How to Bake a Quantum Π

We construct a computationally universal quantum programming language <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mtext>Quantum</mml:mtext> <mml:mtext>Π</mml:mtext> </mml:math> from two copies of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mtext>Π</mml:mtext> </mml:math> , the internal language of rig groupoids. The first step constructs a pure (measurement-free) term language by interpreting each copy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mtext>Π</mml:mtext> </mml:math> in a generalisation of the category 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.