Fast gradient-free optimization of excitations in variational quantum eigensolvers

We introduce ExcitationSolve, a fast globally-informed gradient-free optimizer for physically-motivated ans\"atze constructed of excitation operators, a common choice in variational quantum eigensolvers. ExcitationSolve is to be classified as an extension of quantum-aware and hyperparameter-free optimizers such as Rotosolve, from parameterized unitaries with generators \(G\) of the form \(G^2=I\), e.g., rotations, to the more general class of \(G^3=G\) exhibited by the physically-inspired excitation operators such as in the unitary coupled cluster approach. ExcitationSolve is capable of finding the global optimum along each variational parameter using the same quantum resources that gradient-based optimizers require for a single update step. We provide optimization strategies for both fixed- and adaptive variational ans\"atze, as well as a multi-parameter generalization for the simultaneous selection and optimization of multiple excitation operators. Finally, we demonstrate the utility of ExcitationSolve by conducting electronic ground state energy calculations of molecular systems and thereby outperforming state-of-the-art optimizers commonly employed in variational quantum algorithms. Across all tested molecules in their equilibrium geometry, ExcitationSolve remarkably reaches chemical accuracy in a single sweep over the parameters of a fixed ansatz. This sweep requires only the quantum circuit executions of one gradient descent step. In addition, ExcitationSolve achieves adaptive ans\"atze consisting of fewer operators than in the gradient-based adaptive approach, hence decreasing the circuit execution time.