  We describe an efficient approximation algorithm for evaluating the
ground-state energy of the classical Ising Hamiltonian with linear terms on an
arbitrary planar graph. The running time of the algorithm grows linearly with
the number of spins and exponentially with 1/epsilon, where epsilon is the
worst-case relative error. This result contrasts the well known fact that exact
computation of the ground-state energy for the two-dimensional Ising spin glass
model is NP-hard. We also present a classical approximation algorithm for the
Local Hamiltonian Problem or Quantum Ising Spin Glass problem on a planar graph
with bounded degree which is known to be a QMA-complete problem. Using a
different technique we find a classical approximation algorithm for the quantum
Ising spin glass problem on the simplest planar graph with unbounded degree,
the star graph.
