Work

Quantum Algorithms for Hopcroft's problem

Year: 2026

Type: article

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. 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.

Source: ACM Transactions on Quantum Computing

DOI: 10.1145/3787461

Authors: Vladimirs Andrejevs, Aleksandrs Belovs, Jevgēnijs Vihrovs

Institution: University of Latvia

Language: English

Topic: Quantum Computing Algorithms and Architecture

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Funders: European Commission, QuantERA

Award: G5615427228