In this work we study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. Given $n$ points and $n$ lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n^{4/3})$ time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity $\widetilde O(n^{5/6})$. The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.

翻译：本文研究计算几何基本问题——霍普克罗夫特问题的量子算法。给定平面上的𝑛个点和𝑛条直线，任务是判断是否存在点线关联。该问题的经典复杂度已被充分研究，已知最优算法运行时间为$O(n^{4/3})$，且在部分受限场景中存在匹配的下界。我们给出了两种时间复杂度为$\widetilde O(n^{5/6})$的不同量子算法。第一种算法基于划分树与量子回溯算法。第二种算法采用量子游走，并利用一种与历史无关的动态数据结构存储线排列，该结构支持高效的点定位查询。当点与直线数量不同时，基于量子游走的算法具有渐进更优的复杂度。上述数据结构所实现的量子加速可能对其他几何问题具有参考价值。