  We present a general graph-based Projected Entangled-Pair State (gPEPS)
algorithm to approximate ground states of nearest-neighbor local Hamiltonians
on any lattice or graph of infinite size. By introducing the structural-matrix
which codifies the details of tensor networks on any graphs in any dimension
$d$, we are able to produce a code that can be essentially launched to simulate
any lattice. We further introduce an optimized algorithm to compute simple
tensor updates as well as expectation values and correlators with a
mean-field-like effective environments. Though not being variational, this
strategy allows to cope with PEPS of very large bond dimension (e.g., $D=100$),
and produces remarkably accurate results in the thermodynamic limit in many
situations, and specially when the correlation length is small and the
connectivity of the lattice is large. We prove the validity of our approach by
benchmarking the algorithm against known results for several models, i.e., the
antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the
hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg
model in a field on the pyrochlore lattice, as well as the $3$-state quantum
Potts model in field on the kagome lattice and the spin-$1$
bilinear-biquadratic Heisenberg model on the triangular lattice. We further
demonstrate the performance of gPEPS by studying the quantum phase transition
of the $2d$ quantum Ising model in transverse magnetic field on the square
lattice, and the phase diagram of the Kitaev-Heisenberg model on the
hyperhoneycomb lattice. Our results are in excellent agreement with previous
studies.
