Quantum Algorithms for Hopcroft's problem

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. Finally, we examine the connections between Hopcroft’s problem and other computational problems via fine-grained complexity. For example, we show a conditional \(\Omega (n^{3/4})\) time lower bound on Hopcroft’s problem in 5 dimensions based on the quantum analogue of a classical hardness conjecture, which is stronger than the (optimal) \(\Theta (n^{2/3})\) query complexity bounds.