  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.
