Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction

2025-04-08

AI Summary

This paper addresses the convex hull and maximal point set problems for two-dimensional sorted point sets, presenting the first non-recursive quantum algorithm for these tasks. Methodologically, it introduces a “Quantum Combine-and-Conquer” paradigm, strategically relocating critical quantum operations—such as quantum superposition, amplitude amplification, and output-sensitive query optimization—to the merge phase, thereby eliminating recursive overhead. It further proposes the first quantum adaptation of the “marriage-before-conquest” strategy. The resulting algorithm achieves, with high probability, a sublinear time complexity of Õ(√(nh)), where *n* is the input size and *h* the output size—breaking the classical Ω(*n*) lower bound for sorted inputs. This improves significantly over the optimal classical O(*n* log *n*) algorithm. The work establishes a novel quantum acceleration paradigm for computational geometry, demonstrating that sorting-aware quantum design can yield provable super-classical speedups for fundamental geometric problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the"marriage-before-conquest"technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in $ ilde{O}(sqrt{nh})$ time, w.h.p., where $h$ is the size of the output.

Problem

• Introduces quantum combine and conquer algorithm paradigm
• Applies to sublinear quantum convex hull construction
• Solves 2D maxima set problem efficiently

Innovation

• Quantum combine and conquer paradigm
• Non-recursive quantum algorithm design
• Sublinear time convex hull construction