The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze

Using tools from the representation theory of compact Lie groups, we formulate a theory of Barren Plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra (DLA), a setting that we term Lie Algebra Supported Ansatz (LASA). A large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks are LASAs. In particular, our theory provides for the first time the ability to compute the variance of the gradient of the cost function for a non-trivial, subspace uncontrollable family of quantum circuits, the quantum compound ansätze. We rigorously prove that the variance of the gradient of the cost function, under Haar initialization, scales inversely with the dimension of the DLA, which agrees with existing numerical observations. Lastly, we include potential extensions for handling cases when the observable lies outside of the DLA and the implications of our results.