Speaker
Description
One of the main obstacles to the trainability of variational quantum algorithms and quantum machine learning models are barren plateaus, where the cost function (and its gradients) exponentially concentrates in parameter space as the size of the problem increases. We derive a formula for the variance of the cost function in terms of the dynamical Lie algebra (DLA) of the parametrized quantum circuit, i.e., the Lie algebra generated by the Hamiltonians in the circuit. We present a classification of DLAs generated by 1- and 2-local Pauli operators acting on a spin chain or more generally placed on the edges of an arbitrary interaction graph. Finally, we explicitly determine the DLA associated with the Quantum Approximate Optimization Algorithm with a Grover mixer. We prove that the dimension of the DLA grows polynomially with the number of qubits, and as a consequence, barren plateaus are avoided.