{"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/t-count-and-t-depth-of-any-multi-qubit","title":"T-count and T-depth of any multi-qubit unitary","arxiv_id":"2110.10292","date":"2021-10-19","proceeding":null,"authors":["Vlad Gheorghiu","Michele Mosca","Priyanka Mukhopadhyay"],"abstract":"While implementing a quantum algorithm it is crucial to reduce the quantum resources, in order to obtain the desired computational advantage. For most fault-tolerant quantum error-correcting codes the cost of implementing the non-Clifford gate is the highest among all the gates in a universal fault-tolerant gate set. In this paper we design provable algorithm to determine T-count of any $n$-qubit ($n\\geq 1$) unitary $W$ of size $2^n\\times 2^n$, over the Clifford+T gate set. The space and time complexity of our algorithm are $O\\left(2^{2n}\\right)$ and $O\\left(2^{2n\\mathcal{T}_{\\epsilon}(W)+4n}\\right)$ respectively. $\\mathcal{T}_{\\epsilon}(W)$ ($\\epsilon$-T-count) is the (minimum possible) T-count of an exactly implementable unitary $U$ i.e. $\\mathcal{T}(U)$, such that $d(U,W)\\leq\\epsilon$ and $\\mathcal{T}(U)\\leq\\mathcal{T}(U')$ where $U'$ is any exactly implementable unitary with $d(U',W)\\leq\\epsilon$. $d(.,.)$ is the global phase invariant distance. Our algorithm can also be used to determine the (minimum possible) T-depth of any multi-qubit unitary and the complexity has exponential dependence on $n$ and $\\epsilon$-T-depth. This is the first algorithm that gives T-count or T-depth of any multi-qubit ($n\\geq 1$) unitary. For small enough $\\epsilon$, we can synthesize the T-count and T-depth-optimal circuits. Our results can be used to determine the minimum count (or depth) of non-Clifford gates required to implement any multi-qubit unitary with a universal gate set consisting of Clifford and non-Clifford gates like Clifford+CS, Clifford+V, etc. To the best of our knowledge, there were no such optimal-synthesis algorithm for arbitrary multi-qubit unitaries in any universal gate set.","url_abs":"https://arxiv.org/abs/2110.10292v5","url_pdf":"https://arxiv.org/pdf/2110.10292v5.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":"t-count-and-t-depth-of-any-multi-qubit","repo_url":"https://github.com/vsoftco/approx-t","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}