Identifying overparameterization in Quantum Circuit Born Machines

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.