With a Few Square Roots, Quantum Computing Is as Easy as Pi

Rig groupoids provide a semantic model of $\Pi$ , 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 $\Pi$ , called $\sqrt{\Pi }$ , 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 $\le 2$ qubits, and the computationally universal Gaussian Clifford+T gate set.