Local Reasoning about Probabilistic Behaviour for Classical-Quantum Programs

Verifying the functional correctness of programs with both classical and quantum constructs is a challenging task. The presence of probabilistic behaviour entailed by quantum measurements and unbounded while loops complicate the verification task greatly. We propose a new quantum Hoare logic for local reasoning about probabilistic behaviour by introducing distribution formulas to specify probabilistic properties. We show that the proof rules in the logic are sound with respect to a denotational semantics. To demonstrate the effectiveness of the logic, we formally verify the correctness of non-trivial quantum algorithms including the HHL and Shor's algorithms. Moreover, we embed our logic into the proof assistant Coq. The resulting logical framework, called CoqQLR, can facilitate semi-automated reasoning about classical--quantum programs.