Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks

• URL: http://arxiv.org/abs/2106.02005v1
• Date: Thu, 3 Jun 2021 17:22:08 GMT
• Title: Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks
• Authors: Harry Buhrman, Bruno Loff, Subhasree Patro, Florian Speelman
• Abstract summary: We prove optimal 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Many computational problems are subject to a quantum speed-up: one might find that a problem having an O(n^3)-time or O(n^2)-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 (arXiv:1911.05686), and by Aaronson, Chia, Lin, Wang, and Zhang (arXiv:1911.01973). 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-a}), for any a>0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear O(n^{1-b})-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.

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