  Solving the electronic Schrodinger equation for strongly correlated ground
states is a long-standing challenge. We present quantum algorithms for the
variational optimization of wavefunctions correlated by products of unitary
operators, such as Local Unitary Cluster Jastrow (LUCJ) ansatzes, using
stochastic reconfiguration and the linear method. While an implementation on
classical computing hardware would require exponentially growing compute cost,
the cost (number of circuits and shots) of our quantum algorithms is polynomial
in system size. We find that classical simulations of optimization with the
linear method consistently find lower energy solutions than with the L-BFGS-B
optimizer across the dissociation curves of the notoriously difficult N$_2$ and
C$_2$ dimers; LUCJ predictions of the ground-state energies deviate from exact
diagonalization by 1 kcal/mol or less at all points on the potential energy
curve. While we do characterize the effect of shot noise on the LM
optimization, these noiseless results highlight the critical but often
overlooked role that optimization techniques must play in attacking the
electronic structure problem (on both classical and quantum hardware), for
which even mean-field optimization is formally NP hard. We also discuss the
challenge of obtaining smooth curves in these strongly correlated regimes, and
propose a number of quantum-friendly solutions ranging from symmetry-projected
ansatz forms to a symmetry-constrained optimization algorithm.
