Complex-Valued-Matrix Permanents: SPA-based Approximations and Double-Cover Analysis

2026-01-26

AI Summary

Efficient approximation of the permanent of complex-valued matrices is crucial for boson sampling and probabilistic inference. This work proposes a novel approach by integrating bilateral normal factor graphs with the sum-product algorithm (SPA), thereby extending the Bethe approximation to the complex-valued matrix setting. Leveraging graph cover theory, the study analyzes the structural properties and applicability limits of this extension. It further characterizes the fixed-point behavior of SPA over ensembles of complex-valued matrices and rigorously establishes the conditions under which the Bethe approximation remains valid in the complex domain. These contributions provide a new theoretical foundation and algorithmic framework for approximating permanents of complex-valued matrices.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• complex-valued matrix
• permanent
• Bethe approximation
• sum-product algorithm
• factor graph

Innovation

• complex-valued matrix permanent
• sum-product algorithm
• double-edge normal factor graphs
• Bethe approximation
• graph covers