计算几何和其他问题的量子加速限制：通过量子漫步实现细粒度复杂性
Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks
摘要：
许多计算问题都受到量子加速的影响：人们可能会发现，一个具有O（n^3）时间或O（n^2）时间经典算法的问题可以通过已知的O（n^1.5）时间或O（n）时间量子算法来解决。问题自然出现了：量子加速有多大可能？细粒度复杂性的区域允许我们证明各种计算问题的复杂性的最佳下限，基于某些自然的，经过充分研究的问题的复杂性。这个理论最近在Buhrman，Patro和Speelman（arXiv：1911.05686）以及Aaronson，Chia，Lin，Wang和Zhang（arXiv：1911.01973）的两篇独立论文中被扩展到量子环境。本文将细粒度复杂性理论进一步推广到量子环境。经典环境中的一个基本猜想是，对于任何a>0，3SUM问题都不能在O（n^{2-a}）的时间内通过（经典）算法解决。我们提出了一个类似的猜想，量子3SUM猜想，它指出不存在O（n^{1-b}）时间的量子算法的3SUM问题。基于量子3SUM猜想，我们给出了几个计算问题的量子算法时间复杂度的新下界。我们的大多数下限都是最优的，因为它们与已知的上限相匹配，因此它们意味着对这些问题可能的量子加速的严格限制。
英文摘要：
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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