  Motivated by its relation to an $\cal{NP}$-hard problem, we analyze the
ground state properties of anti-ferromagnetic Ising-spin networks embedded on
planar cubic lattices, under the action of homogeneous transverse and
longitudinal magnetic fields. This model exhibits a quantum phase transition at
critical values of the magnetic field, which can be identified by the
entanglement behavior, as well as by a Majorization analysis. The scaling of
the entanglement in the critical region is in agreement with the area law,
indicating that even simple systems can support large amounts of quantum
correlations. We study the scaling behavior of low-lying energy gaps for a
restricted set of geometries, and find that even in this simplified case, it is
impossible to predict the asymptotic behavior, with the data allowing equally
good fits to exponential and power law decays. We can therefore, draw no
conclusion as to the algorithmic complexity of a quantum adiabatic ground-state
search for the system.
