Predicting quantum learnability from landscape fluctuation

The conflict between trainability and expressibility is a key challenge in variational quantum computing and quantum machine learning. Resolving this conflict necessitates designing specific quantum neural networks (QNN) tailored for specific problems, which urgently needs a general and efficient method to predict the learnability of QNNs without costly training. In this work, we demonstrate a simple and efficient metric for learnability by comparing the fluctuations of the given training landscape with standard learnable landscapes. This metric shows surprising effectiveness in predicting learnability as it unifies the effects of insufficient expressibility, barren plateaus, bad local minima, and overparametrization. Importantly, it can be estimated efficiently on classical computers via Clifford sampling without actual training on quantum devices. We conduct extensive numerical experiments to validate its effectiveness regarding physical and random Hamiltonians. We also prove a compact lower bound for the metric in locally scrambled circuits as analytical guidance. Our findings enable efficient predictions of learnability, allowing fast selection of suitable QNN architectures for a given problem without training, which can greatly improve the efficiency especially when access to quantum devices is limited.