We construct a computationally universal quantum programming language Quantum Π from two copies of Π , the internal language of rig groupoids. The first step constructs a pure (measurement-free) term language by interpreting each copy of Π 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.

@article{CaretteHKS24a,
  title = {How to Bake a Quantum Π},
  author = {Jacques Carette and Chris Heunen and Robin Kaarsgaard and Amr Sabry},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {8},
  number = {ICFP},
  year = {2024},
  doi = {10.1145/3674625}
}