{"id":947553,"date":"2023-06-08T10:44:18","date_gmt":"2023-06-08T17:44:18","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=947553"},"modified":"2023-06-08T10:44:18","modified_gmt":"2023-06-08T17:44:18","slug":"quantum-speedups-for-zero-sum-games-via-improved-dynamic-gibbs-sampling","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/quantum-speedups-for-zero-sum-games-via-improved-dynamic-gibbs-sampling\/","title":{"rendered":"Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling"},"content":{"rendered":"

We give a quantum algorithm for computing an\u00a0\u03f5<\/span><\/span><\/span><\/span>-approximate Nash equilibrium of a zero-sum game in a\u00a0m<\/span><\/em>\u00d7<\/span>n<\/span><\/em><\/span><\/span><\/span>\u00a0payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time\u00a0O<\/span><\/em>\u02dc<\/span><\/span><\/span><\/span>(\u221a<\/span><\/span><\/span><\/span><\/span>m<\/span><\/em>+<\/span>n<\/span><\/em><\/span><\/span>\u22c5<\/span>\u03f5<\/span>\u2212<\/span>2.5<\/span><\/span><\/span><\/span>+<\/span>\u03f5<\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u00a0and outputs a classical representation of the\u00a0\u03f5<\/span><\/span><\/span><\/span>-approximate Nash equilibrium. This improves upon the best prior quantum runtime of O<\/span><\/em>\u02dc<\/span><\/span><\/span><\/span>(\u221a<\/span><\/span>m<\/span><\/em>+<\/span>n<\/span><\/em><\/span><\/span>\u22c5<\/span>\u03f5<\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span>) <\/span><\/span><\/span><\/span>obtained by [vAG19] and the classic\u00a0O<\/span><\/em>\u02dc<\/span><\/span><\/span><\/span>(<\/span>(<\/span>m<\/span><\/em>+<\/span>n<\/span><\/em>)<\/span>\u22c5<\/span>\u03f5<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u00a0runtime due to [GK95] whenever\u00a0\u03f5<\/span>=<\/span>\u03a9<\/span>(<\/span>(<\/span>m<\/span><\/em>+<\/span>n<\/span><\/em>)<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>. We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.<\/p>\n","protected":false},"excerpt":{"rendered":"

We give a quantum algorithm for computing an\u00a0\u03f5-approximate Nash equilibrium of a zero-sum game in a\u00a0m\u00d7n\u00a0payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time\u00a0O\u02dc(\u221am+n\u22c5\u03f5\u22122.5+\u03f5\u22123)\u00a0and outputs a classical representation of the\u00a0\u03f5-approximate Nash equilibrium. This improves upon the best prior quantum runtime of O\u02dc(\u221am+n\u22c5\u03f5\u22123) obtained by […]<\/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":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"ICML 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