Quantum Algorithm Achieves Scalable Maximum Independent Set Solution Using 20 Qubits

Researchers are tackling the computationally challenging Maximum Independent Set (MIS) problem with a novel hybrid quantum-classical approach. Elisabeth Wybo, Jami Rönkkö, and Olli Hirviniemi, all from IQM Quantum Computers, alongside Jernej Rudi Finžgar and Martin Leib et al., demonstrate a scalable algorithm combining the Quantum Approximate Optimization Algorithm (QAOA) with a classical greedy method. This work is significant because it avoids the need for complex, instance-specific quantum parameter training, utilising pre-computed QAOA angles to guide a greedy search for increasingly large independent sets. They have successfully implemented and tested this method on a 20-qubit superconducting device, finding independent sets in graphs containing thousands of nodes, and simulations suggest substantial performance gains over purely classical algorithms, paving the way for practical quantum-enhanced optimisation.

The research team achieved this by leveraging pre-computed QAOA angles, originating from depth-p QAOA circuits on regular trees, to calculate local expectation values and subsequently guide sequential greedy decisions in building an independent set. This innovative hybrid method maintains shallow quantum circuits and crucially avoids instance-specific parameter training, positioning it as a strong candidate for implementation on existing quantum hardware. Researchers successfully implemented the algorithm on a 20 qubit IQM superconducting device, finding independent sets within graphs containing thousands of nodes.

To evaluate performance beyond current quantum capabilities, the study employed tensor network simulations, comparing the results to established classical heuristics. Findings reveal that even at a low depth of p = 4, the quantum-enhanced greedy method significantly surpasses purely classical greedy baselines and more complex approximation algorithms. The algorithm’s modular structure and relatively low quantum resource demands make it a compelling option for scalable, hybrid optimization in the NISQ era and beyond. This work establishes a new approach to utilizing quantum computation not as a complete solver, but as a subroutine to enhance existing classical heuristics.

The central idea behind this breakthrough is to improve the decisions made by greedy algorithms, which, while simple and scalable, can overlook intricate problem structures. By integrating QAOA, the team provides structured guidance derived from entangled quantum states to inform each greedy step, allowing the classical algorithm to retain control over the iterative reduction of the problem. A key contribution of this work is an efficient and scalable version of the quantum-enhanced greedy algorithm that circumvents the need for variational parameter optimization, maintaining an overall linear-time complexity. Avoiding the challenges of parameter optimization, which are often hindered by non-convex energy landscapes and barren plateaus, was achieved by employing fixed-angle QAOA circuits with parameters derived from regular tree models.

This allows for a plug-and-play implementation of QAOA without requiring instance-specific tuning, resulting in a constant number of local expectation values needing evaluation for each greedy update. Consequently, the total runtime of the algorithm scales linearly with the number of nodes in the problem graph. Tensor network simulations were used to benchmark the method, enabling the study of shallow-depth regimes and revealing the exponential cost in p induced by light-cone growth. Numerical simulations on random 3-regular graphs consistently demonstrated superior performance compared to the classical greedy baseline, a trend further validated by experimental implementation on the IQM Garnet quantum processor at depths p = 2 and p = 3. Finally, comparisons with the linear-time prioritized search algorithm of Marino et al. The research team engineered a method leveraging pre-computed QAOA angles, originating from depth-p QAOA circuits on regular trees, to calculate local expectation values and subsequently guide sequential greedy decisions that progressively construct an independent set. This innovative approach maintains shallow quantum circuits and circumvents instance-specific parameter training, rendering it suitable for implementation on contemporary quantum hardware. Experiments employed a 20 qubit IQM superconducting device to determine independent sets within graphs containing thousands of nodes, demonstrating practical application of the algorithm.

Researchers harnessed tensor network simulations to assess the algorithm’s performance beyond the limitations of current quantum hardware, facilitating comparison with established classical heuristics. The study pioneered a technique where, even at a low depth of p = 4, the quantum-enhanced greedy method demonstrably outperformed purely classical greedy baselines and more complex approximation algorithms. This performance boost highlights the potential of combining quantum and classical approaches for optimization problems. The team bypassed the challenge of variational parameter optimization by utilising fixed-angle QAOA circuits, deriving parameters from regular tree models resembling Bethe lattice structures.

