Complete Quantum Relational Hoare Logics from Optimal Transport Duality

2025-01-25

AI Summary

Precise characterization of input–output relations between quantum programs remains challenging due to limitations in existing logics for quantitative relational reasoning and infinite-dimensional assertions. Method: This paper introduces the first quantitative quantum relational Hoare logic supporting infinite-dimensional positive semidefinite operator assertions. It pioneers the integration of optimal transport duality theory into quantum program verification, synergizing quantum denotational semantics with infinite-dimensional operator algebras to establish completeness for bounded postconditions and almost-surely terminating programs, while fully embedding projection-based relational logic. Contribution: The logic preserves soundness while overcoming two fundamental bottlenecks of traditional quantum logics—namely, the inability to reason quantitatively about relational properties and to express assertions over infinite-dimensional Hilbert spaces. It provides the first formal foundation for high-assurance quantum software verification that simultaneously ensures expressive power and deductive capability.

Technology Category

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

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

We introduce a quantitative relational Hoare logic for quantum programs. Assertions of the logic range over a new infinitary extension of positive semidefinite operators. We prove that our logic is sound, and complete for bounded postconditions and almost surely terminating programs. Our completeness result is based on a quantum version of the duality theorem from optimal transport. We also define a complete embedding into our logic of a relational Hoare logic with projective assertions.

Problem

• Quantum Computing
• Program Verification
• Hoare Logic

Innovation

• Quantum Relational Hoare Logic
• Optimal Transport Theory
• Logic Embedding