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/ǫ, where ǫ 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.
In a first result on polynomial time approximation algorithms, Bansal, Bravyi and Terhal REF give a PTAS for a special class of the local Hamiltonian problem, so called quantum Ising spin glasses, for the case where the instance is on a planar graph (and in particular of bounded degree).
Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs
Quant. Inf. Comp. Vol. 9, No.8, p. 0701 (2009)
Mathematics Physics Computer Science