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arXiv:2404.07231 · Communications in Mathematical Physics

Bounds on the Ground State Energy of Quantum p-Spin Hamiltonians

Eric R. Anschuetz, David Gamarnik, Bobak T. Kiani

Published Sep 1, 2025• 7 citations
arXivHTMLPDF
DOI
10.1007/s00220-025-05412-4

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We describe a classical 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/ϵ1/\epsilon1/ϵ, where ϵ\epsilonϵ 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 quantum Local Hamiltonian Problem or Quantum Ising Spin Glass problem on a planar graph {\em 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.