Orthogonal circuits exhibit a barren plateau

For the circuit family whose dynamical Lie algebra is $\mathfrak{so}(2^n)$, the single-ideal loss variance $$\operatorname{Var}_\theta[\ell]=\frac{P_{\mathfrak g}(\rho)\,P_{\mathfrak g}(O)}{\dim\mathfrak g},\qquad \dim\mathfrak g=\frac{4^n-2^n}{2},$$ carries an algebra dimension that grows exponentially in the qubit count $n$. With the purities $P_{\mathfrak g}(\rho),\,P_{\mathfrak g}(O)$ bounded, the variance decays as $C\,b^{-n}$ with $b>1$, so the family exhibits a barren plateau.