  One of the most important properties of neural networks is the clustering of
local minima of the loss function near the global minimum, enabling efficient
training. Though generative models implemented on quantum computers are known
to be more expressive than their traditional counterparts, it has empirically
been observed that these models experience a transition in the quality of their
local minima. Namely, below some critical number of parameters, all local
minima are far from the global minimum in function value; above this critical
parameter count, all local minima are good approximators of the global minimum.
Furthermore, for a certain class of quantum generative models, this transition
has empirically been observed to occur at parameter counts exponentially large
in the problem size, meaning practical training of these models is out of
reach. Here, we give the first proof of this transition in trainability,
specializing to this latter class of quantum generative model. We use
techniques inspired by those used to study the loss landscapes of classical
neural networks. We also verify that our analytic results hold experimentally
even at modest model sizes.
