A hybrid quantum-classical approach can dramatically expand the set of solutions a quantum processor generates for optimisation problems. The idea is simple: quantum hardware supplies the initial solutions and an entirely classical algorithm expands them, increasing the diversity of device measurements by up to several orders of magnitude. The method is applied to the maximum independent set problem, relevant in graph theory with applications in communication networks and logistics.

The starting point is a known limitation of quantum optimisation devices: while they produce high-quality solutions, their sampling capacity is restricted, which is problematic for applications that need a large variety of uncorrelated optimal solutions. The authors turn to the Houdayer cluster update algorithm, a classical Monte Carlo technique used in frustrated systems, to systematically expand the solution sets generated by the quantum computer.

Validation was carried out on 43 problem instances solved with QuEra's Aquila analog annealing device. Because exact ground-state degeneracies are known for these instances, the recovered fraction of the solution manifold can be measured directly rather than merely bounded. Post-processing recovers a median of 21% of the ground-state manifold, and all of it for six instances. The diversity amplification factor — the increase in the number of distinct solutions at a given set size — reaches a maximum of 11,923×, with a median of 24.3×.

In some cases, moreover, the classical post-processing found independent sets larger than those found by the quantum hardware, improving not just diversity but solution quality as well. The method is non-ergodic and cannot guarantee complete manifold coverage, but it consistently discovers more high-quality configurations than quantum sampling alone. The work shows how simple classical post-processing can increase the utility of quantum optimisation results.