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.

翻译：我们考虑从伊辛模型采样的难题，该模型底层交互矩阵的特征值位于长度为 $\gamma$ 的区间内。该领域近期研究表明，当 $\gamma< 1$ 时存在多种算法结果，特别是基于经典格劳伯动力学实现了近线性时间复杂度。然而，由于先前针对反铁磁情形（矩阵元素 $\leq 0$）建立的不可逼近性结果仅适用于 $\gamma>2$，$\gamma$ 取值范围的优化性尚不明确。为此，Kunisky（SODA'24）基于随机矩阵推断问题的低阶计算困难性，近期提出证据表明当 $\gamma>1$ 时该问题已具有计算困难性，并据此推测在相同 $\gamma$ 范围内从伊辛模型采样是NP困难问题。本文证实了该猜想，特别通过证明当 $\gamma>1$ 时近似计数与采样问题具有NP困难性（包含强不可逼近性保证），对现有算法结果形成了补充；我们还针对每行仅允许常数个非零元素的矩阵获得了更精细的困难性结果。归约证明中的核心观测是：当 $\gamma>1$ 时，在完全图与随机正则图上，若所有相互作用均为正（铁磁性），则格劳伯动力学会因底层分布的双峰特性而混合缓慢。虽然铁磁相互作用通常阻碍NP困难性结论的建立，但我们通过引入适度反铁磁相互作用（以特定方式保持谱范围基本不变）来规避此障碍。这使我们能够利用前述图结构的双峰特性，通过适当调整针对反铁磁系统开发的既有不可逼近性技术，最终证明目标问题的NP困难性。