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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.31
URN: urn:nbn:de:0030-drops-156273
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15627/
Buhrman, Harry ;
Loff, Bruno ;
Patro, Subhasree ;
Speelman, Florian
Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks
Abstract
Many computational problems are subject to a quantum speed-up: one might find that a problem having an O(n³)-time or O(n²)-time classic algorithm can be solved by a known O(n^{1.5})-time or O(n)-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible?
The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro and Speelman [Buhrman et al., 2021], and by Aaronson, Chia, Lin, Wang, and Zhang [Aaronson et al., 2020].
In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time O(n^{2-ε}), for any ε > 0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear O(n^{1-ε})-time quantum algorithms for the 3SUM problem.
Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems.
These results are proven by adapting to the quantum setting known classical fine-grained reductions from the 3SUM problem. This adaptation is not trivial, however, since the original classical reductions require pre-processing the input in various ways, e.g. by sorting it according to some order, and this pre-processing (provably) cannot be done in sublinear quantum time.
We overcome this bottleneck by combining a quantum walk with a classical dynamic data-structure having a certain "history-independence" property. This type of construction has been used in the past to prove upper bounds, and here we use it for the first time as part of a reduction. This general proof strategy allows us to prove tight lower bounds on several computational-geometry problems, on Convolution-3SUM and on the 0-Edge-Weight-Triangle problem, conditional on the Quantum-3SUM-Conjecture.
We believe this proof strategy will be useful in proving tight (conditional) lower-bounds, and limits on quantum speed-ups, for many other problems.
BibTeX - Entry
@InProceedings{buhrman_et_al:LIPIcs.ITCS.2022.31,
author = {Buhrman, Harry and Loff, Bruno and Patro, Subhasree and Speelman, Florian},
title = {{Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks}},
booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
pages = {31:1--31:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-217-4},
ISSN = {1868-8969},
year = {2022},
volume = {215},
editor = {Braverman, Mark},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15627},
URN = {urn:nbn:de:0030-drops-156273},
doi = {10.4230/LIPIcs.ITCS.2022.31},
annote = {Keywords: complexity theory, fine-grained complexity, 3SUM, computational geometry problems, data structures, quantum walk}
}
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Keywords: |
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complexity theory, fine-grained complexity, 3SUM, computational geometry problems, data structures, quantum walk |
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Collection: |
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13th Innovations in Theoretical Computer Science Conference (ITCS 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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25.01.2022 |
