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: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12068/
Ambainis, Andris ;
Larka, Nikita
Quantum Algorithms for Computational Geometry Problems
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}
}
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Keywords: |
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Quantum algorithms, quantum search, computational geometry, 3Sum problem, amplitude amplification |
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Collection: |
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15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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08.06.2020 |
