Polynomial PEPS approximation of gapped square-grid ground states

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.