Quantum algorithms for computational geometry problems

TL;DR

This paper presents a quantum algorithm that significantly improves the runtime for solving 3SUM-HARD computational geometry problems, including POINT-ON-3-LINES, from near-quadratic to near-linear time.

Contribution

It introduces a quantum algorithm that solves 3SUM-HARD geometry problems in near-linear time, surpassing classical lower bounds and previous quantum approaches.

Findings

Quantum algorithm solves POINT-ON-3-LINES in O(n^{1 + o(1)}) time.

The same approach applies to other 3SUM-HARD geometric problems.

Demonstrates quantum advantage in computational geometry.

Abstract

We study quantum algorithms for problems in computational geometry, such as POINT-ON-3-LINES problem. In this problem, we are given a set of lines and we are asked to find a point that lies on at least $3$ of these lines. POINT-ON-3-LINES and many other computational geometry problems are known to be 3SUM-HARD. That is, solving them classically requires time $\Omega(n^{2-o(1)})$, unless there is faster algorithm for the well known 3SUM problem (in which we are given a set $S$ of $n$ integers and have to determine if there are $a, b, c \in S$ such that $a + b + c = 0$). Quantumly, 3SUM can be solved in time $O(n \log n)$ using Grover's quantum search algorithm. This leads to a question: can we solve POINT-ON-3-LINES and other 3SUM-HARD problems in $O(n^c)$ time quantumly, for $c<2$? We answer this question affirmatively, by constructing a quantum algorithm that solves POINT-ON-3-LINES in time $O(n^{1 + o(1)})$. The algorithm combines recursive use of amplitude amplification with geometrical ideas. We show that the same ideas give $O(n^{1 + o(1)})$ time algorithm for many 3SUM-HARD geometrical problems.