Result 275, Mathematical physics

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.

Lean formalization Algorithm or complexity result

The bigger picture

Why it matters

Even an idealized collection of electrons and fixed nuclei can encode difficult quantum computational problems. The manuscript reports this barrier for approximating the lowest possible electronic energy, even when every nucleus has charge one.

What changes?

The unit-charge manuscript treats three-dimensional electrons around fixed nuclei of charge one. It minimizes Coulomb energy over all continuum states obeying fermionic exchange rules, including spin, without restricting them to an orbital basis. Its deterministic classical polynomial-time reduction uses polynomially many nuclei and electrons, distinct rational nuclear positions with polynomial bit lengths, and energy thresholds separated by at least one. No magnetic field, extra external potential or assumption that the electrons form a bound system is imposed.

What does that help mathematicians do?

QMA-hardness means these energy questions are at least as hard as every decision problem whose quantum proofs can be checked efficiently by a quantum computer. The claimed reduction locates that difficulty in the continuum Coulomb model itself, rather than in a chosen finite orbital approximation or exceptionally large nuclear charges. The threshold separation shows that the barrier does not require distinguishing exponentially close energy thresholds.

Are there practical applications?

The immediate value is foundational for electronic-structure theory: the result identifies a worst-case computational barrier in a model using only electrons and fixed unit-charge nuclei. It does not show that typical molecules are intractable or that useful approximation methods fail. Researchers can use the distinction to separate universal algorithmic guarantees from success on restricted physical systems.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Continuum Coulomb hardness with binary nuclear charges

September 24, 2026 47 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

QMA-hardness of continuum Coulomb energy with unit nuclear charges

September 24, 2026 22 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/275.md.

QMA-hardness of continuum Coulomb energy

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves QMA-hardness of approximating the electronic Coulomb spectral infimum in three-dimensional continuum space when positive integer nuclear charges are encoded in binary. The instances have distinct rational nuclear positions and a unary electron count, and the energy ranges over antisymmetric continuum states and all spin sectors. A deterministic polynomial-time many-one reduction produces the promise gap with threshold separation at least one, while keeping the complete output length polynomial. The same Comparator file also contains the companion unit-charge result.

The formalization proves QMA-hardness of approximating the electronic ground-energy infimum for clamped unit-charge nuclei in the full spinful fermionic continuum. The reduction is deterministic and polynomial time, with rational nuclear positions and separated rational energy thresholds. No orbital basis, magnetic field, additional external potential, or binding premise is part of the input.

The same Comparator file also includes the companion hardness result when positive integer nuclear charges are encoded in binary.

Comparator links

Result Comparator statement
QMA-hardness of continuum Coulomb energy with binary charges ContinuumCoulombHardness.lean
QMA-hardness of continuum Coulomb ground-energy approximation ContinuumCoulombHardness.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.