On incorrectness logic for Quantum programs

Abstract

Bug-catching is important for developing quantum programs. Motivated by the incorrectness logic for classical programs, we propose an incorrectness logic towards a logical foundation for static bug-catching in quantum programming. The validity of formulas in this logic is dual to that of quantum Hoare logics. We justify the formulation of validity by an intuitive explanation from a reachability point of view and a comparison against several alternative formulations. Compared with existing works focusing on dynamic analysis, our logic provides sound and complete arguments. We further demonstrate the usefulness of the logic by reasoning several examples, including Grover's search, quantum teleportation, and a repeat-until-success program. We also automate the reasoning procedure by a prototyped static analyzer built on top of the logic rules.

Local Completeness Logic on Kleene Algebra with Tests

“…(seq-err) : By induction we have q ≤ t 1 ok p, r ≤ t 1 err p and s ≤ t 2 err q, thus s ≤ t 2 err q ≤ t 2 err t 1 ok p. By (31) we have that t 1 • t 2 err p = t 1 err p + t 2 err t 1 ok p ≥ r + s, thus implying (1). By induction we have t 1…”

“…Incorrectness logic IL has been introduced by O'Hearn [24] as a natural under-approximating counterpart of the pivotal Hoare correctness logic [14], and quickly attracted a lot of research interest [19,25,26,27,31]. Incorrectness logic distinguishes two postconditions corresponding to normal and erroneous/abnormal program termination.…”

An Automata-based Framework for Verification and Bug Hunting in Quantum Circuits (Technical Report)

“…Quantum Hoare logic [37,48,65,76,83]) allows verification against complex correctness properties and rich program constructs such as branches and loops, but requires significant manual work. On the other hand, quantum incorrectness logic [75] is a dual of quantum Hoare logic that allows showing the existence of a bug, but cannot prove its absence. The Q [21] approach alleviates the difficulty of proof search by combining state-of-the-art theorem provers with decision procedures, but, according to their experiments, still requires a significant amount of human intervention.…”

Calculational design of a regular model checker by abstract interpretation

“…The paper provides ample citations in the text and mainly owes to the extensive work on Hoare logic [2][3][4][5]49], a previous study of proof methods [22], abstract interpretation [18] (for the design of semantics by abstraction [16] and the specification of program properties by Galois connections [26]), fixpoint induction [17] to handle termination, and the nonconformist idea of Peter O'Hearn [67] originating the interest in incorrectness [7,9,27,28,32,41,55,58,64,76,78,93,[96][97][98]. The abstraction post(⊇, ⊆) comes from the specification of reverse Hoare logic by 𝑄 ⊆ post S 𝑃 in [32] and then a.o.…”

“…[67,76,97]. [22] and [96,97] also consider (39.d). [58] incorporate reasoning about non-terminating specifications.…”