Holographic Transformation, Belief Propagation and Loop Calculus for Generalized Probabilistic Theories

TL;DR

This paper extends holographic transformation, belief propagation, and loop calculus to generalized probabilistic theories, including quantum mechanics, by representing partition functions as inner products of decomposable vectors and deriving related algorithms.

Contribution

It introduces a unified framework for applying holographic transformation, belief propagation, and loop calculus to quantum and generalized probabilistic theories using tensor decompositions.

Findings

Partition functions represented as inner products of low-dimensional tensor products

Holographic transformation understood via adjoint linear maps

Belief propagation derived from loop calculus in the generalized setting

Abstract

The holographic transformation, belief propagation and loop calculus are generalized to problems in generalized probabilistic theories including quantum mechanics. In this work, the partition function of classical factor graph is represented by an inner product of two high-dimensional vectors both of which can be decomposed to tensor products of low-dimensional vectors. On the representation, the holographic transformation is clearly understood by using adjoint linear maps. Furthermore, on the formulation using inner product, the belief propagation is naturally defined from the derivation of the loop calculus formula. As a consequence, the holographic transformation, the belief propagation and the loop calculus are generalized to measurement problems in quantum mechanics and generalized probabilistic theories.