A Duality Theorem for Classical-Quantum States with Applications to Complete Relational Program Logics

2025-10-08

AI Summary

Classical-quantum hybrid programs lack a unified relational logic for rigorous verification, hindering sound reasoning about correlations between classical and quantum states. Method: We establish the first duality theorem for classical-quantum states over infinite-dimensional classical state spaces—overcoming dimensional limitations of prior optimal transport and semidefinite programming (SDP) approaches—and develop a dimension-agnostic joint modeling framework integrating convex optimization, limit analysis, and generalized SDP. Contributions/Results: (1) We introduce cqOTL—the first complete, classical-quantum unified relational logic supporting full relational verification; (2) we substantially extend both the applicability and completeness of existing logics eRHL and qOTL; and (3) we provide a rigorous mathematical foundation for reliability verification of hybrid computational systems, enabling precise, compositional reasoning about classical-quantum correlations.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Duality theorems play a fundamental role in convex optimization. Recently, it was shown how duality theorems for countable probability distributions and finite-dimensional quantum states can be leveraged for building relatively complete relational program logics for probabilistic and quantum programs, respectively. However, complete relational logics for classical-quantum programs, which combine classical and quantum computations and operate over classical as well as quantum variables, have remained out of reach. The main gap is that while prior duality theorems could readily be derived using optimal transport and semidefinite programming methods, respectively, the combined setting falls out of the scope of these methods and requires new ideas. In this paper, we overcome this gap and establish the desired duality theorem for classical-quantum states. Our argument relies critically on a novel dimension-independent analysis of the convex optimization problem underlying the finite-dimensional quantum setting, which, in particular, allows us to take the limit where the classical state space becomes infinite. Using the resulting duality theorem, we establish soundness and completeness of a new relational program logic, called $mathsf{cqOTL}$, for classical-quantum programs. In addition, we lift prior restrictions on the completeness of two existing program logics: $mathsf{eRHL}$ for probabilistic programs (Avanzini et al., POPL 2025) and $mathsf{qOTL}$ for quantum programs (Barthe et al., LICS 2025).

Problem

• Establishing duality theorem for classical-quantum states
• Developing complete relational logic for classical-quantum programs
• Lifting completeness restrictions on existing program logics

Innovation

• Novel duality theorem for classical-quantum states
• Dimension-independent convex optimization analysis method
• Complete relational logic for classical-quantum programs