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.

References

Identifying and validating the exact ground state of the Ising Hamiltonian generally remains an unsolved problem.

— Continuous Approximation of the Ising Hamiltonian: Exact Ground States and Applications to Fidelity Assessment in Ising Machines  (2411.19604 - Rezaei et al., 2024) in Section 1 (Introduction)

This is because, even with the proposed scheme, convergence to the global optimum is still not guaranteed, and the search stagnated at a local minimum.

— Deterministic Ground-State Search in a Spatial Photonic Ising Machine by Phase Retrieval  (2609.16474 - Shimomura et al., 15 Sep 2026) in Discussion of Fig. 2(g), paragraph beginning “Moreover, the number of iterations required for the convergence was estimated”

Although thermodynamically trivial, no exact solution is known for J2 = −1/4 (in contrast to the case J2 → −∞) and even the ground-state entropy per spin is unknown.

— Meandering stripes in the frustrated $J_1$-$J_2$ Ising model on the honeycomb lattice  (2608.21261 - Gessert et al., 21 Aug 2026) in Section IV A