Bayesian Inference in Quantum Programs

2025-04-29

AI Summary

Quantum programming languages lack formal conditionalization mechanisms, hindering rigorous Bayesian inference over quantum processes. Method: We introduce conditional semantics into the quantum while language, enabling modeling of observational data and computation of posterior distributions. We develop both denotational and operational semantics over infinite-dimensional Hilbert spaces and prove their equivalence. Furthermore, we establish sufficient conditions for the existence of weakest liberal precondition (wlp) transformers and provide an inductive characterization thereof. Contributions: (1) The first formalization of conditionalization and Bayesian inference in a quantum programming language; (2) A semantic equivalence framework yielding a computable, inductive representation of wlp transformers; (3) Support for quantitative analysis of Bayesian effects—even for potentially divergent quantum programs—thereby laying a semantic foundation for quantum probabilistic programming and quantum machine learning.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Probabilistic Programming
• Knowledge Representation and Reasoning: Logic Programming

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Conditioning is a key feature in probabilistic programming to enable modeling the influence of data (also known as observations) to the probability distribution described by such programs. Determining the posterior distribution is also known as Bayesian inference. This paper equips a quantum while-language with conditioning, defines its denotational and operational semantics over infinite-dimensional Hilbert spaces, and shows their equivalence. We provide sufficient conditions for the existence of weakest (liberal) precondition-transformers and derive inductive characterizations of these transformers. It is shown how w(l)p-transformers can be used to assess the effect of Bayesian inference on (possibly diverging) quantum programs.

Problem

• Enable Bayesian inference in quantum programs
• Define semantics for quantum while-language with conditioning
• Characterize weakest precondition-transformers for quantum programs

Innovation

• Quantum while-language with conditioning
• Denotational and operational semantics equivalence
• Weakest precondition-transformers for Bayesian inference