# Double-Edge Factor Graphs: Definition, Properties, and Examples

## TL;DR

This paper introduces double-edge factor graphs (DE-FGs), a new class allowing complex-valued local functions with positive semi-definiteness, and explores their properties and applications, including quantum information processing.

## Contribution

It defines DE-FGs, analyzes their sum-product algorithm behavior, and demonstrates promising numerical results with connections to quantum information.

## Key findings

- SPA can be effectively applied to DE-FGs.
- DE-FGs accommodate complex-valued functions with positive semi-definiteness.
- Numerical experiments show promising results in quantum information contexts.

## Abstract

Some of the most interesting quantities associated with a factor graph are its marginals and its partition sum. For factor graphs \emph{without cycles} and moderate message update complexities, the sum-product algorithm (SPA) can be used to efficiently compute these quantities exactly. Moreover, for various classes of factor graphs \emph{with cycles}, the SPA has been successfully applied to efficiently compute good approximations to these quantities. Note that in the case of factor graphs with cycles, the local functions are usually non-negative real-valued functions. In this paper we introduce a class of factor graphs, called double-edge factor graphs (DE-FGs), which allow local functions to be complex-valued and only require them, in some suitable sense, to be positive semi-definite. We discuss various properties of the SPA when running it on DE-FGs and we show promising numerical results for various example DE-FGs, some of which have connections to quantum information processing.