With a Few Square Roots, Quantum Computing is as Easy as {\Pi}

TL;DR

This paper introduces a new semantic model for quantum computing using rig groupoids extended with two specific maps, leading to a universal quantum programming language with a complete equational theory.

Contribution

It demonstrates that adding just two maps to rig groupoids yields a computationally universal and sound model of quantum computing, with an associated language and equational theory.

Findings

Model is sound and complete for Clifford and Clifford+T gate sets.

Extension of ng with two maps achieves quantum universality.

Semantic construction leads to a universal quantum programming language.

Abstract

Rig groupoids provide a semantic model of \PiLang, 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 $8^{\text{th}}$ 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 \PiLang, called \SPiLang, 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.