  The optimization of circuit parameters of variational quantum algorithms such
as the variational quantum eigensolver (VQE) or the quantum approximate
optimization algorithm (QAOA) is a key challenge for the practical deployment
of near-term quantum computing algorithms. Here, we develop a hybrid
quantum/classical optimization procedure inspired by the Jacobi diagonalization
algorithm for classical eigendecomposition, and combined with Anderson
acceleration. In the first stage, analytical tomography fittings are performed
for a local cluster of circuit parameters via sampling of the observable
objective function at quadrature points in the circuit angles. Classical
optimization is used to determine the optimal circuit parameters within the
cluster, with the other circuit parameters frozen. Different clusters of
circuit parameters are then optimized in "sweeps,'' leading to a
monotonically-convergent fixed-point procedure. In the second stage, the
iterative history of the fixed-point Jacobi procedure is used to accelerate the
convergence by applying Anderson acceleration/Pulay's direct inversion of the
iterative subspace (DIIS). This Jacobi+Anderson method is numerically tested
using a quantum circuit simulator (without noise) for a representative test
case from the multistate, contracted variant of the variational quantum
eigensolver (MC-VQE), and is found to be competitive with and often faster than
Powell's method and L-BFGS.
