Quantum Hoare logic with classical variables

Hoare logic provides a syntax-oriented method to reason about program correctness, and has been proven effective in the verification of classical and probabilistic programs. Existing proposals for quantum Hoare logic either lack completeness or support only quantum variables, thus limiting their capability in practical use. In this paper, we propose a quantum Hoare logic for a simple while language which involves both classical and quantum variables. Its soundness and relative completeness are proven for both partial and total correctness of quantum programs written in the language. Remarkably, with novel definitions of classical-quantum states and corresponding assertions, the logic system is quite simple and similar to the traditional Hoare logic for classical programs. Furthermore, to simplify reasoning in real applications, auxiliary proof rules are provided which support the introduction of disjunction and quantifiers in the classical part of assertions, and of super-operator application and superposition in the quantum part. Finally, a series of practical quantum algorithms, in particular the whole algorithm of Shor's factorisation, are formally verified to show the effectiveness of the logic

Reasoning about states of probabilistic sequential programs

Abstract. A complete and decidable propositional logic for reasoning about states of probabilistic sequential programs is presented. The state logic is then used to obtain a sound Hoare-style calculus for basic probabilistic sequential programs. The Hoare calculus presented herein is the first probabilistic Hoare calculus with a complete and decidable state logic that has truth-functional propositional (not arithmetical) connectives. The models of the state logic are obtained exogenously by attaching sub-probability measures to valuations over memory cells. In order to achieve complete and recursive axiomatization of the state logic, the probabilities are taken in arbitrary real closed fields.