A Quantum Information Theoretic Approach to Tractable Probabilistic Models

TL;DR

This paper introduces positive unital circuits (PUnCs), a quantum information theoretic generalization of probabilistic circuits, enabling circuit evaluations over positive semi-definite matrices and broadening the scope of tractable probabilistic models.

Contribution

It presents PUnCs, a novel quantum-inspired framework that extends probabilistic circuits to matrix evaluations, unifying and generalizing existing circuit classes.

Findings

PUnCs strictly generalize probabilistic circuits.

PUnCs encompass PSD circuits as a special case.

The framework enables new tractable probabilistic modeling approaches.

Abstract

By recursively nesting sums and products, probabilistic circuits have emerged in recent years as an attractive class of generative models as they enjoy, for instance, polytime marginalization of random variables. In this work we study these machine learning models using the framework of quantum information theory, leading to the introduction of positive unital circuits (PUnCs), which generalize circuit evaluations over positive real-valued probabilities to circuit evaluations over positive semi-definite matrices. As a consequence, PUnCs strictly generalize probabilistic circuits as well as recently introduced circuit classes such as PSD circuits.