View source for QAOA and the Sherrington–Kirkpatrick ground-state energy

'''QAOA and the Sherrington–Kirkpatrick ground-state energy''' concerns whether the quantum approximate optimization algorithm, QAOA, can reach the ground-state energy per spin of the Sherrington–Kirkpatrick spin glass when the number of spins tends to infinity before the circuit depth. Two manuscripts in OpenAI's public mathematics collection, dated 25 and 27 September 2026, claim this limit, which Basso, Farhi, Marwaha, Villalonga and Zhou conjectured, and a companion result on the zero-temperature Parisi order parameter. The collection's Lean formalization covers classical inputs only, not the QAOA theorem.

== Background ==
* As recounted in the manuscripts' introductions:
** ''QAOA on the SK model.'' Farhi, Goldstone and Gutmann introduced QAOA. Farhi, Goldstone, Gutmann and Zhou described its performance at every fixed depth in the infinite-size limit and proved concentration at fixed parameters. Basso, Farhi, Marwaha, Villalonga and Zhou related that limit to QAOA on large-degree trees and conjectured that increasing depth eventually attains the Parisi optimum. Boulebnane and coauthors gave a fixed-depth spin–boson representation with numerical evidence for convergence, and Sels and Morone studied the question with a semiclassical approximation.
** ''Order of limits.'' For each fixed instance, increasing the depth approaches its optimum, a result of Farhi, Goldstone and Gutmann; this gives no single finite depth that works as the number of spins tends to infinity.
** ''The Parisi order parameter.'' Auffinger, Chen and Zeng proved that the zero-temperature order parameter has infinitely many support points. H.-B. Chen proved full support on [0, 1) together with a smooth density, in a preprint of July 2026 that the companion manuscript cites. Montanari, and El Alaoui, Montanari and Sellke, developed message-passing algorithms whose optimality depends on such support properties; at positive temperature, Lopatto proved full interval support.

== The original results ==
* '''QAOA, 25 September 2026.''' The main theorem of ''QAOA attains the SK ground-state energy in the thermodynamic-first limit'' states that, for the zero-field SK model with the standard cost Hamiltonian and transverse-field mixer, the best limiting expected energy per spin at depth ''p'' tends to the Parisi ground-state value as ''p'' tends to infinity: for every accuracy there are a finite depth and deterministic angles, independent of the system size and of the disorder, that reach it. The manuscript states that this proves the conjecture of Basso, Farhi, Marwaha, Villalonga and Zhou in its fixed-parameter thermodynamic formulation, and gives a corollary for MaxCut on random regular graphs of large degree, with the size tending to infinity before the degree.
* '''Full support, 27 September 2026.''' ''Full support of the zero-temperature Sherrington–Kirkpatrick order parameter'' states that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure zero-field SK model has full relative Stieltjes support on [0, 1), so its support has no gap. It acknowledges that H.-B. Chen proves the same full-support conclusion and describes its own proof as excluding gaps through an explicit coercive identity.
* ''Scope, as stated:'' the QAOA manuscript gives no quantitative bound on the required depth and no efficient procedure for choosing the angles; the final circuit may contain zero angles, inverse gates and very large angles.
* ''Method, as described:'' the scalar diffusion of the companion yields a finite function of independent Gaussian variables whose value approximates the ground-state energy. The QAOA manuscript realizes this calculation on large-degree trees with quantum gates, first with temporary Gaussian seed controls that a later construction removes, and then fixes one finite word of gates before letting the number of spins tend to infinity. An independent positive-temperature route uses Lopatto's full-support theorem and the analysis of El Alaoui, Montanari and Sellke.
* '''Sources.''' Commit <code>adc7f12</code> of the openai/math repository: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/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''] and [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026 ''Full support of the zero-temperature Sherrington–Kirkpatrick order parameter''].
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/281.md lean/docs/281.md] states that the formalization supplies variational identities for the SK ground-state energy used in the QAOA argument, conditional on an admissible Parisi minimizer and its diffusion, and that it does not itself assert convergence of QAOA circuit energies. It also proves the full-support theorem, conditional on the order parameter being a minimizer. Its Comparator statements are [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SKValue.lean SKValue.lean] and [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SKFullSupport.lean SKFullSupport.lean].
* Within the collection, other manuscripts on the SK model are described on [[The low-temperature Sherrington–Kirkpatrick fluctuation law]] and [[Glauber dynamics of the Sherrington–Kirkpatrick model]].

