We investigate an unsuspected connection between logical connectives with non-harmonious deduction rules, such as Prior's tonk, and quantum computing. We argue that these connectives model the informationerasure, the non-reversibility, and the non-determinism that occur, among other places, in quantum measurement. We introduce an intuitionistic propositional logic with a non-harmonious logical connective sup and two interstitial rules, and show that the proof language of this logic forms the core of a quantum programming language.