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 $\widetilde O(r\sqrt{mn}/\varepsilon)$. 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.

Lay Summary:

Hypergraphs — mathematical structures that model complex multi-way relationships, such as group collaborations in social networks or chemical bonds in molecules — are critical tools for modern algorithm design. However, efficiently simplifying them (a process called hypergraph sparsification) while retaining their essential properties remains a foundational challenge in theoretical computer science. Classical methods for this task scale linearly for large problems, and whether quantum algorithms could solve it faster has been an open question for years.In this theoretical result, we propose the first quantum algorithm for hypergraph sparsification. By harnessing quantum superposition, our method achieves a runtime that provably outperforms classical approaches for common cases. Our algorithm matches fundamental quantum speed limits (lower bounds) for constant-rank hypergraphs — a result that establishes quantum computing’s inherent advantage for this problem.This work advances our understanding of quantum algorithms for hypergraph problems, and paves the way for faster quantum techniques in optimization and machine learning—fields where hypergraphs play a crucial role.