Result 281, Mathematical physics

QAOA attains the SK optimum in the thermodynamic-first limit

Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.

Lean formalization Proof

The bigger picture

Why it matters

The manuscript reports that a layered quantum optimization method can approach the lowest possible average energy in a benchmark model of disordered magnets. The key qualification is the order in which its limits are taken.

What changes?

The Quantum Approximate Optimization Algorithm, or QAOA, alternates cost-based quantum operations with a transverse-field mixer that rotates spins. The manuscript studies the Gaussian zero-field Sherrington-Kirkpatrick model: spins interact in pairs through random Gaussian couplings, with no external field. It reports that, for every accuracy, a finite circuit depth and deterministic angle settings, independent of system size and the particular random couplings, attain that accuracy in limiting expected energy per spin. System size tends to infinity before circuit depth increases.

What does that help mathematicians do?

This would establish that fixed-parameter QAOA is not separated from this model's ground-state energy by a permanent accuracy gap in the stated limit. Researchers could distinguish a question of expressiveness from a question of resources: suitable circuits exist, but the manuscript gives neither a quantitative depth bound nor an efficient angle-selection procedure. It does not establish the same conclusion with the limits reversed.

Are there practical applications?

The summary also reports leading-order optimal expected MaxCut values on random regular graphs, where every vertex has the same degree, as size tends to infinity before degree. MaxCut partitions vertices to maximize edges crossing between the groups. This connects the result to graph optimization, but its immediate value is foundational: an asymptotic performance guarantee, not a demonstrated practical speedup.

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

QAOA attains the SK ground-state energy in the thermodynamic-first limit

September 25, 2026 70 pages

We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer. This proves the eventual Parisi-optimality conjecture of Basso, Farhi, Marwaha, Villalonga, and Zhou in its fixed-parameter thermodynamic formulation. We give no quantitative bound on the required depth or efficient angle-selection procedure.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026,
  author = {{OpenAI}},
  title = {{QAOA attains the SK ground-state energy in the thermodynamic-first limit}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf}{OAI:QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026}},
  year = {2026}
}

Full support of the zero-temperature Sherrington-Kirkpatrick order parameter

September 27, 2026 27 pages

We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on [0,1)[0,1). Thus its support has no gaps at any overlap scale below one. We use the covariance normalization ξ(t)=t2/2\xi(t)=t^2/2.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026,
  author = {{OpenAI}},
  title = {{Full support of the zero-temperature Sherrington--Kirkpatrick order parameter}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf}{OAI:Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/281.md.

QAOA attains the SK optimum in the thermodynamic-first limit

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

Scope

The linked formalization supplies variational identities for the Gaussian zero-field Sherrington–Kirkpatrick ground-state energy used in the paper's QAOA argument. Conditional on an admissible Parisi minimizer and its associated diffusion, it proves convergence of the finite-system ground-state energy, identifies its limit with the Parisi value, and gives equivalent terminal-martingale and integrated-curvature formulas.

It also proves convergence of finite Gaussian coefficient sums to the curvature integral, which approaches the ground-state value as the terminal time tends to one. The selected statement covers these value and approximation results; it does not itself assert convergence of QAOA circuit energies.

The formalization proves that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on [0,1)[0,1). Equivalently, the order parameter increases strictly between every two overlap values below one, so its support has no gap. The covariance normalization is ξ(t)=t2/2\xi(t)=t^2/2.

The statement also constructs the associated diffusion and proves its selected self-consistency moment identities. It is conditional on the order parameter being a minimizer and does not separately assert existence of a minimizer.

Comparator links

Result Comparator statement
Parisi ground-state value identities and finite Gaussian approximation SKValue.lean
Full support of zero-temperature SK minimizers SKFullSupport.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.