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Polynomial-bond-dimension PEPS approximation of gapped ground states on an open square grid

A projected entangled-pair state (PEPS) on a lattice places a tensor at each site, with one virtual index of dimension DeD_e per incident edge, and contracts the virtual indices. In one dimension, gapped ground states have matrix-product approximations of polynomial bond dimension (Hastings 2007). Cirac, Garre-Rubio and Perez-Garcia (2019, Section 2.2, Questions 7-8) asked whether a spectral gap likewise ensures polynomial bond dimension at inverse-polynomial Hilbert-space error in two dimensions, for translation-invariant finite-range Hamiltonians on a torus; the best known general bound was quasipolynomial under an extra density-of-states hypothesis. Does the unique ground state of a uniformly gapped local Hamiltonian on an L×LL\times L lattice admit a PEPS approximation with bond dimension polynomial in LL and error inverse-polynomial in LL?

Result
Proved(see note)
Status
Variant only
AI contribution
AI-discovered
Method
Argument
Field
Tensor networks; projected entangled-pair states
Posed by
J. Ignacio Cirac, Jose Garre-Rubio and David Perez-Garcia (Questions 7-8, Rev. Mat. Complut. 2019)
Year posed
2019
Years open
7y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for fixed q≥2q\ge2, J,Δ>0J,\Delta>0 there are C,cC,c such that for every L≥2L\ge2 and every Hamiltonian on the open grid {1,…,L}2\{1,\dots,L\}^2 with on-site and nearest-neighbour terms of norm at most JJ, a unique ground vector Ω\Omega and global gap Δ\Delta, some PEPS Φ\Phi on the same grid with bond dimension at most CLcCL^c satisfies min⁡θ∥Φ/∥Φ∥−eiθΩ∥≤L−1\min_\theta\|\Phi/\|\Phi\|-e^{i\theta}\Omega\|\le L^{-1}. It does not treat the torus or translation-invariant formulation of the posed question, longer-range interactions, the structural (injective, gapped parent) PEPS conjecture of Schwarz-Buerschaper-Eisert, or efficient algorithms.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. It is the companion of 'A two-dimensional area law from a global spectral gap' (same date), whose entropy bound and collar estimates it uses as inputs.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Questions 7-8 of Cirac, Garre-Rubio and Perez-Garcia as the manuscript reports them. The posed formulation is translation-invariant on a torus; the theorem instead treats arbitrary (not necessarily translation-invariant) on-site and nearest-neighbour interactions on an open square, so it neither contains nor is contained in the posed case, hence Variant. It is an existence result: no efficient construction or contraction is claimed. It depends on the companion area-law paper, which is also unreviewed. No Lean formalization.

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