QAOA energy bounds at large interaction order
A proof candidate for the large odd-order fixed-depth Gaussian obstruction, with a logarithmic-depth bound in the signed sparse model.
Yuxuan Zhang · EPFL; Princeton University
Manuscript, version 1, 26 October 2026 (PDF, 7 pages) · LaTeX source, original target, proof records and audit (ZIP)
How much positive energy can fixed-depth QAOA obtain in a random spin model when the interaction order becomes large, including odd orders?
Result
For the ordered Gaussian q-spin model normalized to have variance n at each assignment, the candidate theorem bounds the limiting expected achieved energy per spin by any prescribed positive α once q exceeds a threshold depending only on α. That threshold is independent of the fixed number of QAOA layers. The thermodynamic limit comes before optimization over deterministic, instance-independent angles.
In the signed ordered Poisson hypergraph model, the manuscript also gives a bound of α + n⁻¹⁄⁸ for up to κ log n layers, with fixed q and density and a sufficiently small positive κ.
What the argument adds
The proof computes the covariance of masked instances at different interpolation times, uses an exact Poisson moment bound, and glues QAOA outputs along a tree while preserving every Born marginal. An elementary metric lemma then selects a forbidden configuration using only m² branches.
Local quantum coupling and concentration are established background in Chou–Love–Sandhu–Shi. The dense conclusion uses the fixed-depth correspondence of Basso–Gamarnik–Mei–Zhou. Mints–Anschuetz supply related tree/Ramsey methodology for a different quantum model. General odd-order overlap gaps are already known from Gamarnik–Jagannath–Kızıldağ.
The proposed contribution is the precise mixed-time composition for this signed model. The manuscript does not claim a new general overlap-gap principle or the first logarithmic-depth QAOA obstruction.
Scope and review
The Gaussian result does not cover q = 3, angles learned from the instance, or logarithmic depth in the dense model. The sparse depth extension is stated separately. Setting all angles to zero gives a zero-energy baseline consistent with the bound; this is a limitation of a specified QAOA family, not of all quantum algorithms.
The full manuscript received a sequential audit by the assembling agent. The audit checked the normalization, covariance mask, marginal coupling, tail constants, growing-depth costs and source-theorem quantifiers. No specific unresolved defect or direct covering theorem was identified in those checks. Independent review of the complete manuscript and its novelty remains outstanding. Publication is not external peer review, formal proof verification, or acceptance of M26 as solved by a problem catalog.
The source package preserves the earlier arguments and identifies the mixed-time proof that supersedes the older equal-time Ramsey route.
This paper belongs to the 26 October 2026 two-paper release.
Human and AI contributions. The work was developed in an author-directed research workflow. AI systems assisted mathematical derivation, primary-source comparison, checks and scientific writing. The research log describes the author’s direction and research harness.