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

Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs