Matchgate circuits have no barren plateau

For the matchgate family whose dynamical Lie algebra is $\mathfrak{so}(2n)$ (with $n\ge 3$), the single-ideal loss variance $$\operatorname{Var}_\theta[\ell]=\frac{P_{\mathfrak g}(\rho)\,P_{\mathfrak g}(O)}{\dim\mathfrak g},\qquad \dim\mathfrak g=n(2n-1),$$ carries an algebra dimension that grows only polynomially in $n$. When the purity product $P_{\mathfrak g}(\rho)\,P_{\mathfrak g}(O)$ stays bounded below by an inverse polynomial in $n$, the variance is likewise inverse-polynomially bounded below and does not decay as $C\,b^{-n}$ with $b>1$, so the family has no barren plateau. The transverse-field Ising chain realizes this concretely: for the highest-weight state $\rho=|0\rangle\!\langle0|^{\otimes n}$ and a two-body Ising observable $O$ the purities are $P_{\mathfrak g}(\rho)=n/2^n$ and $P_{\mathfrak g}(O)=2^n$, giving the exact inverse-linear variance $\operatorname{Var}=1/(2n-1)$.