A simple proof for the equivalence of cost function concentration and gradient vanishing

Barren plateau problems are usually studied based on the vanishing of the gradient for parametrized quantum circuits. However, an explicit parametrization is not necessary and a (parametrization-free) Riemannian optimization has various advantages. Here, we report on a simple proof, establishing the equivalence of cost function concentration and gradient vanishing in the Riemannian optimization of quantum circuits. When the variable unitaries are sampled according to the uniform Haar measure, the cost function variance is strictly equal to half the variance of the Riemannian gradient. Thus, the barren plateau problem can be diagnosed equally well by studying cost function concentration. In many cases, the cost variance is easier to assess than the gradient variance. This report complements arXiv.2104.05868, which proves the equivalence of cost concentration and gradient vanishing in Euclidean space, and has the same implication as arXiv.2011.12245, namely that neither gradient-based optimization nor gradient-free optimization can resolve the barren plateau problem.