On the Relative Completeness of Satisfaction-based Quantum Hoare Logic

TL;DR

This paper introduces the first relatively complete satisfaction-based quantum Hoare logic with while-loops, providing a formal framework for verifying quantum programs' correctness, including complex algorithms like Deutsch's and quantum teleportation.

Contribution

It presents a novel, relatively complete satisfaction-based quantum Hoare logic with while-loops, along with a proof system and semantics for quantum programs.

Findings

Verified correctness of Deutsch's algorithm

Verified quantum teleportation protocol

Established semantics and proof system for quantum Hoare logic

Abstract

Quantum Hoare logic (QHL) is a formal verification tool specifically designed to ensure the correctness of quantum programs. There has been an ongoing challenge to achieve a relatively complete satisfaction-based QHL with while-loop since its inception in 2006. This paper presents a solution by proposing the first relatively complete satisfaction-based QHL with while-loop. The completeness is proved in two steps. First, we establish a semantics and proof system of Hoare triples with quantum programs and deterministic assertions. Then, by utilizing the weakest precondition of deterministic assertion, we construct the weakest preterm calculus of probabilistic expressions. The relative completeness of QHL is then obtained as a consequence of the weakest preterm calculus. Using our QHL, we formally verify the correctness of Deutsch's algorithm and quantum teleportation.