  By exploiting the invariance of the molecular Hamiltonian by a unitary
transformation of the orbitals it is possible to significantly shorter the
depth of the variational circuit in the Variational Quantum Eigensolver (VQE)
algorithm by using the Wavefunction Adapted Hamiltonian Through Orbital
Rotation (WAHTOR) algorithm. This work introduces a non-adiabatic version of
the WAHTOR algorithm and compares its efficiency with three implementations by
estimating Quantum Processing Unit (QPU) resources in prototypical benchmarking
systems. Calculating first and second-order derivatives of the Hamiltonian at
fixed VQE parameters does not introduce a significant QPU overload, leading to
results on small molecules that indicate the non-adiabatic Newton-Raphson
method as the more convenient choice. On the contrary, we find out that in the
case of Hubbard model systems the trust region non-adiabatic optimization is
more efficient. The preset work therefore clearly indicates the best
optimization strategies for empirical variational ansatzes, facilitating the
optimization of larger variational wavefunctions for quantum computing.
