The Dynamical Lie Algebra of QAOA-MaxCut on the Complete Graph

We give an analytical expression for the dynamical Lie algebra corresponding to the QAOA-MaxCut problem on complete graphs, and show that the variance of the associated loss function scales linearly in the number of qubits. This solves an open problem from [ASYZ26] and confirms that such systems do not exhibit barren plateaus. The proof is based on projecting the dynamical Lie algebra generators onto subspaces given by the Schur-Weyl duality between irreducible representations of the unitary and symmetric groups.