Arbitrary Polynomial Separations in Trainable Quantum Machine Learning

Recent theoretical results in quantum machine learning have demonstrated a general trade-off between the expressive power of quantum neural networks (QNNs) and their trainability; as a corollary of these results, practical exponential separations in expressive power over classical machine learning models are believed to be infeasible as such QNNs take a time to train that is exponential in the model size. We here circumvent these negative results by constructing a hierarchy of efficiently trainable QNNs that exhibit unconditionally provable, polynomial memory separations of arbitrary constant degree over classical neural networks in performing a classical sequence modeling task. Furthermore, each unit cell of the introduced class of QNNs is computationally efficient, implementable in constant time on a quantum device. The classical networks we prove a separation over include well-known examples such as recurrent neural networks and Transformers. We show that quantum contextuality is the source of the expressivity separation, suggesting that other classical sequence learning problems with long-time correlations may be a regime where practical advantages in quantum machine learning may exist.