Quantum Hoare logic with classical variables

TL;DR

This paper introduces a new quantum Hoare logic that integrates classical and quantum variables, providing a sound and complete framework for verifying quantum programs, demonstrated through practical algorithms like Shor's factorization.

Contribution

It presents a simple, complete quantum Hoare logic supporting both classical and quantum variables, with practical proof rules and verification of real quantum algorithms.

Findings

Proven soundness and relative completeness for the logic.

Verified quantum algorithms including Shor's factorization.

Provided auxiliary proof rules for classical and quantum assertions.

Abstract

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 standard logical operation in the classical part of assertions, and of super-operator application 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.