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Quantum computing and machine learning convergence enable powerful new approaches for optimizing mobile edge computing (MEC) networks. This paper uses Lyapunov optimization theory to propose a novel quantum machine learning framework for stabilizing computation offloading in next-generation MEC systems. Our approach leverages hybrid quantum-classical neural networks to learn optimal offloading policies that maximize network performance while ensuring the stability of data queues, even under dynamic and unpredictable network conditions. Rigorous mathematical analysis proves that our quantum machine learning controller achieves close-to-optimal performance while bounding queue backlogs. Extensive simulations demonstrate that the proposed framework significantly outperforms conventional offloading approaches, improving network throughput by up to 30% and reducing power consumption by over 20%. These results highlight the immense potential of quantum machine learning to revolutionize next-generation MEC networks and support emerging applications at the intelligent network edge. [ABSTRACT FROM AUTHOR]