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Polynomial PEPS approximation of gapped square-grid ground states
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Abstract
We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an $L\times L$ square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L−1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.
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Notes
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work in progress
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.
Formalization
- LionSR/TNLean, Tracking: polynomial PEPS approximation of gapped square-grid ground states (issue #8736) (GitHub, opened 7 Oct 2026, closed 18 Oct 2026). The maintainer's closing comment reports Theorem 1.1 as
TNLean.PEPS.Approximation.polynomialPEPSApproximation, using only the standard axioms. An earlier comment (17 Oct) listed three requirements of Proposition 7.1 that the work then in progress had not yet established. - TNLean contributors, Polynomial PEPS approximation of gapped square-grid ground states (formalization blueprint, accessed 2026-10-21). This chapter links Lean declarations for the paper’s distribution, column-selection and dyadic-routing arguments, explicitly separating intermediate hypotheses from their verification elsewhere in the development. [AI-generated per source]
- TNLean, Product-Reference Estimates and a Column on the Open Square (formalization scope note, 2026-10-13; marked resolved 2026-10-17). The note explains the formal passage from product-reference and density-compression estimates to column extraction and routing on the original open square.
- hz-xiaxz in LionSR/TNLean, Formalize the two-dimensional area law and polynomial PEPS approximation from a global spectral gap (issue #8732) (GitHub, 7 Oct 2026). Proposes the formalization and states that the Lean code in the OpenAI release proves neither headline theorem. [family-level]
- TNLean contributors, Tensor-network theory, formalized in Lean 4 (Lean 4 repository, accessed 2026-10-21). The repository hosting the family-specific formal developments describes a predominantly agent-produced formalization workflow and notes that some library results remain unfinished or assumed as axioms. [AI-generated per source] [family-level]
- LionSR/QICLean, doc(paper-gaps): update the area-law and PEPS gap notes after the TNLean proofs (PR #856) (GitHub, merged 18 Oct 2026). Updates three "paper-gap" notes on the area-law and PEPS manuscripts to resolved status, citing
polynomialPEPSApproximationandadaptivePatchConstraint(Proposition 4.1). [family-level] - LionSR/TNLean, Tracking: explicit constants in the area law and the polynomial PEPS approximation (issue #9638) (GitHub, opened 18 Oct 2026, closed). Aims to make the constants C, c of Theorem 1.1 explicit; the thread records candidate values, some of them marked heuristic or unverified. [family-level]
- LionSR/TNLean, Tracking: polynomial PEPS approximation for interactions of range R (issue #9641) (GitHub, opened 18 Oct 2026, open). Targets an extension of Theorem 1.1 from nearest-neighbour to range-R interactions, explicitly described as "an extension of the source, not a source theorem"; it was in progress at collection time.
- LionSR/TNLean, Matrix product state approximation of gapped chains by the protocol route of the polynomial PEPS paper (issue #9555) (GitHub, 16 Oct 2026, open). Proposes a one-dimensional analogue of this paper's distributed-protocol construction, giving polynomial-bond-dimension MPS approximations of gapped chain ground states.
Related prior literature mentioned by others
- J. Ignacio Cirac, José Garre-Rubio and David Pérez-García, Mathematical open problems in Projected Entangled Pair States (arXiv:1903.09439; Revista Matemática Complutense, 2019). Identified by VibeMathed as prior context: the survey's Questions 7–8 on polynomial-bond-dimension approximation relate to the manuscript’s open-square theorem.
Possibly relevant literature (found by the agent, not checked)
- Yichen Huang, Two-dimensional local Hamiltonian problem with area laws is QMA-complete (J. Comput. Phys. 443, 110534 (2021)). Relevant to the paper's remark that Theorem 1.1 is an existence statement with no efficient construction or contraction, since it shows that 2D area-law ground states need not have efficient classical descriptions unless MA = QMA; it is not in the bibliography.
- Tomotaka Kuwahara, The Rényi area law conjecture is false below α=1 in high dimensions (arXiv:2610.15199, Oct 2026). Constructs uniformly gapped 2D ground states that violate a Rényi-α area law for every α<1, which bears on the paper's discussion of Huang's PEPS result assuming such a Rényi area law (Section 1.2).
News and blogs
- Azmat, 2D Area Law: How OpenAI’s Proof Connects the Global Gap to Entanglement (Binary Verse AI, 2026-10-13). This exposition outlines the proposed area-law argument and distinguishes the companion’s polynomial-bond PEPS existence claim from efficient tensor construction or contraction. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 265
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026,
author = {{OpenAI}},
title = {{Polynomial PEPS approximation of gapped square-grid ground states}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf}{OAI:Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026}},
year = {2026}
}Family 265: Area laws and tensor networks for two-dimensional gapped systems
Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open $L\times L$ squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.
Other papers in this family:
- A two-dimensional area law from a global spectral gap · September 24, 2026