Approximating the permanent of a complex-valued matrix is a fundamental problem with applications in Boson sampling and probabilistic inference. In this paper, we extend factor-graph-based methods for approximating the permanent of non-negative-real-valued matrices that are based on running the sum-product algorithm (SPA) on standard normal factor graphs, to factor-graph-based methods for approximating the permanent of complex-valued matrices that are based on running the SPA on double-edge normal factor graphs. On the algorithmic side, we investigate the behavior of the SPA, in particular how the SPA fixed points change when transitioning from real-valued to complex-valued matrix ensembles. On the analytical side, we use graph covers to analyze the Bethe approximation of the permanent, i.e., the approximation of the permanent that is obtained with the help of the SPA. This combined algorithmic and analytical perspective provides new insight into the structure of Bethe approximations in complex-valued problems and clarifies when such approximations remain meaningful beyond the non-negative-real-valued settings.

翻译：近似复数值矩阵的积和式是一个基础性问题，在玻色采样与概率推断中具有重要应用。本文将基于标准正态因子图上运行和积算法（SPA）的非负实值矩阵积和式近似方法，扩展为基于双重边正态因子图上运行SPA的复数值矩阵积和式近似方法。在算法层面，我们研究了SPA的行为特性，重点分析了从实值矩阵系综过渡到复值矩阵系综时SPA不动点的变化规律。在分析层面，我们运用图覆盖理论解析积和式的Bethe近似——即借助SPA获得的积和式近似值。这种算法与分析相结合的研究视角，为理解复数值问题中Bethe近似的结构提供了新见解，并阐明了此类近似在超越非负实值设定时仍保持有效性的条件。