We present a description of our {\em ab initio} nuclear structure formalism and its implementation in our highly parallel codebase. We discuss our strategy for incorporating hardware acceleration with Graphics Processing Units and the timing improvements that it produced. Finally, we use these systems to perform an extensive range of {\em ab initio} calculations for neutron drops in 10 and 20 MeV external harmonic-oscillator traps using chiral nucleon-nucleon plus three-nucleon interactions. We present ground state energies and energy differences, radii, internal energies, and level splitting physics results for neutron numbers $N$ = $2 - 40$ using the no-core full configuration model. We compare with quantum Monte Carlo results using AV8$^\prime$ with Urbana IX and Illinois-7 3N forces, where available, and with the nonlocal NN interaction JISP16. These results lead to insights on the distinctions between candidates for the nuclear strong interaction and inform applications to develop new generations of energy density functionals for nuclei. In addition, in light of a promising new correlation between neutron drop and nuclear observables, we present expanded $N=20$ neutron drop results and use the correlation to compare them with the experimental result for the difference in neutron and proton radii of $^{48}$Ca.