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

Theorem 1.1: for fixed $q\ge2$, $J,\Delta>0$ there are $C,c$ such that for every $L\ge2$ and every Hamiltonian on the open grid $\{1,\dots,L\}^2$ with on-site and nearest-neighbour terms of norm at most $J$, a unique ground vector $\Omega$ and global gap $\Delta$, some PEPS $\Phi$ on the same grid with bond dimension at most $CL^c$ satisfies $\min_\theta\|\Phi/\|\Phi\|-e^{i\theta}\Omega\|\le L^{-1}$. 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.

Theorem 1.1: for fixed $q\ge2$, $J,\Delta>0$ there are $C,c$ such that for every $L\ge2$ and every Hamiltonian on the open grid $\{1,\dots,L\}^2$ with on-site and nearest-neighbour terms of norm at most $J$, a unique ground vector $\Omega$ and global gap $\Delta$, some PEPS $\Phi$ on the same grid with bond dimension at most $CL^c$ satisfies $\min_\theta\|\Phi/\|\Phi\|-e^{i\theta}\Omega\|\le L^{-1}$. 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.