Weakest precondition calculus and relative completeness of satisfaction-based quantum Hoare logic

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 a while loop since its inception in 2006. This paper presents a solution. We prove the relative completeness in two steps. First, we establish a semantics and proof system of Hoare triples with quantum programs and deterministic assertions. Then, 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.