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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?

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Your concept to avoid solving a differential equation (or eigenvalue problems) is correct. One problem is that "evolving from separated system method" may give results far away from the statistical values. Even so, your idea is great, and many scientists in statistical physics try to find such a way.

Recently, surprising articles were presented,

S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012) and

S. Sugiura and A. shimizu, arXiv:1302.3138.

In these papers, authors invented the way to avoid solving eigenvalue problems. Authors proved that we can get the statistical values just by acting the Hamiltonian to randomly selected quantum pure states.

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This is not so different from what is done in Green Function Monte Carlo calculations (GFMC). The problem is that the wave function is a function of 3N coordinates. Just to store it is a headache (take N=10, incredibly poor resolution of 20 points per coordinate, get $20^{10}$ points). Then, in each time step you have to apply a two-body Hamiltonian to this wave function (a $20^{10}$ $\times$ $20^{10}$ matrix). Not good ..

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