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A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schrödinger equation by solving its integration (or contraction) to the 2-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 wavefunction 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 non-linear conjugate gradient approaches. Practically these algorithms result in superlinear convergence of the wavefunction 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.