We study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$, which corresponds to the case of the Ising model; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations.
翻译:我们研究了近似美元-州波茨模型分割功能的复杂程度,以及与基本参数的复杂值密切相关的图特多元分子值。除了与量子计算和统计物理阶段过渡的古老联系外,最近的近似计算工作表明,复杂平面的行为,更确切地说零的位置,与近似问题的复杂性密切相关,即使是对正实际价值参数也是如此。戈德贝格和古奥以往在复杂平面上的工作重点是$=2美元,这与Ising模型的情况相对应;对于$>2美元,复杂平面上的行为不那么容易理解,而且大部分工作只适用于实际价值的图特平面。我们的主要结果是对参数所有非真实值的近似问题的复杂性进行彻底分类,确定“P-硬度”结果,即使局限于平面图示时也适用。我们的技术适用于所有$\ge2美元,进一步补充/补充了以前在Ising模型和图特平面上的结果,对于实际价值的图特平面的行为则不甚为理解,而且大部分工作仅适用于实际价值的图特平面。我们的主要结果是对参数所有非实际值的精确值问题作了完整分类,由Berdez 和Freez的计算,在Berdz 和Servicesa 上提出了一个特定的答案。