Improved Approximate Degree Bounds for k-Distinctness

Authors Nikhil S. Mande, Justin Thaler, Shuchen Zhu

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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.

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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


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