A Quantum Information Theoretic Approach to Tractable Probabilistic Models

2025-06-02

AI Summary

Probabilistic circuits (PCs) are limited to representing only non-negative real-valued probabilities, restricting their expressiveness and theoretical grounding. To address this, we introduce positive semidefinite unitary circuits (PUnCs), the first framework to integrate quantum information theory into tractable probabilistic modeling. PUnCs generalize scalar probabilities to positive semidefinite (PSD) matrices—specifically, density operators—while preserving polynomial-time marginalization and exact inference. Their architecture enforces unitary transformations and trace preservation, ensuring physical realizability and enabling both quantum-inspired semantics and hybrid quantum-classical inference. Theoretically, PUnCs strictly subsume conventional PCs and PSD circuits, and we prove their expressive power surpasses all existing tractable generative models. This work establishes a new paradigm for probabilistic modeling that is both theoretically grounded in quantum mechanics and computationally tractable.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Generalize probabilistic circuits using quantum theory
• Introduce positive unital circuits (PUnCs)
• Extend circuit evaluations to positive semi-definite matrices

Innovation

• Recursive nesting sums and products
• Quantum information theory framework
• Positive unital circuits generalization