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Continuum Coulomb hardness with binary nuclear charges
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Abstract
We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.
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Formalization
- OpenAI, QMA-hardness of continuum Coulomb energy — Lean scope notes (OpenAI Math Release, 6 October 2026). The release's scope document says the formalization proves QMA-hardness over the full spinful continuum for binary-encoded positive integer charges, with distinct rational positions, unary electron count and threshold separation at least one, in one Comparator file (
ContinuumCoulombHardness.lean) that also covers the unit-charge result. [family-level]
Related prior literature mentioned by others
- Bryan O’Gorman, Sandy Irani, James Whitfield and Bill Fefferman, Intractability of Electronic Structure in a Fixed Basis (PRX Quantum 3, 020322, 2 May 2022). Identified in VibeMathed’s family entry, this paper proves fixed-basis QMA-completeness and asks in Section V about hardness in the complete infinite-dimensional space when the external potential comes solely from positive nuclei at fixed positions. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Barry Simon, Semiclassical analysis of low lying eigenvalues. II. Tunnelling (Annals of Mathematics 120 (1984), 89–118). The paper cites only part I of this series [Sim83]; part II treats exponentially small tunnelling splittings between wells, which bears on the paper's relative tunnelling calibration for widely separated hydrogenic wells.
- B. Helffer and J. Sjöstrand, Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation (Ann. IHP Phys. Théor. 42 (1985), 127–212). This is a multiple-well interaction-matrix framework related to the paper's derivation of an effective Hubbard hopping from localized one-electron wells.
- Conyers Herring and Michael Flicker, Asymptotic Exchange Coupling of Two Hydrogen Atoms (Phys. Rev. 134, A362, 1964). This is the classical large-separation asymptotics of the singlet–triplet exchange splitting for two hydrogen atoms, relevant to the paper's realization of Heisenberg exchange through superexchange between near-hydrogenic sites.
Elsewhere
- Emergent Mind: family 275
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026,
author = {{OpenAI}},
title = {{Continuum Coulomb hardness with binary nuclear charges}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf}{OAI:Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/275.md. The Lean statements have not been checked by people against the paper's claims.
Family 275: QMA-hardness of continuum Coulomb energy
Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.
Other papers in this family:
- QMA-hardness of continuum Coulomb energy with unit nuclear charges · September 24, 2026