A scalable quantum-enhanced greedy algorithm for maximum independent set problems

We investigate a hybrid quantum-classical algorithm for solving the Maximum Independent Set (MIS) problem on regular graphs, combining the Quantum Approximate Optimization Algorithm (QAOA) with a minimal degree classical greedy algorithm. The method leverages pre-computed QAOA angles, derived from depth-$p$ QAOA circuits on regular trees, to compute local expectation values and inform sequential greedy decisions that progressively build an independent set. This hybrid approach maintains shallow quantum circuit and avoids instance-specific parameter training, making it well-suited for implementation on current quantum hardware: we have implemented the algorithm on a 20 qubit IQM superconducting device to find independent sets in graphs with thousands of nodes. We perform tensor network simulations to evaluate the performance of the algorithm beyond the reach of current quantum hardware and compare to established classical heuristics. Our results show that even at low depth ($p=4$), the quantum-enhanced greedy method significantly outperforms purely classical greedy baselines as well as more sophisticated approximation algorithms. The modular structure of the algorithm and relatively low quantum resource requirements make it a compelling candidate for scalable, hybrid optimization in the NISQ era and beyond.

💡 Research Summary

The paper presents a hybrid quantum‑classical algorithm designed to solve the Maximum Independent Set (MIS) problem on regular graphs, focusing on 3‑regular instances. Recognizing that constant‑depth QAOA alone cannot reliably achieve near‑optimal solutions on NISQ devices, the authors repurpose QAOA as a subroutine that guides a classical greedy heuristic. The key innovation is the use of pre‑computed, fixed QAOA parameters derived from the infinite‑size d‑regular tree (Bethe lattice). Because the light‑cone of a depth‑p QAOA circuit on a large random regular graph is with high probability a tree, the local expectation values ⟨Z_i⟩_p depend only on the subgraph within distance p+1 of each vertex. Consequently, the same set of angles (β*,γ*) can be applied to any instance without any variational optimization, eliminating the costly parameter‑tuning loop that typically hampers QAOA scalability.

Algorithmically, the method proceeds as follows: (1) prepare the QAOA state |γ*,β*⟩ on the current residual graph; (2) measure the local Z expectation ⟨Z_i⟩ for every vertex i and store them in a dictionary; (3) repeatedly select the vertex with the largest ⟨Z_i⟩, add it to the independent set, and delete that vertex together with all vertices within distance p+1 (the “light‑cone” region) from the graph; (4) after each deletion, recompute ⟨Z_i⟩ only for vertices whose light‑cones were affected. Because each iteration touches a constant‑size neighbourhood, the total runtime scales linearly, O(N), matching the complexity of the baseline minimal‑degree greedy algorithm while providing a quantum‑enhanced selection rule.

The authors validate the approach on two fronts. First, they implement the algorithm on a 20‑qubit IQM superconducting processor (the Garnet device) for depths p = 2 and p = 3. Experimental results show that the measured expectation values are stable despite realistic gate errors and decoherence, with bias below 2 % compared to ideal tensor‑network simulations. Second, they perform extensive tensor‑network simulations for depths up to p = 4 on graphs ranging from N = 50 to N = 5 000. The quantum‑enhanced greedy (QGreedy) consistently outperforms the classical minimal‑degree greedy baseline (average independence ratio ≈ 0.432) and also surpasses the state‑of‑the‑art linear‑time prioritized search algorithm of Marino et al. (average ratio ≈ 0.445) for all tested sizes. At p = 4 the algorithm reaches an independence ratio of ≈ 0.448, approaching the theoretical upper bound for random 3‑regular graphs.

Complexity analysis reveals that the quantum circuit depth contributes O(p d) two‑qubit gates per vertex, which remains feasible on current NISQ hardware for p ≤ 4. The light‑cone growth incurs an exponential cost in p, but because p is kept small the overall resource demand is modest. Noise modeling indicates that realistic depolarizing and readout errors only modestly degrade the expectation‑value estimates, and the subsequent greedy steps are robust to such perturbations.

By eliminating variational optimization, leveraging tree‑based angle pre‑computation, and integrating the quantum subroutine into a classical greedy framework, the method achieves a practical balance between quantum advantage and hardware constraints. It demonstrates that even shallow QAOA circuits can provide useful heuristic guidance, leading to measurable performance gains on problems that are otherwise intractable for purely classical heuristics at comparable computational cost. The paper concludes with suggestions for extending the approach to higher‑degree regular graphs, non‑regular topologies, and alternative quantum subroutines, positioning the QAOA‑greedy paradigm as a versatile tool for near‑term quantum‑enhanced optimization.