{"id":5782,"date":"2023-04-24T09:00:49","date_gmt":"2023-04-24T00:00:49","guid":{"rendered":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=5782"},"modified":"2023-04-24T09:00:49","modified_gmt":"2023-04-24T00:00:49","slug":"%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/","title":{"rendered":"\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3068ADMM\u306e\u30cf\u30a4\u30d6\u30ea\u30c3\u30c9\u65b9\u5f0f\u306b\u3088\u308b\u4e0d\u7b49\u5f0f\u5236\u7d04\u3078\u306e\u5bfe\u51e6"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\" >\u6587\u732e\u60c5\u5831<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E6%A6%82%E8%A6%81\" >\u6982\u8981<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E8%AA%B2%E9%A1%8C\" >\u8ab2\u984c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E8%83%8C%E6%99%AF\" >\u80cc\u666f<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E6%9C%80%E9%81%A9%E5%8C%96%E3%83%A2%E3%83%BC%E3%83%89%E3%81%A8%E3%82%B5%E3%83%B3%E3%83%97%E3%83%AA%E3%83%B3%E3%82%B0%E3%83%A2%E3%83%BC%E3%83%89\" >\u6700\u9069\u5316\u30e2\u30fc\u30c9\u3068\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E6%96%B9%E6%B3%95\" >\u65b9\u6cd5<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E6%8B%A1%E5%BC%B5%E3%83%A9%E3%82%B0%E3%83%A9%E3%83%B3%E3%82%B8%E3%83%A5%E6%B3%95\" >\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#ADMM\" >ADMM<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E6%96%B0%E3%81%97%E3%81%84%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0\" >\u65b0\u3057\u3044\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#_%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0%E3%81%AE%E6%89%8B%E9%A0%86\" >[ \u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u624b\u9806 ]<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E5%AE%9F%E9%A8%93\" >\u5b9f\u9a13<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E5%AE%9F%E9%A8%93%E6%96%B9%E6%B3%95\" >\u5b9f\u9a13\u65b9\u6cd5<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#QKP\" >QKP<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#MAPE\" >MAPE<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E5%AE%9F%E9%A8%93%E7%B5%90%E6%9E%9C\" >\u5b9f\u9a13\u7d50\u679c<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#QKP_%E3%81%AB%E5%AF%BE%E3%81%99%E3%82%8B%E8%A7%A3%E3%81%AE%E7%B2%BE%E5%BA%A6%E3%81%AE%E6%A4%9C%E8%A8%BC\" >QKP \u306b\u5bfe\u3059\u308b\u89e3\u306e\u7cbe\u5ea6\u306e\u691c\u8a3c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E8%A8%88%E7%AE%97%E6%99%82%E9%96%93%E3%81%AE%E6%A4%9C%E8%A8%BC\" >\u8a08\u7b97\u6642\u9593\u306e\u691c\u8a3c<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E7%B5%90%E8%AB%96\" >\u7d50\u8ad6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E3%81%82%E3%81%A8%E6%9B%B8%E3%81%8D\" >\u3042\u3068\u66f8\u304d<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E8%A3%9C%E8%B6%B3\" >\u88dc\u8db3<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E8%A3%9C%E8%B6%B3_1_embedding_%E3%81%A8_unembedding\" >\u88dc\u8db3 1 : embedding \u3068 unembedding<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E5%AF%86%E3%82%B0%E3%83%A9%E3%83%95%E3%81%A8%E7%96%8E%E3%82%B0%E3%83%A9%E3%83%95\" >\u5bc6\u30b0\u30e9\u30d5\u3068\u758e\u30b0\u30e9\u30d5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#embedding\" >embedding<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#unembedding\" >unembedding<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E8%A3%9C%E8%B6%B3_2_%E3%82%B9%E3%83%A9%E3%83%83%E3%82%AF%E5%A4%89%E6%95%B0%E3%81%AE%E4%BA%8C%E5%80%A4%E5%A4%89%E6%95%B0%E3%81%AB%E3%82%88%E3%82%8B%E5%B1%95%E9%96%8B\" >\u88dc\u8db3 2 : \u30b9\u30e9\u30c3\u30af\u5909\u6570\u306e\u4e8c\u5024\u5909\u6570\u306b\u3088\u308b\u5c55\u958b<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2023\/04\/24\/%e4%b8%8d%e7%ad%89%e5%bc%8f%e5%88%b6%e7%b4%84%e3%81%ab%e5%af%be%e3%81%99%e3%82%8badmm\/#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE\" >\u53c2\u8003\u6587\u732e<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\"><\/span>\u6587\u732e\u60c5\u5831<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul>\n<li>\u30bf\u30a4\u30c8\u30eb <span>: Solving Inequality-Constrained Binary Optimization Problems on Quantum Annealer<\/span><\/li>\n<li>\u8457\u8005 <span>: Kouki Yonaga<\/span>\u3001<span>Masamichi Miyama<\/span>\u3001<span>Masayuki Ohzeki<\/span><\/li>\n<li>\u66f8\u8a8c\u60c5\u5831 ( DOI )<span> : <\/span><a href=\"https:\/\/doi.org\/10.48550\/arXiv.2012.06119\">https:\/\/doi.org\/10.48550\/arXiv.2012.06119<\/a><\/li>\n<\/ul>\n\n\n\n\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%A6%82%E8%A6%81\"><\/span>\u6982\u8981<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u6700\u9069\u5316\u554f\u984c\u306e\u4e2d\u306b\u306f\u3001\u4e0d\u7b49\u5f0f\u306e\u5f62\u3067\u8868\u3055\u308c\u308b\u5236\u7d04 ( <span> =<\/span> \u4e0d\u7b49\u5f0f\u5236\u7d04 ) \u3092\u542b\u3080\u554f\u984c\u304c\u591a\u304f\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u4e0d\u7b49\u5f0f\u5236\u7d04\u3092\u6271\u3046\u969b\u306b\u3001\u5f93\u6765\u306e\u624b\u6cd5\u3067\u306f\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3068\u547c\u3070\u308c\u308b\u88dc\u52a9\u5909\u6570\u304c\u7528\u3044\u3089\u308c\u3066\u3044\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u306f\u4e8c\u5024\u5909\u6570\u3092\u7528\u3044\u3066\u8868\u3057\u76f4\u3055\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u305f\u3081\u3001\u591a\u6570\u306e\u7269\u7406\u91cf\u5b50\u30d3\u30c3\u30c8\u304c\u8ffd\u52a0\u3067\u5fc5\u8981\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u308c\u306b\u3088\u308a\u3001\u73fe\u884c\u306e\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u30de\u30b7\u30f3\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3053\u3068\u306e\u3067\u304d\u308b\u554f\u984c\u306f\u5c0f\u898f\u6a21\u306a\u3082\u306e\u306b\u9650\u5b9a\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u305d\u3053\u3067\u3001\u672c\u8ad6\u6587\u3067\u306f\u3001\u4ea4\u4e92\u65b9\u5411\u4e57\u6570\u6cd5<span> ( ADMM ) <\/span>\u3068\u3044\u3046\u65e2\u5b58\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3092\u7d44\u307f\u5408\u308f\u305b\u305f\u65b0\u305f\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u63d0\u6848\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u308a\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04\u3092\u30b9\u30e9\u30c3\u30af\u5909\u6570\u306a\u3057\u3067\u6271\u3046\u3053\u3068\u304c\u3067\u304d\u3001\u5f93\u6765\u3088\u308a\u3082\u5927\u898f\u6a21\u306a\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u53ef\u80fd\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u6027\u80fd\u3092\u691c\u8a3c\u3059\u308b\u305f\u3081\u306b\u3001\u672c\u8ad6\u6587\u3067\u306f <span>2 <\/span>\u6b21\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c ( <span>QKP<\/span> ) \u3092\u7528\u3044\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u305f <span>QKP<\/span> \u306b\u5bfe\u3059\u308b\u89e3\u306e\u7cbe\u5ea6\u3092 <span>Gubori<\/span> \u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u308b\u53b3\u5bc6\u89e3\u3068\u6bd4\u8f03\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%AA%B2%E9%A1%8C\"><\/span>\u8ab2\u984c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0 ( QA : Quantum Annealing ) \u306f\u3001\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306e\u6280\u8853\u3067\u3059\u3002\u305d\u3057\u3066\u3001QA \u3092\u5b9f\u884c\u3059\u308b\u30de\u30b7\u30f3\u3092\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u30de\u30b7\u30f3\u3068\u547c\u3073\u307e\u3059\u3002\u672c\u8ad6\u6587\u3067\u306f\u3001D-Wave