Learning to Maximize Quantum Neural Network Expressivity via Effective Rank

Abstract Quantum neural networks (QNNs) are widely used as trainable models for solving variational problems, where their ability to represent complex functions directly determines performance. However, accurately quantifying this expressivity remains a major challenge, limiting the understanding of the expressive power of QNNs. Here, we introduce the effective rank $\kappa$ as a quantitative measure of expressivity. We show that the expressivity of a QNN is determined by the interplay of three factors: the dataset, the measurement operators, and the circuit structure. Unlike conventional metrics based on parameter sampling or entanglement structure, $\kappa$ captures the number of independent parameters that effectively contribute to the model output, providing an operational and data-dependent characterization of expressivity while avoiding costly parameter sampling. We demonstrate that $\kappa$ can reach its theoretical maximum $4^n - 1$ for an $n$-qubit system when the circuit architecture, input distribution, and measurement protocol are jointly optimized. Leveraging this insight, we employ $\kappa$ as a design objective within a reinforcement learning framework with a self-attention transformer agent to automatically discover highly expressive circuit architectures. Our results show that the proposed approach efficiently identifies high-performing circuit configurations, outperforming heuristic designs and random search in terms of sample efficiency. By bridging theoretical characterization with automated design, our approach provides a practical, scalable, and computationally efficient route for constructing expressive quantum circuits, advancing the development of quantum machine learning models.