== 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/281-qaoa-sk-parisi-support ''QAOA and the SK optimum: Parisi support and the order of limits''], an article of the Latent Seminar blog at lowrank.github.io || Reading, exposition || Explains family 281: spin-glass energy and quantum optimization; the Parisi variational problem; contact conditions and the exclusion of gaps; the integrated curvature certificate; from the diffusion to a finite Gaussian calculation; rooted energy and quantum synthesis; fixing the word before taking limits; source and formalization boundaries. || 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 "supporting-only": no QAOA statement exists in the Lean challenges, which state the Parisi value identities for the finite-size ground-state energy, conditional on a minimizer and a diffusion, and the full support of zero-temperature minimizers. It lists the hand-written Parisi and stochastic-control objects as definitions to check, and records sandboxed Comparator passes for both challenges on 8 October 2026. || Review repository. Read on 31 October 2026.
|-
| 2026-10-12 || P. M. Aronow and P. Lopatto, [https://hexagonmath.org/abs/2610.00177v1 ''An N<sup>−7/10</sup> bound for the finite-size bias of the Sherrington–Kirkpatrick free energy''], Hexagon 2610.00177v1 || Citation, related result || In its related-work section, after crediting H.-B. Chen with full support and a smooth density at zero temperature, notes that full support at zero temperature is also stated in the companion's Theorem 1.1. Its own theorem uses interval support at a fixed inverse temperature above one. || Preprint on Hexagon. Read on 31 October 2026.
|-
| 2026-10-13 || J. Shi, [https://github.com/jnshi/math-lineage/tree/da8c0e694f00c2575ccf9d26c0ea560543ef0a1b/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026 precedence review of the companion] in [https://github.com/jnshi/math-lineage jnshi/math-lineage] || Prior work, attribution || Lists seven places where earlier work already contains part of the companion manuscript, four of them classed as attribution gaps. It classes the main full-support result as the same result as H.-B. Chen's July 2026 preprint, which the manuscript cites after its theorem but not in its abstract, and names earlier work of Zhou, Lopatto, Auffinger, Chen and Zeng, El Alaoui, Montanari and Sellke, Chen, Handschy and Lerman, and Auffinger and Chen for supporting steps. || Report generated by LLM agents running a precedence-review skill; the repository states that it makes no claim of completeness and has not been vetted by a human. Read on 31 October 2026.
|-
| 2026-10-16 || J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_154.json rank_154] of the QAOA manuscript and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_287.json rank_287] of the companion in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] || Review, AI-assisted audit || Both reviews report no gap. The QAOA review notes that the main route depends on the companion's full-support theorem and finite Gaussian coefficient formula, which it checked as non-circular, and records as a reference-integrity point that the cited theorem numbers of the companion do not match its labels; it did not re-derive the long gate-calculus estimates. The companion review reports a symbolic check of the rational identity and the table of the certificate. || Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0.
|-
| 2026-10-18 || ''AI Proved What?'', episode 281 [https://aiprovedwhat.com/episodes/281-qaoa-sk-optimum/ QAOA reaches the optimum, proved at last?], remade version of 23 October 2026 || Reading, exposition || Video explainer of the two preprints, 97 pages: for every accuracy, a finite depth and fixed angles reach the SK optimum on average as the size grows, with a MaxCut corollary on random regular graphs. It states that the preprints give no depth bound, angle recipe, finite-size guarantee or speedup, that a classical algorithm of Montanari already reaches any accuracy, and that the companion's result was posted earlier by H.-B. Chen, as the paper acknowledges. || Episode status, last checked 30 October 2026: posted as preprints, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, in part, classical inputs only. The site states that it is made with Anthropic's Claude. 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]]