My (rather incomplete) understanding of the NP-hard fermion/numerical sign problem is that it occurs when attempting to converge on a wavefunction for many-body fermion systems (for example, a small system of carbon atoms). And the problem is the noise introduced by the near-cancellation of large positive and negative amplitudes.
It sounds to me like most of the difficulty comes in trying to solve a large collection of differential equations for a system that already exists; the computational cost goes with $2^N$, where $N$ is the number of particles.
Now this may be naive, but why couldn't you simply start with a system of electrons and a positive "nucleus" charge that are far apart (so you've got a bunch of uncorrelated/unentangled single-particle wavefunctions), combine these into one multi-body wavefunction fairly easily at this point, and then evolve that overall wavefunction according to the time-dependent Schrödinger equation?
I mean, since that wavefunction would numerically evolve the same way a similar one in reality would, I assume such a "separated" system would in real-life coalesce into an atom at some point in time. And it's much easier to evolve a function over time than solve a differential equation. Thus, electron-electron correlations would naturally develop in the many-body wavefunction without any additional computational effort.
I imagine if it was this easy though, someone would have already done it. Thoughts?