{"id":430905,"date":"2018-11-06T16:45:33","date_gmt":"2018-11-07T00:45:33","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=430905"},"modified":"2018-11-06T16:45:33","modified_gmt":"2018-11-07T00:45:33","slug":"quantum-speed-ups-semidefinite-programming","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/quantum-speed-ups-semidefinite-programming\/","title":{"rendered":"Quantum Speed-ups for Semidefinite Programming"},"content":{"rendered":"

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n^<\/span>1\/<\/span>2 <\/span><\/span><\/span><\/span><\/span>m^<\/span>1\/<\/span>2 <\/span><\/span><\/span><\/span><\/span>s^<\/span>2 <\/span><\/span>poly<\/span>(<\/span>log<\/span><\/span>(<\/span>n<\/span>)<\/span>,<\/span>log<\/span><\/span>(<\/span>m<\/span>)<\/span>,<\/span>R<\/span>,<\/span>r<\/span>,<\/span>1<\/span>\/<\/span><\/span><\/span>\u03b4<\/span>)<\/span><\/span><\/span><\/span> , with n<\/span><\/span><\/span><\/span> and s<\/span><\/span><\/span><\/span> the dimension and row-sparsity of the input matrices, respectively, m<\/span><\/span><\/span><\/span> the number of constraints, \u03b4<\/span><\/span><\/span><\/span> the accuracy of the solution, and R<\/span>,<\/span>r<\/span><\/span><\/span><\/span> a upper bounds on the size of the optimal primal and dual solutions. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in n<\/span><\/span><\/span><\/span> and m<\/span><\/span><\/span><\/span> . We prove the algorithm cannot be substantially improved (in terms of n<\/span><\/span><\/span><\/span> and m<\/span><\/span><\/span><\/span> ) giving a \u03a9<\/span>(<\/span>n^<\/span>1\/<\/span>2<\/span><\/span><\/span><\/span><\/span>+<\/span>m^<\/span>1\/<\/span>2<\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> quantum lower bound for solving semidefinite programs with constant s<\/span>,<\/span>R<\/span>,<\/span>r<\/span><\/span><\/span><\/span> and \u03b4<\/span><\/span><\/span><\/span> . The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.<\/p>\n","protected":false},"excerpt":{"rendered":"

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n^1\/2 m^1\/2 s^2 poly(log(n),log(m),R,r,1\/\u03b4) , with n and s the dimension and row-sparsity of the input matrices, respectively, m the number of constraints, \u03b4 the accuracy of the solution, and R,r a upper bounds on the size of the optimal […]<\/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":[243138],"msr-publication-type":[193715],"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-430905","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-quantum","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2016-09-18","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":"https:\/\/arxiv.org\/abs\/1609.05537","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"https:\/\/arxiv.org\/abs\/1609.05537","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1609.05537"}],"msr-author-ordering":[{"type":"text","value":"Fernando 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