{"id":168248,"date":"2013-06-01T00:00:00","date_gmt":"2013-06-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/synthesis-of-unitaries-with-cliffordt-circuits\/"},"modified":"2018-10-16T20:11:37","modified_gmt":"2018-10-17T03:11:37","slug":"synthesis-of-unitaries-with-cliffordt-circuits","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/synthesis-of-unitaries-with-cliffordt-circuits\/","title":{"rendered":"Synthesis of Unitaries with Clifford+T Circuits"},"content":{"rendered":"
\n

We describe a new method for approximating an arbitrary n<\/span><\/span><\/span><\/span> qubit unitary with precision \u03b5<\/span><\/span><\/span><\/span> using a Clifford and T circuit with O<\/span>(<\/span>4<\/span>n<\/span><\/span><\/span><\/span>n<\/span>(<\/span>log<\/span><\/span>(<\/span>1<\/span>\/<\/span><\/span><\/span>\u03b5<\/span>)<\/span>+<\/span>n<\/span>)<\/span>)<\/span><\/span><\/span><\/span> gates. The method is based on rounding off a unitary to a unitary over the ring Z<\/span><\/span><\/span>[<\/span>i<\/span>,<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span><\/span>\u2013\u221a<\/span>]<\/span><\/span><\/span><\/span> and employing exact synthesis. We also show that any n<\/span><\/span><\/span><\/span> qubit unitary over the ring Z<\/span><\/span><\/span>[<\/span>i<\/span>,<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span><\/span>\u2013\u221a<\/span>]<\/span><\/span><\/span><\/span> with entries of the form (<\/span>a<\/span>+<\/span>b<\/span>2<\/span><\/span>\u2013\u221a<\/span>+<\/span>i<\/span>c<\/span>+<\/span>i<\/span>d<\/span>2<\/span><\/span>\u2013\u221a<\/span>)<\/span>\/<\/span><\/span><\/span>2<\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span> can be exactly synthesized using O<\/span>(<\/span>4<\/span>n<\/span><\/span><\/span><\/span>n<\/span>k<\/span>)<\/span><\/span><\/span><\/span> Clifford and T gates using two ancillary qubits. This new exact synthesis algorithm is an improvement over the best known exact synthesis method by B. Giles and P. Selinger requiring O<\/span>(<\/span>3<\/span>2<\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>k<\/span>)<\/span><\/span><\/span><\/span> elementary gates.<\/p>\n<\/div>\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

We describe a new method for approximating an arbitrary n qubit unitary with precision \u03b5 using a Clifford and T circuit with O(4nn(log(1\/\u03b5)+n)) gates. The method is based on rounding off a unitary to a unitary over the ring Z[i,1\/2\u2013\u221a] and employing exact synthesis. We also show that any n qubit unitary over the ring […]<\/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":[13546],"msr-publication-type":[193716],"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-168248","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"arxiv","msr_affiliation":"","msr_published_date":"2013-06-01","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_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","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":"","msr_publicationurl":"http:\/\/arxiv.org\/abs\/1306.3200","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"http:\/\/arxiv.org\/abs\/1306.3200","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"http:\/\/arxiv.org\/abs\/1306.3200"}],"msr-author-ordering":[{"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":[],"publication":[],"video":[],"download":[],"msr_publication_type":"inproceedings","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/168248"}],"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":1,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/168248\/revisions"}],"predecessor-version":[{"id":413558,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/168248\/revisions\/413558"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=168248"}],"wp:term":[{"taxonomy":"msr-content-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-content-type?post=168248"},{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=168248"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=168248"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=168248"},{"taxonomy":"msr-product-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-product-type?post=168248"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=168248"},{"taxonomy":"msr-platform","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-platform?post=168248"},{"taxonomy":"msr-download-source","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-download-source?post=168248"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=168248"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=168248"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=168248"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=168248"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=168248"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=168248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}