[xmcqdpt2]

To start, we'll neglect the bosonic degrees of freedom (the displacement of the atoms themselves) and look at only the fermionic DoF (the electrons).

Water has 8 electrons (which QC can treat exactly without any extra work) in a number of orbital. In general we need 2 qubits per orbital.

Most QC demonstrations so far were performed using so-called minimal basis sets, which have a small number of orbitals and thus give inaccurate results. A better approach would be to take a large orbital basis, do a classical relatively expensive Hartree-Fock calculation then use the orbitals from that to do the QC. This technique when done on classical computers is called MRCI (multi-reference configuration interaction) and is the gold standard in Quantum Chemistry.

So, provided we can pay the cost of doing a large orbital HF calculation (and we can do that for fairly large molecules), we can get pretty good result using n electrons in n orbitals MRCI. So production electronic calculations of water molecules would take about 16 qubits per molecule.

The more frustrating problem is that the number of electronic interaction terms is N^4 the number of orbitals so that we would very rapidly need extremely deep circuits which are not feasible without error correction (which involve using like 8 actual qubits for every calculation qubit). There are proposal to use plane wave basis sets (N^2 interactions) but then we need many more orbitals and thus many more qubits.

We are in practice very far from QC having a significant impact on real-life quantum chemistry. It's not at all clear that we'll ever be able to do QC on a molecule the size of a typical drug, let alone a protein.