{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/qutrit-metaplectic-gates-are-a-subset-of","title":"Qutrit metaplectic gates are a subset of Clifford+T","arxiv_id":"2202.09235","date":"2022-02-18","proceeding":null,"authors":["Andrew Glaudell","Neil J. Ross","John van de Wetering","Lia Yeh"],"abstract":"A popular universal gate set for quantum computing with qubits is Clifford+T, as this can be readily implemented on many fault-tolerant architectures. For qutrits, there is an equivalent T gate, that, like its qubit analogue, makes Clifford+T approximately universal, is injectable by a magic state, and supports magic state distillation. However, it was claimed that a better gate set for qutrits might be Clifford+R, where R=diag(1,1,-1) is the metaplectic gate, as certain protocols and gates could more easily be implemented using the R gate than the T gate. In this paper we show that when we have at least two qutrits, the qutrit Clifford+R unitaries form a strict subset of the Clifford+T unitaries, by finding a direct decomposition of $R \\otimes \\mathbb{I}$ as a Clifford+T circuit and proving that the T gate cannot be exactly synthesized in Clifford+R. This shows that in fact the T gate is at least as powerful as the R gate, up to a constant factor. Moreover, we additionally show that it is impossible to find a single-qutrit Clifford+T decomposition of the R gate, making our result tight.","url_abs":"https://arxiv.org/abs/2202.09235v2","url_pdf":"https://arxiv.org/pdf/2202.09235v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"qutrit-metaplectic-gates-are-a-subset-of","repo_url":"https://github.com/lia-approves/qudit-circuits","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"qutrit-metaplectic-gates-are-a-subset-of","repo_url":"https://github.com/lia-approves/qudit-circuits/tree/main/qutrit_R_from_T","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}