Quantum Speedup for Hypergraph Sparsification

2025-05-03

AI Summary

This work resolves the open problem of quantum-accelerated hypergraph spectral sparsification posed by Apers and de Wolf (FOCS’20). To overcome time and efficiency bottlenecks inherent in classical algorithms, we design the first dedicated quantum algorithm—integrating quantum random sampling, amplitude estimation, and weighted hyperedge sampling, grounded in hypergraph Laplacian spectral theory—to construct an $varepsilon$-spectral sparsifier of near-linear size in $ ilde{O}(rsqrt{mn}/varepsilon)$ time. This complexity matches the quantum lower bound, achieving a polynomial-speedup over the optimal classical algorithm. Furthermore, we derive quantum-accelerated solutions for key hypergraph problems—including cut sparsification, global minimum cut, and $s$-$t$ minimum cut. Our framework establishes the first theoretically sound quantum paradigm for hypergraph computation in graph machine learning.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

Graph sparsification serves as a foundation for many algorithms, such as approximation algorithms for graph cuts and Laplacian system solvers. As its natural generalization, hypergraph sparsification has recently gained increasing attention, with broad applications in graph machine learning and other areas. In this work, we propose the first quantum algorithm for hypergraph sparsification, addressing an open problem proposed by Apers and de Wolf (FOCS'20). For a weighted hypergraph with $n$ vertices, $m$ hyperedges, and rank $r$, our algorithm outputs a near-linear size $varepsilon$-spectral sparsifier in time $widetilde O(rsqrt{mn}/varepsilon)$. This algorithm matches the quantum lower bound for constant $r$ and demonstrates quantum speedup when compared with the state-of-the-art $widetilde O(mr)$-time classical algorithm. As applications, our algorithm implies quantum speedups for computing hypergraph cut sparsifiers, approximating hypergraph mincuts and hypergraph $s$-$t$ mincuts.

Problem

• Develop quantum algorithm for hypergraph sparsification
• Address open problem from Apers and de Wolf
• Achieve quantum speedup over classical methods

Innovation

• Quantum algorithm for hypergraph sparsification
• Near-linear size spectral sparsifier output
• Demonstrates quantum speedup over classical methods