  A contracted quantum eigensolver (CQE) finds a solution to the many-electron
Schr\"odinger equation by solving its integration (or contraction) to the
2-electron space -- a contracted Schr\"odinger 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
Schr\"odinger 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.