This design choice is particularly well-suited for large d-regular graphs exhibiting local tree-like characteristics, enabling a plug-and-play implementation of QAOA without instance-specific tuning. Each greedy update necessitates the evaluation of a constant number of local expectation values alongside a constant amount of classical post-processing, ensuring a linear time complexity of O(N) with N representing the number of nodes in the problem graph. Furthermore, the modular structure of the algorithm and its relatively low quantum resource requirements position it as a promising candidate for scalable, hybrid optimization within the NISQ era and beyond. The work demonstrates that QAOA can function effectively as a subroutine enhancing classical heuristics, rather than a standalone optimizer, offering a pragmatic pathway towards quantum utility in the near term. The team implemented this algorithm on a 20 qubit IQM superconducting device, successfully finding independent sets in graphs containing thousands of nodes. Experiments revealed that pre-computed QAOA angles, derived from depth-QAOA circuits on regular trees, were effectively used to compute local expectation values and inform sequential greedy decisions. This hybrid approach maintains shallow quantum circuits and avoids instance-specific parameter training, making it suitable for current quantum hardware.

Researchers performed tensor network simulations to evaluate the algorithm’s performance beyond the limitations of existing quantum hardware and compared it to established classical heuristics. Results demonstrate that even at a low depth of p = 4, the quantum-enhanced greedy method significantly outperforms purely classical greedy baselines and more sophisticated approximation algorithms. Specifically, the team observed improved performance up to N ≈ 5000 nodes when compared to the linear-time prioritized search algorithm of Marino et al, despite utilizing only shallow circuits to estimate local expectation values. Measurements confirm a substantial benefit from incorporating quantum components into classical optimization strategies, even in the near term.

The modular structure of the algorithm and its relatively low quantum resource requirements position it as a compelling candidate for scalable, hybrid optimization in the NISQ era and beyond. Scientists achieved a linear-time complexity, O(N), with N representing the number of nodes in the problem graph, due to the use of fixed-angle QAOA circuits derived from regular tree models. Each greedy update required the evaluation of a constant number of local expectation values and a constant amount of classical post-processing. Tests prove that this approach partially mitigates current hardware limitations that prevent the use of much deeper circuits required by a standalone quantum algorithm.

Data shows that the quantum-enhanced greedy algorithm consistently outperformed the classical greedy baseline in numerical simulations on random 3-regular graphs. The breakthrough delivers a method where the quantum processor provides structured guidance derived from an entangled quantum state, while the classical algorithm controls the iterative reduction of the problem. The method utilises pre-calculated QAOA angles, obtained from analysing regular trees, to determine local expectation values and guide sequential greedy decisions in constructing an independent set. This approach employs shallow quantum circuits and avoids the need for instance-specific parameter training, making it suitable for implementation on current quantum hardware. Researchers successfully implemented this algorithm on a 20-qubit IQM superconducting device, finding independent sets in graphs containing thousands of nodes, and validated its performance using tensor network simulations.

Results demonstrate that, even at low QAOA depths, the quantum-enhanced greedy method surpasses purely classical greedy algorithms and established approximation techniques. The algorithm’s modular design and limited quantum resource requirements position it as a viable option for scalable, hybrid optimisation both presently and in the future. The authors acknowledge that classical simulation remains feasible at low depths using tensor-network methods, but computational cost increases rapidly with QAOA depth, potentially creating a regime where quantum processors are essential for efficient subgraph evaluation. They also note that while simulated annealing can achieve comparable results, it lacks the principled exploitation of structural locality inherent in their quantum approach. Future work may extend this class of non-variational, quantum-enhanced methods to other combinatorial optimisation problems on sparse graphs, such as MaxCut, minimum vertex cover, and constraint satisfaction problems, by leveraging local expectation values within classical heuristics. These findings suggest that even modest quantum resources can offer valuable guidance for hybrid strategies, paving the way for improved quantum-classical heuristics for MIS and related challenges.