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QAOA attains the SK ground-state energy in the thermodynamic-first limit
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Abstract
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.
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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.
Comments and discussion
- Ruslan Shaydulin, with a comment by Junyu Liu, QAOA Convergence in Higher-Order Spin Glasses (LinkedIn, accessed 2026-10-21). Shaydulin discusses OpenAI's claimed resolution of thermodynamic-first SK optimality and relates it to his group's spin-boson work on higher-order pure and mixed spin glasses; in a comment, Liu reports difficulty following the manuscript's mathematical language and explicitly leaves its correctness unresolved.
Formalization
- OpenAI math release, QAOA attains the SK optimum in the thermodynamic-first limit: Lean scope notes (lean/docs/281.md, 2026-10-06). The Lean scope covers Parisi ground-state value identities and finite Gaussian approximations (conditional on an admissible minimizer) and the companion's full-support theorem; per the notes it "does not itself assert convergence of QAOA circuit energies." [family-level]
Related prior literature mentioned by others
- S. Boulebnane, A. Khan, P. Subrahmanya, D. Herman, E. Farhi, B. Villalonga, R. Shaydulin, Decoupling of the QAOA into independent spin-boson systems and high-depth performance on pure and mixed spin glasses (arXiv:2609.40309, 30 Sep 2026). Discussed by Shaydulin alongside OpenAI's claim, it extends the spin-boson description of infinite-size QAOA (cited in the paper via arXiv:2505.07929) to mixed spin glasses and reports high-depth numerical optimization.
Possibly relevant literature (found by the agent, not checked)
- M. Goh, The Overlap Gap Property limits limit swapping in the QAOA (arXiv:2404.06087, Quantum Inf. Comput. 25 (2025)). Studies when QAOA limits in size and depth can be exchanged; relevant to the paper's emphasis on the order of limits (size first, then depth).
- J. Basso, D. Gamarnik, S. Mei, L. Zhou, Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models (arXiv:2204.10306, FOCS 2022). Analyzes fixed-depth QAOA on mixed spin models in the infinite-size limit and shows obstructions for even pure q-spin with q ≥ 4; context for the fixed-word SK limit and the regular-MaxCut corollary.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 281
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/281.md. The Lean statements have not been checked by people against the paper's claims.
Family 281: 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.
Other papers in this family:
- Full support of the zero-temperature Sherrington-Kirkpatrick order parameter · September 27, 2026