QMA-hardness of continuum Coulomb energy

The explanation

Start with one proton and one electron, a hydrogen atom. Its lowest electronic energy is −1/2 in atomic units, the hartree. Now pose a yes-or-no question with two thresholds, a = −0.4 and b = 0.6. Is the lowest energy at most a, or at least b? For hydrogen the answer is clearly yes, since −0.5 is below −0.4.

The general problem lists nuclei at fixed points with their charges, and a number of electrons. The electrons feel only Coulomb forces: attraction to the nuclei and repulsion from each other. The lowest energy E₀ is taken over every allowed electron state in ordinary three-dimensional space, with spin. A YES instance has E₀ ≤ a; a NO instance has E₀ ≥ b. The thresholds are promised to be far enough apart.

QMA is the quantum version of NP. A problem is in QMA when a quantum computer can check a proposed quantum state as evidence. A problem is QMA-hard when every QMA problem converts to it by an efficient classical recipe. So a fast general method for a QMA-hard problem would give fast methods for all of QMA.

Earlier results changed the question. O'Gorman, Irani, Whitfield and Fefferman proved QMA-completeness in a supplied finite basis of orbitalsSource 1. Schuch and Verstraete used designed potentials and local magnetic fieldsSource 2. The obstacle is the continuum. Restricting to orbitals gives an upper bound on the energy. A NO answer needs a lower bound over every state outside those orbitals as wellSource 3.

The first paper proves hardness when positive integer charges are written in binarySource 4. The charges may then be exponentially large, but every output has polynomial lengthSource 5. The second paper proves hardness when every nucleus has charge oneSource 6. Its recipe outputs polynomially many nuclei and electrons, with thresholds at least one hartree apartSource 7. No orbital basis, magnetic field or extra potential is givenSource 8. The papers do not claim the problem lies in QMASource 9.

The hardness is worst case. It says some carefully built arrangements of nuclei are hard. It does not say typical molecules are hard. The gap between thresholds is a precision promise, not an energy gap in the moleculeSource 10.

How sure is this? Lean, a computer proof checker, has formal proofs of both hardness statements as given in the papers. They cover binary charges and unit charges, both over the full spinful continuum. These statements rely on a standard QMA-hard spin problem as their starting point. No specialist review of the papers has been published.

What it doesn't show. This does not show the problem belongs to QMA, that typical real molecules are hard, or that any particular chemistry method fails.