Quantum algorithms for computational geometry problems

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 ?(n²-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 ? 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(n¹⁺o⁽¹⁾). The algorithm combines recursive use of amplitude amplification with geometrical ideas. We show that the same ideas give O(n¹⁺o⁽¹⁾) time algorithm for many 3Sum-Hard geometrical problems.