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.CCC.2020.16
URN: urn:nbn:de:0030-drops-125681
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12568/
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Aaronson, Scott ; Chia, Nai-Hui ; Lin, Han-Hsuan ; Wang, Chunhao ; Zhang, Ruizhe

On the Quantum Complexity of Closest Pair and Related Problems

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LIPIcs-CCC-2020-16.pdf (0.8 MB)

Abstract

The closest pair problem is a fundamental problem of computational geometry: given a set of n points in a d-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in O(n log n) time in constant dimensions (i.e., when d = O(1)). This paper asks and answers the question of the problem’s quantum time complexity. Specifically, we give an Õ(n^(2/3)) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference.
In polylog(n) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover’s algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover’s algorithm is optimal for CNF-SAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an n^o(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.

BibTeX - Entry

@InProceedings{aaronson_et_al:LIPIcs:2020:12568,
  author =	{Scott Aaronson and Nai-Hui Chia and Han-Hsuan Lin and Chunhao Wang and Ruizhe Zhang},
  title =	{{On the Quantum Complexity of Closest Pair and Related Problems}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{16:1--16:43},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Shubhangi Saraf},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12568},
  URN =		{urn:nbn:de:0030-drops-125681},
  doi =		{10.4230/LIPIcs.CCC.2020.16},
  annote =	{Keywords: Closest pair, Quantum computing, Quantum fine grained reduction, Quantum strong exponential time hypothesis, Fine grained complexity}
}

Keywords: Closest pair, Quantum computing, Quantum fine grained reduction, Quantum strong exponential time hypothesis, Fine grained complexity
Collection: 35th Computational Complexity Conference (CCC 2020)
Issue Date: 2020
Date of publication: 17.07.2020

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