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Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks Buhrman, Harry Loff, Bruno Patro, Subhasree Speelman, Florian complexity theory fine-grained complexity 3SUM computational geometry problems data structures quantum walk 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Harry Buhrman and Bruno Loff and Subhasree Patro and Florian Speelman 2022 Is Part Of LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2022.31 urn:nbn:de:0030-drops-156273 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.31 eng https://creativecommons.org/licenses/by/4.0/legalcode