Ambainis, Andris ;
Larka, Nikita
Quantum Algorithms for Computational Geometry Problems
Abstract
We study quantum algorithms for problems in computational geometry, such as PointOn3Lines 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. PointOn3Lines and many other computational geometry problems are known to be 3SumHard. That is, solving them classically requires time Ω(n^{2o(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 PointOn3Lines and other 3SumHard problems in O(n^c) time quantumly, for c<2? We answer this question affirmatively, by constructing a quantum algorithm that solves PointOn3Lines 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 3SumHard 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 = {9:19:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771467},
ISSN = {18688969},
year = {2020},
volume = {158},
editor = {Steven T. Flammia},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12068},
URN = {urn:nbn:de:0030drops120687},
doi = {10.4230/LIPIcs.TQC.2020.9},
annote = {Keywords: Quantum algorithms, quantum search, computational geometry, 3Sum problem, amplitude amplification}
}
08.06.2020
Keywords: 

Quantum algorithms, quantum search, computational geometry, 3Sum problem, amplitude amplification 
Seminar: 

15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Issue date: 

2020 
Date of publication: 

08.06.2020 