Complete Quantum Relational Hoare Logics from Optimal Transport Duality

TL;DR

This paper develops a new quantitative relational Hoare logic for quantum programs, leveraging optimal transport duality to establish soundness and completeness for certain classes of quantum programs.

Contribution

It introduces a novel quantum relational Hoare logic with an infinitary assertion language, and proves its completeness using a quantum optimal transport duality theorem.

Findings

Logic is sound for bounded postconditions

Logic is complete for almost surely terminating programs

Embeds a relational Hoare logic with projective assertions

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.