  Suppose we would like to approximate all local properties of a quantum
many-body state to accuracy $\delta$. In one dimension, we prove that an area
law for the Renyi entanglement entropy $R_\alpha$ with index $\alpha<1$ implies
a matrix product state representation with bond dimension
$\mathrm{poly}(1/\delta)$. For (at most constant-fold degenerate) ground states
of one-dimensional gapped Hamiltonians, it suffices that the bond dimension is
almost linear in $1/\delta$. In two dimensions, an area law for
$R_\alpha(\alpha<1)$ implies a projected entangled pair state representation
with bond dimension $e^{O(1/\delta)}$. In the presence of logarithmic
corrections to the area law, similar results are obtained in both one and two
dimensions.
