On the quantum complexity of closest pair and related problems

Scott Aaronson, Nai Hui Chia, Han Hsuan Lin, Chunhao Wang, Ruizhe Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Scopus citations


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 Oe(n2/3) 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 no(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.

Original languageEnglish (US)
Title of host publication35th Computational Complexity Conference, CCC 2020
EditorsShubhangi Saraf
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771566
StatePublished - Jul 1 2020
Event35th Computational Complexity Conference, CCC 2020 - Virtual, Online, Germany
Duration: Jul 28 2020Jul 31 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference35th Computational Complexity Conference, CCC 2020
CityVirtual, Online

All Science Journal Classification (ASJC) codes

  • Software


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