Fig. 1: Illustration of the main result.
From: Characterizing barren plateaus in quantum ansätze with the adjoint representation
For the gradient variance, when the observable is in the dynamical Lie algebra (DLA), as in the case of Lie algebra supported ansatz, only the components of ρ in the DLA matter, and everything can be computed in the adjoint representation. Specifically, the subscript α for operators corresponds to their orthogonal projection onto the simple ideal \({{\mathfrak{g}}}_{\alpha }\). When the DLA has multiple ideals, each ideal individually contributes a term to the variance.
