Simple-DLA barren plateau (g ≃ su(d))

For a parametrized quantum circuit whose dynamical Lie algebra $\mathfrak{g}$ is simple, $\mathfrak{g} \simeq \mathfrak{su}(d)$ (dimension $d^2-1$, centerless), the loss variance reduces to a single term $$\operatorname{Var}_\theta[\ell] = \frac{P_{\mathfrak{g}}(\rho)\, P_{\mathfrak{g}}(O)}{d^2-1}.$$ In particular, for $d = 2^n$ the dimension $\dim\mathfrak{g} = 4^n-1$ grows exponentially in the qubit count $n$, so the loss exhibits an exponential barren plateau (under the Haar second-moment / Schur hypothesis carried as a named input).