  Current quantum computing hardware is restricted by the availability of only
few, noisy qubits which limits the investigation of larger, more complex
molecules in quantum chemistry calculations on quantum computers in the
near-term. In this work, we investigate the limits of their classical and
near-classical treatment while staying within the framework of quantum circuits
and the variational quantum eigensolver. To this end, we consider naive and
physically motivated, classically efficient product ansatz for the parametrized
wavefunction adapting the separable pair ansatz form. We combine it with
post-treatment to account for interactions between subsystems originating from
this ansatz. The classical treatment is given by another quantum circuit that
has support between the enforced subsystems and is folded into the Hamiltonian.
To avoid an exponential increase in the number of Hamiltonian terms, the
entangling operations are constructed from purely Clifford or near-Clifford
circuits. While Clifford circuits can be simulated efficiently classically,
they are not universal. In order to account for missing expressibility,
near-Clifford circuits with only few, selected non-Clifford gates are employed.
The exact circuit structure to achieve this objective is molecule-dependent and
is constructed using simulated annealing and genetic algorithms. We demonstrate
our approach on a set of molecules of interest and investigate the extent of
our methodology's reach. Empirical validation of our approach using numerical
simulations shows a reduction of the qubit count of up to a 50\% at a similar
accuracy as compared to the separable-pair ansatz.
