We construct a computationally universal quantum programming language $\mathrm{Quantum}\Pi$

from two copies of $\Pi$, the internal language of rig groupoids. The first

step constructs a pure (measurement-free) term language by

interpreting each copy of $\Pi$ in a generalisation of the category

$\mathbf{Unitary}$ in which every morphism is ``rotated'' by a particular angle, and

the two copies are amalgamated using a free categorical construction expressed

as a computational effect. The amalgamated language only exhibits quantum

behaviour for specific values of the rotation angles, a property which is

enforced by imposing a small number of equations on the resulting category. The

second step in the construction introduces measurements by layering an

additional computational effect.