A VC-Dimension View of Capacity and Generalization in Quantum Machine Learning with Classical Data

Quantum machine learning (QML) models are often motivated by the large Hilbert spaces accessible to parametrized quantum circuits, but their capacity is ultimately determined by their trainable parameters. We study model capacity through the Vapnik-Chervonenkis (VC) dimension for several QML families. For purely univariate QML classifiers, we derive linear VC dimensions, analogous to univariate classical models. Multivariate uniform re-uploading analyzes the VC dimension from the encoding side, providing an upper bound of $O\left(\right(\frac{d}{p}+1{)}^p)$ for $d$ trainable parameters and input dimension $p$, which is consistent with empirical evaluations. Finally, the bounded-reuse trainable Pauli classifier analyzes the VC dimension from the trainable parameter side, obtaining a $O(d\log d)$ upper bound, similar to a feedforward neural net. We conclude that asymptotic scaling depends on input dimension and architecture rather than on the model's classical or quantum nature. Empirically, we find that QML models trained on classical data exhibit generalization trends consistent with what VC-bounds predict. Conversely, previous studies have observed strong generalization for very few quantum-data, suggesting that QML generalization behavior is not yet fully understood.