Rethinking Expressibility-Trainability Trade-off in Hybrid Quantum Neural Networks

Abstract: Hybrid quantum neural networks (HQNNs) integrate parameterized quantum circuits (PQCs) within classical networks, where the behavior of the underlying PQCs is often the primary focus of analysis. In this context, expressibility and trainability are widely used to characterize PQC's performance and are commonly assumed to exhibit a trade-off, where highly expressive circuits are more susceptible to barren plateaus. However, the validity of this relationship in HQNNs remains unclear. In this paper, we systematically analyze the expressibility--trainability relationship in HQNNs across varying circuit depths, qubit counts, entanglement topologies. We consider different training configurations, including pure PQCs, quantum-only training in hybrid setting, and full end-to-end training of hybrid models. Our results show that pure PQCs exhibit only a weak and regime-dependent trade-off, while hybrid architectures increasingly disrupt and can eliminate this relationship under full hybrid training. This indicates that classical components reshape the optimization landscape, decoupling trainability from PQC expressibility. We further propose a multi-objective neural architecture search (NAS) framework that jointly optimizes expressibility, trainability, and task performance over a combined classical--quantum design space, revealing different Pareto-optimal solutions under full end-to-end and quantum only training in hybrid setting. different trainability definitions. Our results suggest that hybridization is not just an implementation detail, but a defining factor in the performance of quantum machine learning models.