797, July 2001. URL:
  6. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and Weaknesses of Quantum Computing. SIAM Journal of Computing, 26(5):1510-1523, October 1997. URL:
  7. D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 439(1907):553-558, 1992. URL:
  8. Samuel Kutin. Quantum Lower Bound for the Collision Problem with Small Range. Theory of Computing, 1(2):29-36, 2005. URL:
  9. Peter W. Shor. Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In IEEE Symposium on Foundations of Computer Science, pages 124-134, 1994. URL:
  10. Daniel R. Simon. On the Power of Quantum Cryptography. In 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, USA, 20-22 November 1994, pages 116-123, 1994. URL:
  11. Mark Zhandry. A Note on the Quantum Collision and Set Equality Problems. Quantum Info. Comput., 15(7-8):557-567, May 2015. URL:
  12. Mark Zhandry. How to Record Quantum Queries, and Applications to Quantum Indifferentiability. IACR Cryptology ePrint Archive, 2018:276, 2018. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail