We present an algorithm for e\ufb03ciently approximating of qubit unitaries over gate sets derived from totally de\ufb01nite quaternion algebras. It achieves \u03b5-approximations using circuits of length O(log(1\/\u03b5)), which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Cli\ufb00ord+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Cli\ufb00ord+\u03c0\/12 [arXiv:1409.3552], running on average in time polynomial in O(log(1\/\u03b5)) (conditional on a number-theoretic conjecture). Ours is the \ufb01rst such algorithm that works for a wide range of gate sets and provides insight into what should constitute a \u201cgood\u201d gate set for a fault-tolerant quantum computer.<\/p>\n","protected":false},"excerpt":{"rendered":"
We present an algorithm for e\ufb03ciently approximating of qubit unitaries over gate sets derived from totally de\ufb01nite quaternion algebras. It achieves \u03b5-approximations using circuits of length O(log(1\/\u03b5)), which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Cli\ufb00ord+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Cli\ufb00ord+\u03c0\/12 [arXiv:1409.3552], running on average in time […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,243138],"msr-publication-type":[193718],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-168848","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-quantum","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2015-10-13","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"MSR-TR-2015-80","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"arXiv preprint arXiv:1510.03888","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"204140","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"Kliuchnikov%20Bocharov%20Roetteler%20Yard%201510.03888.pdf","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/Kliuchnikov20Bocharov20Roetteler20Yard201510.03888.pdf","id":204140,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":204140,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/Kliuchnikov20Bocharov20Roetteler20Yard201510.03888.pdf"}],"msr-author-ordering":[{"type":"text","value":"V. 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