Optimizing Unitary Coupled Cluster Wave Functions on Quantum Hardware: Error Bound and Resource-Efficient Optimizer

Abstract

In this work, we study the projective quantum eigensolver (PQE) approach to optimizing unitary coupled cluster wave functions on quantum hardware, as introduced in arXiv:2102.00345. The projective quantum eigensolver is a hybrid quantum-classical algorithm which, 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, PQE uses projections of the Schrodinger equation to efficiently bring the trial state closer to an eigenstate of the Hamiltonian. In this work, 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 H$_4$, H$_6$, BeH$_2$ and LiH dissociation curves over both the optimization introduced in arXiv:2102.00345 and VQE optimized using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.