With a Few Square Roots, Quantum Computing is as Easy as Π
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914834018205696 |
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| author | Carette, Jacques Heunen, Chris Kaarsgaard, Robin Sabry, Amr |
| author_facet | Carette, Jacques Heunen, Chris Kaarsgaard, Robin Sabry, Amr |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14056 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | With a Few Square Roots, Quantum Computing is as Easy as Π Carette, Jacques Heunen, Chris Kaarsgaard, Robin Sabry, Amr Programming Languages 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. |
| title | With a Few Square Roots, Quantum Computing is as Easy as Π |
| topic | Programming Languages |
| url | https://arxiv.org/abs/2310.14056 |