Accelerated Convergence of Contracted Quantum Eigensolvers through a Quasi-Second-Order, Locally Parameterized Optimization

A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schrödinger equation by solving its integration (or contraction) to the two-electron space─a contracted Schrödinger equation (CSE)─on a quantum computer. When applied to the anti-Hermitian part of the CSE (ACSE), the CQE iterations optimize the wave function, with respect to a general product ansatz of two-body exponential unitary transformations that can exactly solve the Schrödinger equation. In this work, we accelerate the convergence of the CQE and its wave function ansatz via tools from classical optimization theory. By treating the CQE algorithm as an optimization in a local parameter space, we can apply quasi-second-order optimization techniques, such as quasi-Newton approaches or nonlinear conjugate gradient approaches. Practically, these algorithms result in superlinear convergence of the wave function to a solution of the ACSE. Convergence acceleration is important because it can both minimize the accumulation of noise on near-term intermediate-scale quantum (NISQ) computers and achieve highly accurate solutions on future fault-tolerant quantum devices. We demonstrate the algorithm, as well as some heuristic implementations relevant for cost-reduction considerations, comparisons with other common methods such as variational quantum eigensolvers, and a Fermionic-encoding-free form of the CQE.