A Quantum Genetic Algorithm Framework for the MaxCut Problem

TL;DR

This work tackles the NP-hard MaxCut problem by introducing a Quantum Genetic Algorithm (QGA) framework that merges Grover-based search with divide-and-conquer graph partitioning. The method recursively partitions graphs into subgraphs, encodes subproblem solutions in quantum registers, and uses a fitness oracle with Grover diffusion to amplify high-fitness configurations before contracting them into a global solution. Theoretical analysis highlights a quantum max-finding capability with $\mathcal{O}(\sqrt{N})$ iterations, and empirical results show the QGA achieving true optima on complete graphs and competitive performance on Erdős-Rényi graphs compared to Semidefinite Programming (SDP), with advantages expected as quantum hardware advances. The work demonstrates the potential of quantum-enhanced genetic strategies for scalable combinatorial optimization and outlines future directions including hardware deployment, circuit refinements, variational quantum approaches, and extensions to weighted or other NP-hard problems.

Abstract

The MaxCut problem is a fundamental problem in Combinatorial Optimization, with significant implications across diverse domains such as logistics, network design, and statistical physics. The algorithm represents innovative approaches that balance theoretical rigor with practical scalability. The proposed method introduces a Quantum Genetic Algorithm (QGA) using a Grover-based evolutionary framework and divide-and-conquer principles. By partitioning graphs into manageable subgraphs, optimizing each independently, and applying graph contraction to merge the solutions, the method exploits the inherent binary symmetry of MaxCut to ensure computational efficiency and robust approximation performance. Theoretical analysis establishes a foundation for the efficiency of the algorithm, while empirical evaluations provide quantitative evidence of its effectiveness. On complete graphs, the proposed method consistently achieves the true optimal MaxCut values, outperforming the Semidefinite Programming (SDP) approach, which provides up to 99.7\% of the optimal solution for larger graphs. On Erdős-Rényi random graphs, the QGA demonstrates competitive performance, achieving median solutions within 92-96\% of the SDP results. These results showcase the potential of the QGA framework to deliver competitive solutions, even under heuristic constraints, while demonstrating its promise for scalability as quantum hardware evolves.

Figures (1)

• Figure 1: Diagram of the proposed framework.