Cautious optimism for deep parameterized quantum circuits

As quantum machine learning models scale to include more trainable parameters, it remains unclear how their generalization performance on unseen data evolves. Traditional statistical learning theory suggests that increasing model size beyond a certain point leads to overfitting and degraded performance. This paper investigates whether parameterized quantum circuits (PQCs) follow this traditional view or if they exhibit the 'double descent' phenomenon—a non-monotonic relationship between model size and test error—observed in classical deep learning.

The authors employ an 'add-one-in' perturbation analysis to derive theoretical bounds on the expected risk of PQCs. By leveraging spectral properties of random matrices and assuming the model is trained to a local minimum of the empirical risk, they characterize how the risk behaves as the number of parameters $p$ and training samples $N$ scale together. They supplement these analytical results with numerical experiments using data re-uploading PQCs on various datasets, including MNIST-1D, Fashion MNIST, and synthetic regression tasks, to observe the test loss dynamics across the interpolation threshold.

The study demonstrates that the risk bounds for gradient-based PQCs peak at the interpolation threshold ($p = NK$, where $K$ is the output dimension). Beyond this point, the test error decreases, confirming that PQCs can exhibit double descent. The authors show that when training successfully converges, increasing the number of parameters beyond the number of training constraints does not necessarily degrade generalization. This suggests that overparameterization can play a constructive role in quantum machine learning, similar to its role in classical neural networks.