On the Role of Controllability in Pulse-based Quantum Machine Learning Models

TL;DR

This paper explores how controllability affects the expressivity and trainability of pulse-based quantum machine learning models, proposing a universal criterion and design strategies to balance these aspects effectively.

Contribution

It introduces a universal criterion for assessing expressivity based on controllability and demonstrates how low-dimensional manifold design improves trainability.

Findings

Increasing dimensionality enhances expressivity.

Limited controllability on submanifolds avoids barren plateaus.

Designing on low-dimensional manifolds balances expressivity and training difficulty.

Abstract

Pulse-based quantum machine learning (QML) models possess full expressivity when they are ensemble controllable. However, it has also been shown that barren plateaus emerge in such models, rendering training intractable for systems with large dimension. In this paper, we show that the trade-off is closely related to the controllability of the underlying pulse-based models. We first apply the Fliess-series expansion to pulse-based QML models to investigate the effect of control system structure on model expressivity, which leads to a universal criterion for assessing the expressivity of generic QML models. Guided by this criterion, we then demonstrate how designing pulse-based models on low-dimensional manifolds can balance expressivity and trainability. Finally, numerical experiments are carried out to verify the proposed criterion and our analysis, which futher demonstrate that increasing dimensionality enhances expressivity but avoids barren plateaus if the model is designed with limited controllability on a submanifold. Our approach provides a promising path for designing pulse-based QML models that are both highly expressive and trainable.