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Phys-59 · physics · mathematics · Physics

Polynomial-bond PEPS approximation of every gapped square-grid ground vector

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix site dimension $q\geq 2$ and bounds $J,Delta>0$. For every L-by-L square-grid Hamiltonian with site and nearest-neighbor terms of norm at most J, a unique normalized ground vector and global spectral gap at least Delta, does there exist a PEPS of maximum bond dimension at most C $L^c$ whose normalized vector is within $L^{-1}$, up to global phase, of that ground vector, with C,c depending only on q,J,Delta?

Short English statement

Fix site dimension $q\geq 2$ and bounds $J,Delta>0$. For every L-by-L square-grid Hamiltonian with site and nearest-neighbor terms of norm at most J, a unique normalized ground vector and global spectral gap at least Delta, does there exist a PEPS of maximum bond dimension at most C $L^c$ whose normalized vector is within $L^{-1}$, up to global phase, of that ground vector, with C,c depending only on q,J,Delta?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-59-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-59

Uniformly gapped square-grid ground states are claimed to admit PEPS approximations with polynomial bond dimension and normalized vector error at most $L^{-1}$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Polynomial-bond PEPS approximation of every gapped square-grid ground vector

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Uniformly gapped square-grid ground states are claimed to admit PEPS approximations with polynomial bond dimension and normalized vector error at most $L^{-1}$.

Mathematical statement of the claimed result

Result 1.
Fix local dimension $q\ge2$ and $J,\Delta>0$. On an $L$-by-$L$ square grid, let \[ H=\sum_vh_v+\sum_eh_e,\quad\|h_v\|,\|h_e\|\le J,\qquad H-E_0I\ge\Delta(I-|\Omega\rangle\langle\Omega|), \] with site/edge-supported Hermitian terms and a unique normalized ground vector $\Omega$. The claimed constants $C,c>0$, depending only on $q,J,\Delta$, give a nonzero projected entangled-pair state $\Phi$ for every $L\ge2$, with maximum bond dimension $CL^c$, satisfying \[ \min_{\theta\in\mathbb R}\|\Phi/\|\Phi\|-e^{i\theta}\Omega\|\le L^{-1}. \] The assertion is existence of the PEPS, without an efficient construction guarantee.

Outline of the argument

Separated-region information bounds lead to adaptive finite-rank constraints on small patches. Distributed density networks are compressed using encoded frames and controlled ownership changes; a planar distribution protocol assembles the PEPS without assuming gaps for subregions.

Formal verification

Family 265 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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