On Sampling from Ising Models with Spectral Constraints

We consider the problem of sampling from the Ising model when the underlying interaction matrix has eigenvalues lying within an interval of length $\gamma$. Recent work in this setting has shown various algorithmic results that apply roughly when $\gamma< 1$, notably with nearly-linear running times based on the classical Glauber dynamics. However, the optimality of the range of $\gamma$ was not clear since previous inapproximability results developed for the antiferromagnetic case (where the matrix has entries $\leq 0$) apply only for $\gamma>2$. To this end, Kunisky (SODA'24) recently provided evidence that the problem becomes hard already when $\gamma>1$ based on the low-degree hardness for an inference problem on random matrices. Based on this, he conjectured that sampling from the Ising model in the same range of $\gamma$ is NP-hard. Here we confirm this conjecture, complementing in particular the known algorithmic results by showing NP-hardness results for approximately counting and sampling when $\gamma>1$, with strong inapproximability guarantees; we also obtain a more refined hardness result for matrices where only a constant number of entries per row are allowed to be non-zero. The main observation in our reductions is that, for $\gamma>1$, Glauber dynamics mixes slowly when the interactions are all positive (ferromagnetic) for the complete and random regular graphs, due to a bimodality in the underlying distribution. While ferromagnetic interactions typically preclude NP-hardness results, here we work around this by introducing in an appropriate way mild antiferromagnetism, keeping the spectrum roughly within the same range. This allows us to exploit the bimodality of the aforementioned graphs and show the target NP-hardness by adapting suitably previous inapproximability techniques developed for antiferromagnetic systems.