Browse › Mathematical physics › 275 QMA-hardness of continuum Coulomb energy
QMA-hardness of continuum Coulomb energy with unit nuclear charges
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
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 of the ground-energy infimum for clamped unit-charge nuclei in the full spinful fermionic continuum via a deterministic polynomial-time reduction, in one Comparator file (
ContinuumCoulombHardness.lean) that also contains the binary-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)
- 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 H₂ singlet–triplet exchange splitting, a continuum Coulomb benchmark for the paper's claim that Hubbard superexchange between localized modes realizes the prescribed positive spin exchange.
- 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 reduction of the continuum one-particle problem with planar wells to localized modes with effective hopping.
- Barry Simon, Semiclassical analysis of low lying eigenvalues. II. Tunnelling (Annals of Mathematics 120 (1984), 89–118). The paper cites only part I [Sim83]; part II gives tunnelling asymptotics for $-\tfrac12\Delta+\lambda^2V$ with several wells, related to the dilated-coordinate well construction with large scale $\lambda$.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 275
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026,
author = {{OpenAI}},
title = {{QMA-hardness of continuum Coulomb energy with unit nuclear charges}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf}{OAI:QMA-hardness-of-continuum-Coulomb-energy-with-unit-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:
- Continuum Coulomb hardness with binary nuclear charges · September 24, 2026