Hello, thank you for the lead. I tried replying the other day but I had just woken up drunk in the campment of a homeless guy and I was not in the best conditions to go through the math.

I took a look at this worked example: https://dojo.qulacs.org/en/qp_main/notebooks/7.1_quantum_phase_estimation_detailed.html

As far as I understand they use the iterative approach because it requires less qbits and are able to decompose the eigenvalues of a Hamiltonian in more or less a single step.

If it is as I understand it this would be quite huge, since you'd be able to directly apply this to the Hartree-Fock equation or Density Functional Theory without having to come up with new ways to represent molecules.

One thing which appears quite critical is:

Prepare an initial state with sufficient overlap with the ground state

What does sufficient overlap mean? Could we take an AlphaFold model and that's sufficient to then determine the ground state?

I guess this is easy with helium when you have 2 atoms, but when you have hundred of thousands it becomes a difficult task even to get to that point.

Moreover, I'm not exactly sure what they're calculating: they plot an error; but it appears to be an error over the computed energy and not atom positions.

We already have reliable ways, and moderately fast, to compute the energy of a system. What we're missing is a quick way to explore different spatial conformations of atoms to identify the one which leads to the lowest energy.

Another problem which I could not determine is whether the amount of required qbits scales with the dimensions of the molecular system. I suppose it does. In that case, could we estimate how many qbits would be required for a protein or at least a peptide?