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

TL;DR

This paper presents efficient approximation algorithms for estimating the ground-state energy of classical and quantum Ising spin Hamiltonians on planar graphs, providing practical solutions despite NP-hardness and QMA-completeness.

Contribution

It introduces novel approximation algorithms for classical and quantum Ising models on planar graphs, including bounded degree and star graphs, with proven efficiency.

Findings

Linear-time approximation for classical Ising Hamiltonian on planar graphs.

Approximation algorithms for quantum Ising spin glass on planar graphs.

Contrasts NP-hardness of exact computation with efficient approximation methods.

Abstract

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.