License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TQC.2020.2
URN: urn:nbn:de:0030-drops-120613
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12061/
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Mande, Nikhil S. ; Thaler, Justin ; Zhu, Shuchen

Improved Approximate Degree Bounds for k-Distinctness

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LIPIcs-TQC-2020-2.pdf (0.6 MB)

Abstract

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).

BibTeX - Entry

@InProceedings{mande_et_al:LIPIcs:2020:12061,
  author =	{Nikhil S. Mande and Justin Thaler and Shuchen Zhu},
  title =	{{Improved Approximate Degree Bounds for k-Distinctness}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{2:1--2:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Steven T. Flammia},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12061},
  URN =		{urn:nbn:de:0030-drops-120613},
  doi =		{10.4230/LIPIcs.TQC.2020.2},
  annote =	{Keywords: Quantum Query Complexity, Approximate Degree, Dual Polynomials, k-distinctness}
}

Keywords: Quantum Query Complexity, Approximate Degree, Dual Polynomials, k-distinctness
Collection: 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)
Issue Date: 2020
Date of publication: 08.06.2020

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