Optimizing unitary coupled cluster wave functions on quantum hardware: Error bound and resource-efficient optimizer

In this study, we study the projective quantum eigensolver (PQE) approach, a hybrid quantum-classical algorithm that, by optimizing a unitary coupled cluster wave function, aims at computing the ground state of many-body systems. Instead of trying to minimize the energy of the system like the variational quantum eigensolver (VQE), PQE uses projections of the Schrödinger equation to efficiently bring the trial state closer to an eigenstate of the Hamiltonian. In this study, we provide a mathematical study of the algorithm. We derive a bound relating off-diagonal coefficients (residues) of the Hamiltonian to the energy error of the algorithm and the overlap achieved by the obtained wavefunction. These bounds not only give formal guarantees to PQE, but they also allow us to formulate a well-informed convergence criterion for residue-based optimizers. We then study the classical optimization itself and derive convergence guarantees under certain conditions. We propose a new residue-based optimizer, with numerical evidence of the superiority of this new approach for H4, H6, BeH2, and LiH dissociation curves over both the optimization introduced in Stair and Evangelista [PRX Quantum 2, 030301 (2021)] and VQE optimized using the Broyden-Fletcher-Goldfarb-Shanno method.