  We analyze the error of approximating Gibbs states of local quantum spin
Hamiltonians on lattices with Projected Entangled Pair States (PEPS) as a
function of the bond dimension ($D$), temperature ($\beta^{-1}$), and system
size ($N$). First, we introduce a compression method in which the bond
dimension scales as $D=e^{O(\log^2(N/\epsilon))}$ if $\beta<O(\log (N))$.
Second, building on the work of Hastings [Phys. Rev. B 73, 085115 (2006)], we
derive a polynomial scaling relation, $D=\left(N/\epsilon\right)^{O(\beta)}$.
This implies that the manifold of PEPS forms an efficient representation of
Gibbs states of local quantum Hamiltonians. From those bounds it also follows
that ground states can be approximated with $D=N^{O(\log(N))}$ whenever the
density of states only grows polynomially in the system size. All results hold
for any spatial dimension of the lattice.
