Quantum Hypergraph Partitioning

TL;DR

This paper introduces a quantum approach to hypergraph partitioning that produces a probability distribution over solutions, demonstrating potential advantages over classical methods in distributional and fairness objectives.

Contribution

It develops QAOA-based quantum solvers for distributional hypergraph partitioning, connecting quantum algorithms with fairness and community detection problems.

Findings

QAOA can outperform classical algorithms on distributional hypergraph objectives

Quantum solutions naturally represent distributions over partitions

Experiments show advantages on real-world and synthetic hypergraphs

Abstract

Quantum optimization algorithms are inherently probabilistic, yet they are most often used to search for a single high-quality solution. In this paper, we instead study hypergraph partitioning problems in which the desired output is itself a probability distribution over partitions. We introduce a distributional perspective on hypergraph partitioning motivated by maximin and minimax objectives such as Fair Cut Cover, and we show how these objectives align naturally with the measurement distribution produced by QAOA. To motivate the formulation, we introduce a workforce-scheduling-inspired toy problem, the Greatest Expected Imbalance problem, in which the goal is to minimize the worst expected imbalance across hyperedges. We then develop QAOA-based quantum solvers that represent distributional solutions natively through quantum states, together with quadratic hypergraph objectives suitable for standard and multi-objective QAOA. These formulations connect balanced hypergraph partitioning, polarized community discovery, and distributional fairness under a unified quantum optimization framework. For comparison, we provide optimal polynomial-time classical approximation algorithms based on semidefinite programming and hyperplane rounding. Experiments on real-world and synthetic hypergraphs demonstrate that low-depth multi-angle QAOA can outperform these classical approximation baselines on the proposed objectives, highlighting the potential of quantum algorithms for optimization problems where the solution is a distribution rather than a single partition.