View source for QMA-hardness of continuum Coulomb energy

'''QMA-hardness of continuum Coulomb energy''' concerns the computational complexity of the electronic ground-state energy of a molecule with clamped point nuclei. Two manuscripts in OpenAI's public mathematics collection, both dated 24 September 2026, state that deciding whether this energy lies below one threshold or above another, over the full spinful fermionic continuum and with no orbital basis supplied, is QMA-hard: one with positive nuclear charges encoded in binary, the other with every nuclear charge equal to one.

== Background ==
* As recounted in the manuscripts' introductions:
** ''Finite-dimensional versions.'' Liu, Christandl and Verstraete established QMA-completeness of the finite-mode ''N''-representability problem under polynomial-time Turing reductions, using general two-body fermionic Hamiltonians. Whitfield, Love and Aspuru-Guzik explained the distinctions between such formulations and the role of magnetic fields in the early hardness results.
** ''Designed potentials.'' Schuch and Verstraete proved QMA-completeness for a Hubbard model with local magnetic fields and developed a continuum realization with designed scalar and spin-dependent potentials.
** ''A supplied orbital basis.'' O'Gorman, Irani, Whitfield and Fefferman proved QMA-completeness for electronic structure in a supplied finite orbital basis, including a construction with positive unit point charges, and raised the question for the full space. A projection onto orbitals gives only an upper bound on the continuum ground energy, so a reduction for the continuum problem also needs a lower bound that excludes every omitted state.
** ''The spin source.'' Cubitt, Montanaro and Piddock proved QMA-hardness of signed Heisenberg exchange on square-lattice subgraphs, with a spatially sparse construction and polynomially bounded interactions; the singlet mediators of Piddock and Montanaro replace signed by positive exchange.

== The original results ==
* '''Binary nuclear charges (24 September 2026).''' ''Continuum Coulomb hardness with binary nuclear charges'' states in its main theorem that, for the electronic Hamiltonian of ''N'' electrons with distinct rational nuclear positions and positive integer charges encoded in binary, a unary electron number and rational thresholds ''a'' < ''b'' with ''b'' − ''a'' at least the reciprocal of the input length, deciding whether the infimum of the spectrum on the antisymmetric spinful continuum space is at most ''a'' or at least ''b'' is QMA-hard under deterministic classical polynomial-time many-one reductions; the reduction produces instances with ''b'' − ''a'' ≥ 1. Charges and inverse nuclear separations may be exponentially large.
* '''Unit nuclear charges (24 September 2026).''' ''QMA-hardness of continuum Coulomb energy with unit nuclear charges'' states the same hardness when every nuclear charge equals one. Its reduction outputs polynomially many nuclei and electrons, rational positions of polynomial bit length and thresholds at least one apart.
* ''Scope, as stated:'' both manuscripts minimize over every total-spin sector, supply no orbital basis, electronic state, magnetic field or additional external potential, impose no neutrality, binding or spectral-gap promise, and do not assert membership in QMA. The threshold separation is a precision promise, not a spectral gap of the molecule.
* ''Method, as described:'' a QMA-hard Heisenberg model is made positive with singlet mediators; each spin becomes an electron in a localized well, and positive exchange arises from virtual hopping through double occupancy, Anderson's superexchange mechanism, with the direct Coulomb terms kept explicitly. The binary-charge manuscript builds the wells from nuclei of very large charge; the unit-charge manuscript first builds a nonnegative nuclear density with a confining slab and then replaces it by distinct unit point charges through a transported quadrature rule. A lower bound on all states outside the localized modes completes each reduction.
* '''Sources.''' Commit <code>adc7f12</code> of the openai/math repository: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026 ''Continuum Coulomb hardness with binary nuclear charges''] and [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026 ''QMA-hardness of continuum Coulomb energy with unit nuclear charges''].
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/275.md lean/docs/275.md] states that the formalization proves QMA-hardness of approximating the electronic Coulomb spectral infimum in three-dimensional continuum space, both with binary-encoded positive integer charges and with unit charges, over antisymmetric continuum states and all spin sectors. Its Comparator statement is [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ContinuumCoulombHardness.lean ContinuumCoulombHardness.lean].
* Within the collection, the unit-charge manuscript notes that the strict-binding excess-charge bound of the ionization manuscript does not supply the continuum lower bound needed here; see [[The ionization and generalized ionization conjectures]].

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 31 October 2026)
|-
| 2026-10-07 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/275-continuum-coulomb-energy-hardness ''Continuum Coulomb energy: hardness beyond an orbital basis''], an article of the Latent Seminar blog at lowrank.github.io || Reading, exposition || Explains family 275: the continuum electronic energy problem; hardness and the encoding of physical data; why a finite orbital calculation is insufficient; spin interactions and virtual double occupancy; retaining direct Coulomb interactions; two physical realizations of the wells; from positive density to unit point charges; the continuum complement and square completion; thresholds and polynomial output length. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. Read on 31 October 2026.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet], a GitHub review of openai/math, tables at repository commit <code>c76f1e4</code> of 14 October 2026 || Review, fidelity of the formal statements || Rates this family "full": the Lean theorems state that every promise problem in QMA reduces, by deterministic polynomial-time many-one reductions defined with Mathlib's polynomial-time Turing machines, to deciding the ground energy for the unit-charge and for the binary-charge encodings, with the ground energy defined as the infimum of the two-spin Coulomb form over normalized antisymmetric states. It lists the hand-written quantum complexity definitions, namely the gate set, the verifier and the class QMA, as definitions to check, and records a sandboxed Comparator pass of 12 October 2026 after a local Comparator time-out. || Review repository. Read on 31 October 2026.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal], task #348 || Verification, second kernel || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal lists <code>ContinuumCoulombHardness</code> among seven challenges that con-leche rejects because its evaluation fuel runs out, a depth limit in closed-term evaluation on which, the journal notes, Lean's official kernel also works hard, with 62 seconds and 10.5 GB for the declaration; with a raised fuel limit its memory use on the whole export grew past 25 GB. The challenge is listed for a re-run after a rewrite. || Development journal; a resource limit of the second kernel, not a reported error in the proof. Read on 31 October 2026.
|-
| 2026-10-18 || ''AI Proved What?'', episode 275 [https://aiprovedwhat.com/episodes/275-coulomb-qma-hard/ A hard chemistry problem, proved at last?], remade version of 23 October 2026 || Reading, exposition || Video explainer of two OpenAI preprints, 69 pages, dated 24 September 2026: deciding whether the lowest electronic energy over all states with fixed nuclei is below one threshold or above another, at least one hartree apart, is QMA-hard, with large charges written in binary in one paper and unit charges at rational positions in the other. It states that the preprints do not claim membership in QMA or anything about real molecules, and that the hard instances are very large. || Episode status, last checked 30 October 2026: posted as preprints; under expert review, not yet; peer reviewed, not yet; checked in Lean, both theorems. The site states that its scripts and animations are made with AI assistance and that the narration is an AI-generated voice. Read on 31 October 2026.
|}

== Notes ==
* The background summarizes the manuscripts' own accounts of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]