We establish a general framework for developing approximation algorithms for a class of counting problems. Our framework is based on the cluster expansion of abstract polymer models formalism of Koteck\'y and Preiss. We apply our framework to obtain efficient algorithms for (1) approximating probability amplitudes of a class of quantum circuits close to the identity, (2) approximating expectation values of a class of quantum circuits with operators close to the identity, (3) approximating partition functions of a class of quantum spin systems at high temperature, and (4) approximating thermal expectation values of a class of quantum spin systems at high temperature with positive-semidefinite operators. Further, we obtain hardness of approximation results for approximating probability amplitudes of quantum circuits and partition functions of quantum spin systems. This establishes a computational complexity transition for these problems and shows that our algorithmic conditions are optimal under complexity-theoretic assumptions. Finally, we show that our algorithmic condition is almost optimal for expectation values and optimal for thermal expectation values in the sense of zero freeness.

翻译：我们建立了一个通用框架，用于开发一类计数问题的近似算法。该框架基于Kotecký和Preiss提出的抽象聚合物模型簇展开形式。我们将此框架应用于以下问题的有效算法开发：（1）近似一类靠近恒等变换的量子电路的概率幅值；（2）近似一类靠近恒等变换的量子电路与算符的期望值；（3）高温下一类量子自旋系统的配分函数近似；（4）高温下一类量子自旋系统在半正定算符作用下的热期望值近似。此外，我们证明了量子电路概率幅值和量子自旋系统配分函数的近似计算具有困难性。这为上述问题建立了计算复杂性相变，并表明在复杂性理论假设下我们的算法条件是优化最优的。最后，我们展示了在“零自由”意义上，我们的算法条件对于期望值几乎最优，对于热期望值则达到最优。