[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?

[tchitra]

Yes, for sure. In practice (and not at the 50,000 ft. level), you do try to include the multimodalities — you don't _really_ just use E[quantum_energy(r)]. But you ARE still reliant on some computable/smooth/Lipschitz moment and/or expectation from the quantum surface. The semi-heuristic argument for why you get away with this in biological simulation is somewhat heuristic, but of the following form:

- Most quantum field theories are described by of the form L(E), where E is an energy level ["effective field theory"] and the Lagragian changes as E change.

   - When E is low, L(E) is classical mechanics & EM

   - When E is around 1GeV, L(E) is the aforementioned plus QED

   - When E is around 100GeV, L(E) has the aforementioned plus some QCD

   - When E is at 1 TeV, L(E) has the aforementioned plus Higgs-like stuff

Now biology is on the lowest end of that scale, so you mainly have to deal with QED and perturbative electronic expansion. These electronic expansions are the most important part — you need them to get hydrogen bonding + electrodynamic molecular interactions correct — BUT they are highly local.

This locality is what you take advantage of when you normalize — you find from QM that the potentials only matter when the two charged/polar molecules are close, so you try to make a classical potential that has quantum 'jumps' when these things are close.

Do you miss the purely quantum stuff? Aharanov-Bohm, Chern classes, and the like? Of course. But from a practical standpoint, you do get the structures that you measure from experiment to be correct because the 'cool' quantum with 'tons' of states is less important for pedestrian things at low energy scale.

It is still hard to get right though! There's a lot of entropy you need to localize correctly and in some sense, you have to make sure you get the modes as a function of local particle positions correct.

The final thing to point out is that the Wick rotated path integral stuff works for biology much better than for real HEP-type of stuff because molecules are contained at low energies — those tunneling probabilities are O(h E), and log(E) is still dwarfed by -log(h) so you _can_ safely ignore them.

This is not true for things like circuits, however, because the lithography at EUV scales (3nm -_-) does have tunneling issues at high field strengths.

tl;dr: Biology has some saving graces that give you good approximations. Are they perfect? No, but if you find a time that I have to compute a vanishing first Chern class in a noisy, ugly biological system, then you deserve a Nobel Prize!