[madez]

This explanation is interesting. Thanks for sharing it. While reading it, I got the impression that the simulation is not fully quantum mechanical, but rather classical with select quantum mechanical effects.

Which parts of quantum mechanics are idealised away and how do we know that not including them won't significantly reduce the quality of the result?

Are you possibly using stochastical noise in the simulations and repeat them multiple times, in the hope that whatever disturbance caused by the idealisation of the model is covered by the noise?

[tchitra]

That's a good question and there are a number of ways to try to tackle this. One of the main reasons you cannot do QM simulations directly is that the high quality methods can cost Omega(n^6/eps) to get eps. relative accuracy (you can do better with DFT, but then you're making your life hard in other way). At a high-level (and I mean, 50,000 ft. level), here are the simplest way:

1) Do quantum mechanics simulations of interactions of a small number of atoms — two amino acids, two ethanol molecules. Then fit a classical function to the surface E[energy(radius between molecules, angles)], where this expectation operator is the quantum one (over some separable Hilbert space). Now use the approximation for E[energy(r, a)] to act as your classical potential. - Upshot: You use quantum mechanics to decide a classical potential for you (e.g. you chose the classical potential that factors into pairs such that each pair energy is 'closest' in the Hilbert space metric to the quantum surface) - Downside: You're doing this for small N — this ignores triplet and higher interactions. You're missing the variance and other higher moments (which is usually fine for biology, FWIW, but not for, say, the Aharanov-Bohm effect).

2) Path Integral methods: This involves running classical simulation for T timesteps, then sampling the 'quantum-sensitive pieces' (e.g. highly polar parts) in a stochastic way. This works because Wick rotation lets you go from Hamiltonian evolution operator e^{i L}, for a Lagrangian density L, to e^{-L} [0]. You can sample the last density via stochastic methods to add a SDE-like correction to your classical simulation. This way, you simulate the classical trajectory and have the quantum portions 'randomly' kick that trajectory based on a real Lagrangian.

3) DFT-augmented potentials: A little more annoying to describe, but think of this as a combination of the first two methods. A lot of the "Neural Network for MD" stuff falls closer in this category [1]

[0] Yes, assume L is absolutely continuous with regards to whatever metric-measure space and base measure you're defined over :) Physics is more flexible than math, so you can make such assumption and avoid thinking about nuclear spaces and atomic measures until really needed

[1] https://arxiv.org/abs/2002.02948

[madez]

> Upshot: You use quantum mechanics to decide a classical potential for you (e.g. you chose the classical potential that factors into pairs such that each pair energy is 'closest' in the Hilbert space metric to the quantum rest) - Downside: You're missing the variance.

Couldn't the quantum mechanical state become multimodal such that the classical approximation picks a state that is far away from the physical reality?

And, couldn't this multimodality excaberate during the actual physical process and possibly arrive at a number of probable outcomes which are never predicted by the simulation? Is there more than hope that that doesn't happen?