若干几何3SUM困难问题及更广问题的量子加速
Quantum Speedup for Some Geometric 3SUM-Hard Problems and Beyond
AI总结：
本文推广Ambainis-Larka的Grover搜索框架，为最小面积三角形、单位圆盘多点包含及区间平移包含等几何3SUM困难问题给出$O(n^{1+o(1)})$时间量子算法。
摘要：
经典3SUM猜想指出，在经典计算中，3SUM困难问题类别不存在真正次二次的$O(n^{2-δ})$时间算法，其中$δ>0$。几何3SUM困难问题在计算几何中已得到广泛研究，近来这些问题也在量子计算模型下受到检验。例如，Ambainis和Larka [TQC'20]设计了一种量子算法，能够以$O(n^{1+o(1)})$时间求解许多几何3SUM困难问题；而Buhrman [ITCS'22]则在量子3SUM猜想下研究了下界，该猜想声称3SUM问题不存在任何次线性的$O(n^{1-δ})$时间量子算法。Ambainis和Larka的主要思想是将一个3SUM困难问题表述为一个搜索问题：需要在由平面中的直线排列所确定的一组区域中，寻找具有某种性质的一个点。量子加速随后来自在所有区域上应用著名的量子搜索技术，即Grover搜索。\n本文将Ambainis和Larka的技术进一步推广到某些3SUM困难问题，在这些问题中，解不一定对应于单个点，或者搜索区域并不直接对应于由直线排列确定的细分。给定包含$n$个点的集合和一个正数$q$，我们设计了$O(n^{1+o(1)})$时间的量子算法，用于判定这些点中是否存在一个面积至多为$q$的三角形，或者是否存在一个包含至少$q$个点的单位圆盘。我们还给出了一个$O(n^{1+o(1)})$时间的量子算法，用于判定给定的一组区间是否可以平移，使其包含于另一组给定区间之中，并讨论了进一步的推广。
英文摘要：
The classical 3SUM conjecture states that the class of 3SUM-hard problems does not admit a truly subquadratic $O(n^{2-δ})$-time algorithm, where $δ>0$, in classical computing. The geometric 3SUM-hard problems have widely been studied in computational geometry and recently, these problems have been examined under the quantum computing model. For example, Ambainis and Larka [TQC'20] designed a quantum algorithm that can solve many geometric 3SUM-hard problems in $O(n^{1+o(1)})$-time, whereas Buhrman [ITCS'22] investigated lower bounds under quantum 3SUM conjecture that claims there does not exist any sublinear $O(n^{1-δ})$-time quantum algorithm for the 3SUM problem. The main idea of Ambainis and Larka is to formulate a 3SUM-hard problem as a search problem, where one needs to find a point with a certain property over a set of regions determined by a line arrangement in the plane. The quantum speed-up then comes from the application of the well-known quantum search technique called Grover search over all regions. This paper further generalizes the technique of Ambainis and Larka for some 3SUM-hard problems when a solution may not necessarily correspond to a single point or the search regions do not immediately correspond to the subdivision determined by a line arrangement. Given a set of $n$ points and a positive number $q$, we design $O(n^{1+o(1)})$-time quantum algorithms to determine whether there exists a triangle among these points with an area at most $q$ or a unit disk that contains at least $q$ points. We also give an $O(n^{1+o(1)})$-time quantum algorithm to determine whether a given set of intervals can be translated so that it becomes contained in another set of given intervals and discuss further generalizations.
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