A Quantum Genetic Algorithm Framework for the MaxCut Problem

TL;DR

This paper introduces a Quantum Genetic Algorithm framework for solving the MaxCut problem, combining quantum computing principles with evolutionary strategies to improve solution quality and scalability.

Contribution

It presents a novel QGA method using Grover-based evolution and divide-and-conquer, outperforming traditional SDP methods on certain graph types.

Findings

Achieves optimal MaxCut on complete graphs.

Outperforms SDP in solution quality on large graphs.

Provides competitive results on Erdős-Rényi graphs.

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 Erd\H{o}s-R\'{e}nyi 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.