We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda$ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all $|\lambda|\neq 1$ by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens around $\lambda=1$, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda=1$, and more generally on the entire unit circle. For an integer $\Delta\geq 3$ and edge interaction parameter $b\in (0,1)$ we show #P-hardness for approximating the partition function on graphs of maximum degree $\Delta$ on the arc of the unit circle where the Lee-Yang zeros are dense. This result contrasts with known approximation algorithms when $|\lambda|\neq 1$ or when $\lambda$ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee-Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.
翻译:我们研究铁磁系模型与复杂平面中单位圆圆的外部字段参数 $\ lambda$=1美元相匹配的计算复杂性。 复合平面中, 元素模型的复杂值参数对于统计物理的量子电路计算和阶段转换具有相关性, 但对于最近由刘、 辛克莱和斯利瓦斯塔瓦塔瓦提出的所有$ lambda ⁇ neq 1 的确定性近似方案来说, 也是关键。 这里, 我们关注单位圆上尚未解决的复杂图象, 以及围绕 $\ lambda=1 美元左右发生什么的吊动问题。 一方面, Jerrum 和 Sinclair 的经典计算法在真实轴上提供了一种随机化的近似近似方案, 显示利亚黑零指值在计算硬度上, 美元 lamda=1, 整个单位圆圈上一个整方块的计算性转变。 美元和边缘的连接值 $bda $( 0, 1, 1) 。 因此, 当我们以 美元 美元的直方平面的直方平面的直方值计算值计算值运行值 时, 我们的直方的直方的直方值运行的直方的直方值 。