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Quantum Speedup for Hypergraph Sparsification

Chenghua Liu, Minbo Gao, Zhengfeng Ji, Mingsheng Ying

TL;DR

This paper resolves an open question by presenting the first quantum algorithm for hypergraph sparsification, producing an $\varepsilon$-spectral sparsifier of near-linear size for a hypergraph with $n$ vertices, $m$ hyperedges, and rank $r$ in time $\tilde{O}( r\sqrt{mn}/\varepsilon )$. The approach extends quantum graph sparsification techniques by introducing hyperedge leverage score overestimates computed via sparse underlying graphs, and leveraging quantum subroutines for sampling, sum estimation, and state preparation within a chaining framework. Key contributions include a sparsifier of size $\tilde{O}(n/\varepsilon^2)$ and sublinear-time hypergraph mincut and $s$-$t$ mincut algorithms, with stronger speedups in dense regimes ($m=\Omega(n^r)$). The results have broad implications for quantum-accelerated hypergraph-based ML tasks and optimization, offering sublinear dependence on the number of hyperedges in important problems. Overall, the work significantly advances quantum speeds for fundamental hypergraph tasks and opens avenues for further quantum improvements in hypergraph algorithms.

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.

Quantum Speedup for Hypergraph Sparsification

TL;DR

This paper resolves an open question by presenting the first quantum algorithm for hypergraph sparsification, producing an -spectral sparsifier of near-linear size for a hypergraph with vertices, hyperedges, and rank in time . The approach extends quantum graph sparsification techniques by introducing hyperedge leverage score overestimates computed via sparse underlying graphs, and leveraging quantum subroutines for sampling, sum estimation, and state preparation within a chaining framework. Key contributions include a sparsifier of size and sublinear-time hypergraph mincut and - mincut algorithms, with stronger speedups in dense regimes (). The results have broad implications for quantum-accelerated hypergraph-based ML tasks and optimization, offering sublinear dependence on the number of hyperedges in important problems. Overall, the work significantly advances quantum speeds for fundamental hypergraph tasks and opens avenues for further quantum improvements in hypergraph algorithms.

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 vertices, hyperedges, and rank , our algorithm outputs a near-linear size -spectral sparsifier in time . This algorithm matches the quantum lower bound for constant and demonstrates quantum speedup when compared with the state-of-the-art -time classical algorithm. As applications, our algorithm implies quantum speedups for computing hypergraph cut sparsifiers, approximating hypergraph mincuts and hypergraph - mincuts.
Paper Structure (19 sections, 21 theorems, 47 equations, 1 table, 2 algorithms)

This paper contains 19 sections, 21 theorems, 47 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Hypergraph Sparsification
  3. Main Results
  4. Techiques
  5. Related Works
  6. Preliminaries
  7. Notation
  8. Laplacian and Graph Sparsification
  9. Hypergraph Sparsification
  10. Quantum Computing and Speedup
  11. Quantum Algorithm for Leverage Score Overestimates
  12. Quantum Hypergraph Sparsification
  13. Applications
  14. Mincut
  15. $s$-$t$ mincut
  16. ...and 4 more sections

Key Result

Theorem 1.1

There exists a quantum algorithm that, given query access to a hypergraph $H =\left( V,E,w \right)$ with $\left\lvert V \right\rvert=n$, $\left\lvert E \right\rvert=m$, $w \in \mathbb{R}^E_{\geq 0}$, rank $r$ and $\varepsilon> 0$, outputs with high probabilityThroughout this paper, we say something

Theorems & Definitions (41)

  • Theorem 1.1: Informal version of \ref{['thm:quantum-hypergraph-sparsification']}
  • Definition 2.1: Laplacian
  • Definition 2.2: Graph Spectral Sparsifier
  • Definition 2.3: Effective Resistance
  • Definition 2.4: Energy
  • Definition 2.5: Hypergraph Spectral Sparsifier
  • Theorem 2.6: Quantum Graph Sparsification, Theorem 1 in apers2022quantum
  • Theorem 2.7: Preparing Many Copies of a Quantum State, Theorem 1 in hamoudi2022preparing
  • Corollary 2.8
  • Theorem 2.9: Quantum Sum Estimation, Lemma 3.1 in li2019sublinear
  • ...and 31 more