Matching Triangles and Triangle Collection: Hardness based on a Weak Quantum Conjecture

• URL: http://arxiv.org/abs/2207.11068v1
• Date: Fri, 22 Jul 2022 13:16:50 GMT
• Title: Matching Triangles and Triangle Collection: Hardness based on a Weak Quantum Conjecture
• Authors: Andris Ambainis, Harry Buhrman, Koen Leijnse, Subhasree Patro, Florian Speelman
• Abstract summary: We show that an $n1.5-epsilon$ time quantum algorithm for either of these two graph problems would imply faster quantum algorithms for k-SAT, 3SUM, and APSP. We also present quantum algorithms that require careful use of data structures and Ambainis' variable time search.
• Score: 0.8924669503280332
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Classically, for many computational problems one can conclude time lower bounds conditioned on the hardness of one or more of key problems: k-SAT, 3SUM and APSP. More recently, similar results have been derived in the quantum setting conditioned on the hardness of k-SAT and 3SUM. This is done using fine-grained reductions, where the approach is to (1) select a key problem $X$ that, for some function $T$, is conjectured to not be solvable by any $O(T(n)^{1-\epsilon})$ time algorithm for any constant $\epsilon > 0$ (in a fixed model of computation), and (2) reduce $X$ in a fine-grained way to these computational problems, thus giving (mostly) tight conditional time lower bounds for them. Interestingly, for Delta-Matching Triangles and Triangle Collection, classical hardness results have been derived conditioned on hardness of all three mentioned key problems. More precisely, it is proven that an $n^{3-\epsilon}$ time classical algorithm for either of these two graph problems would imply faster classical algorithms for k-SAT, 3SUM and APSP, which makes Delta-Matching Triangles and Triangle Collection worthwhile to study. In this paper, we show that an $n^{1.5-\epsilon}$ time quantum algorithm for either of these two graph problems would imply faster quantum algorithms for k-SAT, 3SUM, and APSP. We first formulate a quantum hardness conjecture for APSP and then present quantum reductions from k-SAT, 3SUM, and APSP to Delta-Matching Triangles and Triangle Collection. Additionally, based on the quantum APSP conjecture, we are also able to prove quantum lower bounds for a matrix problem and many graph problems. The matching upper bounds follow trivially for most of them, except for Delta-Matching Triangles and Triangle Collection for which we present quantum algorithms that require careful use of data structures and Ambainis' variable time search.

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