We consider a family of quantum spin systems which includes, as special cases, the ferromagnetic $XY$ model and ferromagnetic Ising model on any graph, with or without a transverse magnetic field. We prove that the partition function of any model in this family can be efficiently approximated to a given relative error $\epsilon$ using a classical randomized algorithm with runtime polynomial in ${\epsilon }^{-1}$, system size, and inverse temperature. As a consequence, we obtain a polynomial time algorithm which approximates the free energy or ground energy to a given additive error. We first show how to approximate the partition function by the perfect matching sum of a finite graph with positive edge weights. Although the perfect matching sum is not known to be efficiently approximable in general, the graphs obtained by our method have a special structure which facilitates efficient approximation via a randomized algorithm due to Jerrum and Sinclair.

中文翻译：

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量子铁磁体的多项式时间经典模拟

我们考虑一类量子自旋系统，其中包括特殊情况下的铁磁 $Xÿ$有或没有横向磁场的任何图形上的三维模型和铁磁伊辛模型。我们证明该族中任何模型的分区函数都可以有效地近似于给定的相对误差$\epsilon$ 在运行时多项式中使用经典随机算法 ${\epsilon }^{-\mathrm{1个}}$，系统大小和逆温度。结果，我们获得了多项式时间算法，该算法将自由能或地面能近似为给定的加性误差。我们首先展示如何通过具有正边缘权重的有限图的完美匹配和来近似划分函数。尽管通常不知道完美匹配和是否有效逼近，但是由于Jerrum和Sinclair，通过我们的方法获得的图具有特殊的结构，该结构有助于通过随机算法进行有效逼近。