We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters. Our results are partly motivated by the study of the quantum complexity classes BQP and IQP. Recent results show how to encode quantum computations as evaluations of classical partition functions. These results rely on interesting and deep results about quantum computation in order to obtain hardness results about the difficulty in (classically) evaluating the partition functions for certain fixed parameters.The motivation for this paper is to study more comprehensively the complexity of (classically) approximating the Ising and Tutte partition functions with complex parameters. Partition functions are combinatorial in nature, and quantifying their approximation complexity does not require a detailed understanding of quantum computation. Using combinatorial arguments, we give the first full classification of the complexity of multiplicatively approximating the norm and additively approximating the argument of the Ising partition function for complex edge interactions (as well as of approximating the partition function according to a natural complex metric). We also study the norm approximation problem in the presence of external fields, for which we give a complete dichotomy when the parameters are roots of unity. Previous results were known just for a few such points, and we strengthen these results from BQP-hardness to #P-hardness. Moreover, we show that computing the sign of the Tutte polynomial is #P-hard at certain points related to the simulation of BQP. Using our classifications, we then revisit the connections to quantum computation, drawing conclusions that are a little different from (and incomparable to) ones in the quantum literature, but along similar lines.

中文翻译：

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逼近复值 Ising 和 Tutte 分区函数的复杂性

我们研究了近似评估具有复杂参数的 Ising 和 Tutte 分区函数的复杂性。我们的结果部分受到对量子复杂性类 BQP 和 IQP 的研究的启发。最近的结果显示了如何将量子计算编码为经典分配函数的评估。这些结果依赖于有关量子计算的有趣而深入的结果，以获得关于（经典）评估某些固定参数的配分函数困难的硬度结果。 本文的动机是更全面地研究（经典）逼近的复杂性具有复参数的 Ising 和 Tutte 分区函数。划分函数本质上是组合的，量化它们的近似复杂度不需要对量子计算有详细的了解。使用组合参数，我们给出了乘法逼近范数和加法逼近 Ising 分配函数的参数的复杂边缘交互（以及根据自然复数度量逼近分配函数）的复杂性的第一个完整分类。我们还研究了存在外部场时的范数逼近问题，当参数为单位根时，我们给出了完整的二分法。以前的结果只是因为几个这样的点而为人所知，我们将这些结果从 BQP-hardness 加强到 #P-hardness。此外，我们表明在与 BQP 模拟相关的某些点计算 Tutte 多项式的符号是 #P-hard 的。