Quantum factor graphs: Closing-the-box operation and variational approaches

Factor graphs and the sum-product algorithms form a powerful framework for expressing a great variety of algorithms in coding theory, signal processing, artificial intelligence, and other areas. Two different approaches to derive the sum-product algorithm are given by the so-called closing-the-box operation or by a variational approach known as Bethe free energy function minimization. In this paper we consider a generalization of factor graphs known as quantum factor graphs, along with a generalization of the sum-product algorithm known as the quantum sum-product algorithm. We explore the generalization of the closing-the-box operation and the Bethe free energy function from the classical to the quantum setup. Some expressions that hold exactly in the classical case hold only approximately in the quantum case; we give some analytical and numerical characterizations of these approximations.