Quantum Speedup for Some Geometric 3SUM-Hard Problems and Beyond

TL;DR

The paper addresses quantum speedups for geometric $3SUM$-hard problems by extending the Ambainis–Larka framework of searching over plane subdivisions. It introduces a Recursive-Quantum-Search (RQS) paradigm that decomposes problems into subproblems, leverages Grover search for cross-subproblem solutions, and requires controlled subproblem sizes to achieve near-linear quantum time $O(n^{1+o(1)})$ for each target problem. The authors obtain $O(n^{1+o(1)})$ quantum algorithms for $q$-Area Triangle, $q$-Points in a Disk, and Interval Containment, and generalize the approach to a broad Pair/Tuple Search setting with applications to Polygon Cutting and Disjoint Projections, as well as higher-dimensional generalizations. These results demonstrate that a wide class of geometric $3SUM$-hard problems admit efficient quantum speedups and provide a flexible framework for multi-structure search in computational geometry.

Abstract

The classical 3SUM conjecture states that the class of 3SUM-hard problems does not admit a truly subquadratic $O(n^{2-δ})$-time algorithm, where $δ>0$, in classical computing. The geometric 3SUM-hard problems have widely been studied in computational geometry and recently, these problems have been examined under the quantum computing model. For example, Ambainis and Larka [TQC'20] designed a quantum algorithm that can solve many geometric 3SUM-hard problems in $O(n^{1+o(1)})$-time, whereas Buhrman [ITCS'22] investigated lower bounds under quantum 3SUM conjecture that claims there does not exist any sublinear $O(n^{1-δ})$-time quantum algorithm for the 3SUM problem. The main idea of Ambainis and Larka is to formulate a 3SUM-hard problem as a search problem, where one needs to find a point with a certain property over a set of regions determined by a line arrangement in the plane. The quantum speed-up then comes from the application of the well-known quantum search technique called Grover search over all regions. This paper further generalizes the technique of Ambainis and Larka for some 3SUM-hard problems when a solution may not necessarily correspond to a single point or the search regions do not immediately correspond to the subdivision determined by a line arrangement. Given a set of $n$ points and a positive number $q$, we design $O(n^{1+o(1)})$-time quantum algorithms to determine whether there exists a triangle among these points with an area at most $q$ or a unit disk that contains at least $q$ points. We also give an $O(n^{1+o(1)})$-time quantum algorithm to determine whether a given set of intervals can be translated so that it becomes contained in another set of given intervals and discuss further generalizations.

Figures (4)

• Figure 1: (a)--(b) Illustration for a point set and its corresponding lines in the dual plane. (c) Illustration for $T_k$, where $k=2$, with a face $R$ shown in blue shaded region. The dual and supporting edges are in blue and green, respectively. (d) Illustration for the zone of a supporting line.
• Figure 2: Illustration for the proof of Lemma \ref{['triang']}.
• Figure 3: (a) $P$. (b) $Q$. (c) $P\subset Q$. Illustration for (d) Remark \ref{['rem:point']} and (e) Theorem \ref{['thm:intervals']}.
• Figure 4: (a) Illustration for $F_1$ and $F_2$ in red and blue, respectively. (b) A face $F$. (c)--(d) Sweeping a line (shown in dashed orange) to subdivide $P$.

Theorems & Definitions (16)

• Theorem 2.1: Grover Search grover1996fast
• Theorem 2.2: Amplitude Amplification brassard2002quantum
• Theorem 2.3
• Lemma 2.4: Ambainis and Larka ambainis2020quantum
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3