Quantum Speedup for Hypergraph Sparsification

TL;DR

This paper introduces the first quantum algorithm for hypergraph sparsification, achieving near-linear size spectral sparsifiers efficiently and demonstrating quantum speedup over classical methods, with applications in hypergraph cut problems.

Contribution

It presents the first quantum algorithm for hypergraph sparsification, solving an open problem and providing faster spectral sparsifier construction compared to classical algorithms.

Findings

Quantum algorithm outputs near-linear size spectral sparsifiers.

Achieves quantum speedup over classical algorithms for hypergraph sparsification.

Enables faster quantum algorithms for hypergraph cut problems.

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.