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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计算几何和其他问题的量子加速限制：通过量子游走的细粒度复杂性

许多计算问题都受到量子加速的影响：人们可能会发现具有 O(n^3) 时间或 O(n^2) 时间的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。自然会出现一个问题：量子加速可能达到多少？细粒度复杂性区域使我们能够基于某些自然的、经过充分研究的问题的推测硬度，证明各种计算问题复杂性的最佳下限。该理论最近在 Buhrman、Patro 和 Speelman (arXiv:1911.05686) 以及 Aaronson、Chia、Lin、Wang 和 Zhang (arXiv:1911.01973) 的两篇独立论文中扩展到量子设置。在本文中，我们进一步将细粒度复杂性理论扩展到量子设置。经典设置中的一个基本猜想表明，对于任何 a>0，3SUM 问题都无法在时间为 O(n^{2-a}) 的（经典）算法中解决。我们制定了一个类似的猜想，即 Quantum-3SUM-Conjecture，它指出对于 3SUM 问题不存在次线性 O(n^{1-b}) 时间量子算法。基于 Quantum-3SUM 猜想，我们展示了几个计算问题的量子算法时间复杂度的新下限。我们的大多数下界都是最优的，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有严格的限制。它指出对于 3SUM 问题不存在次线性 O(n^{1-b}) 时间量子算法。基于 Quantum-3SUM 猜想，我们展示了几个计算问题的量子算法时间复杂度的新下限。我们的大多数下界都是最优的，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有严格的限制。它指出对于 3SUM 问题不存在次线性 O(n^{1-b}) 时间量子算法。基于 Quantum-3SUM 猜想，我们展示了几个计算问题的量子算法时间复杂度的新下限。我们的大多数下界都是最优的，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有严格的限制。