%0 Generic %1 carette2023square %A Carette, Jacques %A Heunen, Chris %A Kaarsgaard, Robin %A Sabry, Amr %D 2023 %K programming quantum_computing %T With a Few Square Roots, Quantum Computing is as Easy as \Pi %U http://arxiv.org/abs/2310.14056 %X 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 $8^th$ 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 $2$ qubits, and the computationally universal Gaussian Clifford+T gate set.

@misc{carette2023square, 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.}, added-at = {2023-10-24T13:23:23.000+0200}, author = {Carette, Jacques and Heunen, Chris and Kaarsgaard, Robin and Sabry, Amr}, biburl = {https://www.bibsonomy.org/bibtex/2bde0172ca238f9ea6ff0ab2ad7c1b174/tabularii}, description = {[2310.14056] With a Few Square Roots, Quantum Computing is as Easy as Π}, interhash = {64206b9d5201d97446649da1209ca0c1}, intrahash = {bde0172ca238f9ea6ff0ab2ad7c1b174}, keywords = {programming quantum_computing}, note = {cite arxiv:2310.14056}, timestamp = {2023-10-24T13:23:23.000+0200}, title = {With a Few Square Roots, Quantum Computing is as Easy as {\Pi}}, url = {http://arxiv.org/abs/2310.14056}, year = 2023 }