The paper discusses quantum speedup for some geometric 3SUM-hard problems. The key ideas are:
Modeling the problems as a point search problem over a subdivision of the plane with a small number of regions. This allows leveraging the Grover search technique for quantum speedup.
Showing how the technique of Ambainis and Larka can be adapted even for problems where a solution may not correspond to a single point or the search regions do not necessarily correspond to a subdivision determined by an arrangement of straight lines.
Presenting quantum algorithms for the following problems in O(n^(1+o(1))) time:
Generalizing the techniques to a "Pair Search Problem" and "d-Tuple Search Problem", which can be used to obtain quantum speedups for other problems like Polygon Cutting and Disjoint Projections.
The paper also discusses how to create a suitable subdivision of the plane or space to ensure the subproblem sizes are small enough to apply the Grover search technique effectively.
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by J. Mark Keil... at arxiv.org 04-09-2024
https://arxiv.org/pdf/2404.04535.pdfDeeper Inquiries