Quantum Hypergraph Partitioning

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.