QNN overparametrization

A variational quantum neural network with $M$ trainable parameters $\theta$ has, over its optimization landscape, achievable QFIM rank $R(M) = \sup_\theta \operatorname{rank} F(\theta)$, with saturated value $R = \sup_M R(M)$. The network is overparametrized at parameter count $M$ when the achievable rank saturates, $R(M) = R$; the critical parameter count $M_c = \inf\{\,M : R(M) = R\,\}$ is the least such $M$, beyond which adding parameters explores no new state-space directions of the optimization landscape.