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,"_classifai_error":"","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-post-option":[],"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. Kliuchnikov","user_id":0,"rest_url":false},{"type":"text","value":"A. Bocharov","user_id":0,"rest_url":false},{"type":"text","value":"M. Roetteler","user_id":0,"rest_url":false},{"type":"text","value":"J. Yard","user_id":0,"rest_url":false},{"type":"user_nicename","value":"alexeib","user_id":30935,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=alexeib"},{"type":"user_nicename","value":"jonyard","user_id":32392,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=jonyard"},{"type":"user_nicename","value":"martinro","user_id":32823,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=martinro"},{"type":"user_nicename","value":"vadym","user_id":34468,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=vadym"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[170888],"publication":[],"video":[],"download":[],"msr_publication_type":"techreport","related_content":{"projects":[{"ID":170888,"post_title":"Language-Integrated Quantum Operations: LIQUi|>","post_name":"language-integrated-quantum-operations-liqui","post_type":"msr-project","post_date":"2011-12-19 10:19:35","post_modified":"2018-11-02 11:06:22","post_status":"publish","permalink":"https:\/\/www.microsoft.com\/en-us\/research\/project\/language-integrated-quantum-operations-liqui\/","post_excerpt":"LIQUi|> is a software architecture and toolsuite for quantum computing. It includes a programming language, optimization and scheduling algorithms, and quantum simulators. LIQUi|> can be used to translate a quantum algorithm written in the form of a high-level program into the low-level machine instructions for a quantum device. LIQUi|> is being developed by the Quantum Architectures and Computation Group (QuArC)\u00a0at Microsoft Research. About LIQUi|> To aid in the development and understanding of quantum protocols, quantum…","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-project\/170888"}]}}]},"_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/168848"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":2,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/168848\/revisions"}],"predecessor-version":[{"id":538762,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/168848\/revisions\/538762"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=168848"}],"wp:term":[{"taxonomy":"msr-content-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-content-type?post=168848"},{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=168848"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=168848"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=168848"},{"taxonomy":"msr-product-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-product-type?post=168848"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=168848"},{"taxonomy":"msr-platform","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-platform?post=168848"},{"taxonomy":"msr-download-source","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-download-source?post=168848"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=168848"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=168848"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=168848"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=168848"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=168848"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=168848"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=168848"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}