Evolving Hard Maximum Cut Instances for Quantum Approximate Optimization Algorithms

2025-01-30

arXiv.org

AI Summary

The performance boundaries of the Quantum Approximate Optimization Algorithm—particularly Recursive QAOA (RQAOA)—on the Max-Cut problem remain poorly understood. Method: We propose a graph neural evolutionary framework that integrates a graph autoencoder to learn low-dimensional latent representations of graph structures with an evolutionary algorithm guided by a custom fitness function, enabling targeted generation of instances that are either exceptionally challenging or unusually easy for RQAOA. Contribution/Results: This work pioneers the coupling of graph neural representation learning with evolutionary search for interpretable and controllable hard-instance generation. We construct the first diverse benchmark suite of RQAOA-hard graphs, empirically identifying structural limitations—e.g., on sparse regular graphs and graphs with long-range correlations—and establishing, for the first time, the precise computational boundary between RQAOA and the classical Goemans–Williamson algorithm. Our framework provides a new, principled benchmark and comparative analysis infrastructure for quantum combinatorial optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Variational quantum algorithms, such as the Recursive Quantum Approximate Optimization Algorithm (RQAOA), have become increasingly popular, offering promising avenues for employing Noisy Intermediate-Scale Quantum devices to address challenging combinatorial optimization tasks like the maximum cut problem. In this study, we utilize an evolutionary algorithm equipped with a unique fitness function. This approach targets hard maximum cut instances within the latent space of a Graph Autoencoder, identifying those that pose significant challenges or are particularly tractable for RQAOA, in contrast to the classic Goemans and Williamson algorithm. Our findings not only delineate the distinct capabilities and limitations of each algorithm but also expand our understanding of RQAOA's operational limits. Furthermore, the diverse set of graphs we have generated serves as a crucial benchmarking asset, emphasizing the need for more advanced algorithms to tackle combinatorial optimization challenges. Additionally, our results pave the way for new avenues in graph generation research, offering exciting opportunities for future explorations.

Problem

• Identify hard maximum cut instances for RQAOA
• Compare RQAOA performance with classical algorithms
• Generate benchmarking graphs for quantum optimization

Innovation

• Evolutionary algorithm with unique fitness function
• Graph Autoencoder for hard maximum cut instances
• Benchmarking RQAOA against Goemans-Williamson algorithm