Quantum algorithms for Hopcroft's problem

TL;DR

This paper introduces two quantum algorithms that significantly improve the efficiency of solving Hopcroft's problem in computational geometry, achieving sublinear time complexity compared to classical methods.

Contribution

It presents the first quantum algorithms for Hopcroft's problem with complexities of O(n^{5/6}), surpassing classical algorithms, and introduces novel quantum data structures for geometric computations.

Findings

Quantum algorithms achieve O(n^{5/6}) time complexity.

Quantum walk-based algorithm outperforms classical in asymmetric cases.

Quantum data structures may benefit other geometric problems.

Abstract

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.