Exact ground states of general Ising Hamiltonians

Determine the exact ground state energy and spin configuration for general finite-size Ising Hamiltonians H defined by pairwise couplings J_ij and binary spins s_i ∈ {−1, +1}, where H equals the sum over 1 ≤ i < j ≤ N of J_ij s_i s_j, and develop a rigorous procedure to validate that a proposed configuration is the exact ground state for arbitrary coupling matrices.

Background

The paper defines the Ising Hamiltonian with arbitrary real pairwise couplings and notes that determining its exact ground state is generally NP-hard. While the authors introduce a novel continuous reformulation that yields exact solutions for a new, fully connected class of Ising models, they emphasize that, in general, finding and certifying exact ground states remains unresolved.

This broad unresolved problem provides motivation for the authors’ contribution: an analytically solvable subclass that can be used to benchmark and assess the fidelity of Ising machines. Nonetheless, the general task of exact identification and validation of ground states for arbitrary Ising instances persists beyond the specific class addressed here.