We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a $2\times 2$ symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary $2\times 2$ interaction matrices with real entries. We show that in some regions of the parameter space, it's \#P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

翻译：我们研究二态自旋系统中配分函数计算的可近似性问题。该问题由一个$2\times 2$对称矩阵参数化。此前关于该问题的结果仅限于矩阵元素非负的情形，或对角元相等的情形（即伊辛模型）。本文研究该问题向具有实系数的任意$2\times 2$相互作用矩阵的推广。我们证明，在参数空间的某些区域，甚至判定配分函数符号的问题都是#P-难的，而在其他区域则存在配分函数的全多项式近似方案。我们的研究结果揭示了若干新的计算相变现象。