  In machine learning, overparameterization is associated with qualitative
changes in the empirical risk landscape, which can lead to more efficient
training dynamics. For many parameterized models used in statistical learning,
there exists a critical number of parameters, or model size, above which the
model is constructed and trained in the overparameterized regime. There are
many characteristics of overparameterized loss landscapes. The most significant
is the convergence of standard gradient descent to global or local minima of
low loss. In this work, we study the onset of overparameterization transitions
for quantum circuit Born machines, generative models that are trained using
non-adversarial gradient-based methods. We observe that bounds based on
numerical analysis are in general good lower bounds on the overparameterization
transition. However, bounds based on the quantum circuit's algebraic structure
are very loose upper bounds. Our results indicate that fully understanding the
trainability of these models remains an open question.
