{"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/the-solovay-kitaev-algorithm","title":"The Solovay-Kitaev algorithm","arxiv_id":"quant-ph/0505030","date":"2005-05-06","proceeding":null,"authors":["Christopher M. Dawson","Michael A. Nielsen"],"abstract":"This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $\\pi / 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\\sc not}, and $\\pi / 8$ gates. The algorithm runs in $O(\\log^{2.71}(1/\\epsilon))$ time, and produces as output a sequence of $O(\\log^{3.97}(1/\\epsilon))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $\\epsilon > 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from $SU(d)$.","url_abs":"https://arxiv.org/abs/quant-ph/0505030v2","url_pdf":"https://arxiv.org/pdf/quant-ph/0505030v2.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":"the-solovay-kitaev-algorithm","repo_url":"https://github.com/vadym-kl/sqct","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"the-solovay-kitaev-algorithm","repo_url":"https://github.com/virattara/QOSF-Task-3","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}