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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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