We construct a computationally universal quantum programming language $\langle \Pi, \Pi \rangle$ from two copies of $\Pi$, the internal language of rig groupoids. The construction uses free categorical constructions to characterise quantum computing as the combination of two copies of a reversible classical model, glued by the complementarity equations of classical structures. In Quantum$\Pi$, the user’s view of $\langle \Pi, \Pi \rangle$, measurement-free quantum computations are expressed using a computational effect layered on top of classical reversible computations. We further show that measurements can be added to Quantum$\Pi$ by layering an additional effect.