Quantum algorithms for Hopcroft's problem

TL;DR

We address Hopcroft's problem, deciding whether any point lies on any line among $n$ lines and $n$ points in the plane, a problem classically solvable in $O(n^{4/3})$ time. The authors present two quantum algorithms that achieve $ ilde{O}(n^{5/6})$ time by accelerating core geometric data structures: (i) partition-tree based backtracking to speed up hyperplane emptiness queries, and (ii) a quantum walk on a Johnson graph using a history-independent line-arrangement data structure built from skip lists. In 2D, the first method yields $O(n^{1/3} m^{1/2} ext{log } n)$ when $m olinebreak \ge n^{2/3}$ and $O(n^{1/2} m^{1/4} ext{log } n)$ otherwise (with $n=m$ giving $O(n^{5/6} ext{log } n)$), while the second method achieves $O(n^{1/3} m^{1/2} ext{log}^6 n)$ in similar regimes and $O(n^{5/6} ext{log}^6 n)$ for $n=m$, plus analogous bounds when $m$ and $n$ differ. Together, these results demonstrate concrete quantum speedups for fundamental geometric data-structures and provide techniques (backtracking speedups and history-independent dynamic structures) that may inform a broader class of geometric quantum algorithms. The work highlights the potential of quantum algorithms to accelerate core geometric primitives and offers data-structure innovations (skip-list based line-arrangement representations) with potential applicability beyond Hopcroft's problem.

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.

Figures (5)

• Figure 2: The quantum time complexity of Hopcroft's problem in $2$ dimensions on $m$ points and $n$ lines, assuming $m \leq n$. The red line shows the query complexity (Theorem \ref{['thm:lower-bound']}); the blue line shows the complexity of the quantum algorithm based on the partition tree (Theorem \ref{['thm:speedup-backtracking']}); the green line shows the complexity of the quantum walk algorithm with the line arrangement data structure (Theorem \ref{['thm:algo2']}).
• Figure 3: An example of a skip list.
• Figure 4: Path points of a line intersection.
• Figure 5: New edges along the inserted line.
• Figure 6: Updated levels at the intersection of an old line $\ell_j$ with the new line $\ell_i$.

Theorems & Definitions (15)

• Theorem 1: Grover's search with bounded-error inputs ABBLS23HMDw03
• Theorem 2
• proof
• Lemma 3: Hyperplane emptiness query procedure
• proof
• Theorem 4: Partition tree Cha12
• Theorem 5: Quantum algorithm for backtracking Montanaro15AK17
• Theorem 6
• proof
• Theorem 7