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.2020.9
URN: urn:nbn:de:0030-drops-120687
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12068/
Go to the corresponding LIPIcs Volume Portal


Ambainis, Andris ; Larka, Nikita

Quantum Algorithms for Computational Geometry Problems

pdf-format:
LIPIcs-TQC-2020-9.pdf (0.6 MB)


Abstract

We study quantum algorithms for problems in computational geometry, such as Point-On-3-Lines problem. In this problem, we are given a set of lines and we are asked to find a point that lies on at least 3 of these lines. Point-On-3-Lines and many other computational geometry problems are known to be 3Sum-Hard. That is, solving them classically requires time Ω(n^{2-o(1)}), unless there is faster algorithm for the well known 3Sum problem (in which we are given a set S of n integers and have to determine if there are a, b, c ∈ S such that a + b + c = 0). Quantumly, 3Sum can be solved in time O(n log n) using Grover’s quantum search algorithm. This leads to a question: can we solve Point-On-3-Lines and other 3Sum-Hard problems in O(n^c) time quantumly, for c<2? We answer this question affirmatively, by constructing a quantum algorithm that solves Point-On-3-Lines in time O(n^{1 + o(1)}). The algorithm combines recursive use of amplitude amplification with geometrical ideas. We show that the same ideas give O(n^{1 + o(1)}) time algorithm for many 3Sum-Hard geometrical problems.

BibTeX - Entry

@InProceedings{ambainis_et_al:LIPIcs:2020:12068,
  author =	{Andris Ambainis and Nikita Larka},
  title =	{{Quantum Algorithms for Computational Geometry Problems}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{9:1--9:10},
  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/2020/12068},
  URN =		{urn:nbn:de:0030-drops-120687},
  doi =		{10.4230/LIPIcs.TQC.2020.9},
  annote =	{Keywords: Quantum algorithms, quantum search, computational geometry, 3Sum problem, amplitude amplification}
}

Keywords: Quantum algorithms, quantum search, computational geometry, 3Sum problem, amplitude amplification
Collection: 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)
Issue Date: 2020
Date of publication: 08.06.2020


DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI