VibeMathedMath problems solved with AI
Sign in

The Basso-Farhi-Marwaha-Villalonga-Zhou conjecture: QAOA attains the Sherrington-Kirkpatrick ground-state energy as depth grows

For the zero-field Sherrington-Kirkpatrick model the expected QAOA energy per spin at fixed depth pp and fixed angles has an infinite-size limit vp(γ,β)v_p(\gamma,\beta) (Farhi-Goldstone-Gutmann-Zhou), and Qp=sup⁡vpQ_p=\sup v_p cannot exceed the Parisi ground-state value P∗P_*. Fixed-instance optimality as depth grows (Farhi-Goldstone-Gutmann) gives no depth uniform in system size. Basso, Farhi, Marwaha, Villalonga and Zhou, relating the limit to QAOA on large-girth regular graphs, conjectured that increasing depth eventually reaches the Parisi optimum; numerics (Boulebnane et al., Sels-Morone) supported it. Is lim⁡p→∞Qp=P∗\lim_{p\to\infty}Q_p=P_*?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Quantum optimization; spin glasses
Posed by
J. Basso, E. Farhi, K. Marwaha, B. Villalonga and L. Zhou (The QAOA at high depth for MaxCut on large-girth regular graphs and the SK model, TQC 2022, Section 7, Eq. (7.2))
Year posed
2021
Years open
5y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
16 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: lim⁡p→∞Qp=P∗\lim_{p\to\infty}Q_p=P_* for the zero-field SK model; for every ε>0\varepsilon>0 some finite depth and deterministic angles, independent of nn and the disorder, reach expected energy per spin ≥P∗−ε\ge P_*-\varepsilon in the limit n→∞n\to\infty taken first, with the standard cost Hamiltonian and transverse-field mixer. It also gives leading-order optimal expected MaxCut on large-degree random regular graphs (size before degree). Not given: any depth bound, finite-size estimate, efficient angle selection, or concentration beyond expectation. The companion proves full support on [0,1)[0,1) of zero-temperature Parisi minimizers, a result H.-B. Chen had already posted (arXiv 2607.18032), as the companion acknowledges.

What the AI did

Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against the BFMVZ conjecture as quoted; proof not refereed. Lean: the release's Comparator challenges SKValue and SKFullSupport (lean/docs/281.md) formalize Parisi ground-state value identities, a finite Gaussian approximation and the companion's full-support theorem, conditional on an admissible minimizer; the docs say the selected statement does not assert convergence of QAOA circuit energies, so the tier is unreviewed. The primary route imports the companion's support theorem; an alternative route uses Lopatto's and El Alaoui-Montanari-Sellke's results.

Sources

Changelog1 change

Discussion