Rig groupoids provide a semantic model of Π , a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th root of the identity morphism on the unit 1. The second map corresponds to a square root of the symmetry on 1 + 1 . As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of Π , called Π , that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to ≤ 2 qubits, and the computationally universal Gaussian Clifford+T gate set.

@article{paperbot2602,
  title = {With a Few Square Roots, Quantum Computing Is as Easy as Pi},
  author = {Jacques Carette and Chris Heunen and Robin Kaarsgaard and Amr Sabry},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {8},
  number = {POPL},
  year = {2024},
  doi = {10.1145/3632861}
}