Abstract interpretation, Hoare logic, and incorrectness logic for quantum programs

Abstract interpretation, Hoare logic, and incorrectness (or reverse Hoare) logic are powerful techniques for static analysis of computer programs. They have all been successfully extended to the quantum setting, but developed largely in parallel. This paper explores the relationship between these techniques in the context of verifying quantum while-programs, where the abstract domain and the set of assertions for quantum states are well-structured. We show that any complete quantum abstract interpretation induces a quantum Hoare logic and a quantum incorrectness logic, both being sound and relatively complete. Moreover, the induced logic systems operate in a forward manner, making them more suitable for certain applications. Conversely, any sound and relatively complete quantum Hoare logic or incorrectness logic induces a complete quantum abstract interpretation. As an application, we show the non-existence of any sound and relatively complete quantum Hoare logic or incorrectness logic if tuples of local subspaces are taken as assertions.