Systems \u304c\u958b\u767a\u3057\u305f\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u30de\u30b7\u30f3 ( = D-Wave \u30de\u30b7\u30f3 ) \u3092\u4f7f\u7528\u3057\u307e\u3059\u3002\u73fe\u884c\u306e D-Wave \u30de\u30b7\u30f3\u3067\u3042\u308b D-Wave 2000Q \u306f\u3001\u4ee5\u4e0b\u306e\u6700\u5c0f\u5316\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$ \\underset{\\bm{x}} {\\rm{minimize}} \\quad {\\bm{x}}^{\\top} Q {\\bm{x}} \\tag{1} $$<\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001 $\\bm{x} \\in \\{ 0,1 \\}^{N}$ \u306f $N$ \u6b21\u5143\u306e\u4e8c\u5024\u30d9\u30af\u30c8\u30eb\u3001 $Q$ \u306f\u5b9f\u6570\u307e\u305f\u306f\u6574\u6570\u306e\u884c\u5217\u3067\u3059\u3002\u3053\u306e\u554f\u984c\u3092\u4e8c\u6b21\u5236\u7d04\u7121\u3057\u4e8c\u5024\u6700\u9069\u5316\u554f\u984c ( QUBO : Quadratic Unconstrained Binary Optimization ) \u3068\u547c\u3073\u3001 $Q$ \u3092 QUBO \u884c\u5217\u3068\u8a00\u3044\u307e\u3059[1]\u3002<\/p>\n\n\n\n\n\n<p>\u5f93\u3063\u3066\u3001D-Wave 2000Q \u3092\u7528\u3044\u3066\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306b\u306f\u3001\u5bfe\u8c61\u306e\u554f\u984c\u3092 QUBO \u5f62\u5f0f\u3067\u8868\u73fe\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u4eca\u3001\u5bfe\u8c61\u306e\u554f\u984c\u304c\u4ee5\u4e0b\u306e\u7dda\u5f62\u5236\u7d04\u3092\u542b\u3080\u6700\u9069\u5316\u554f\u984c\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$\u00a0 \\begin{split} \\underset{\\bm{x}} {\\rm{minimize}} \\quad\u00a0 &amp;f({\\bm x}) \\\\ {\\rm subject \\ to} \\quad &amp;\\bm{F}^{\\top}_{l} \\bm{x} = C_{l} \\qquad(l=1,\\cdots, L) \\\\ &amp;\\bm{G}^{\\top}_{m}\u00a0 \\bm{x} \\leq\u00a0 D_{m} \\quad (m=1,\\cdots, M) \\\\ \\end{split} \\tag{2}$$<\/p>\n<p class=\"lang-tex s-code-block\">\u3053\u3053\u3067\u3001${\\bm F}_{l}, \\ {\\bm G}_{m} \\in \\mathbb{Z}^{N}$ ( $\\mathbb{Z}$ \u306f\u6574\u6570\u306e\u96c6\u5408 )\u3001$C_{l}, \\ D_{m} \\in \\mathbb{Z}$ \u3067\u3042\u308a\u3001$f(\\bm{x})$ \u306f QUBO \u5f62\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b\u76ee\u7684\u95a2\u6570\u3067\u3059\u3002$l$ \u306f\u7b49\u5f0f\u5236\u7d04\u306e\u756a\u53f7\u3092\u8868\u3059\u6dfb\u3048\u5b57\u3001$m$ \u306f\u4e0d\u7b49\u5f0f\u5236\u7d04\u306e\u756a\u53f7\u3092\u8868\u3059\u6dfb\u3048\u5b57\u3067\u3059\u3002<\/p>\n<p>\u4eca\u3001\u3053\u306e\u554f\u984c\u3092\u5f0f <span>(1) <\/span>\u306e <span>QUBO <\/span>\u5f62\u5f0f\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<span>QUBO <\/span>\u5f62\u5f0f\u306f\u300c\u5236\u7d04\u7121\u3057\u300d\u306e\u6700\u9069\u5316\u554f\u984c\u306a\u306e\u3067\u3001\u7b49\u5f0f\u5236\u7d04 ${\\bm F}^{\\top}_{l} {\\bm x} = C_{l}$ \u3068\u4e0d\u7b49\u5f0f\u5236\u7d04 ${\\bm G}^{\\top}_{m} {\\bm x} \\leq D_{m}$ \u3092\u76ee\u7684\u95a2\u6570\u306b\u542b\u3081\u308b\u5f62\u3067\u8868\u73fe\u3057\u76f4\u3055\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093\u3002\u305d\u306e\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u308b\u65b9\u6cd5\u306e\u4e00\u3064\u306b\u300c\u7f70\u91d1\u6cd5\u300d\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>\u307e\u305a\u306f\u3001\u7b49\u5f0f\u5236\u7d04\u306e\u65b9\u306b\u7f70\u91d1\u6cd5\u3092\u9069\u7528\u3057\u307e\u3059\u3002\u7f70\u91d1\u6cd5\u3067\u306f\u3001\u7b49\u5f0f\u5236\u7d04 ${\\bm F}^{\\top}_{l} {\\bm x} = C_{l}$\u00a0 $(l=1,\\dots,L)$ \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u73fe\u3057\u307e\u3059\u3002<\/p>\n<p>$$ \\displaystyle \\sum_{l=1}^{L} ( \\bm{F}^{\\top}_{l} \\bm{x} -C_l )^2 \\tag{3}$$<\/p>\n<p>\u305d\u3057\u3066\u3001\u5f0f (3) \u3092\u76ee\u7684\u95a2\u6570 $f({\\bm x})$ \u306b\u52a0\u3048\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u5f0f (2) \u306f\u4ee5\u4e0b\u306e\u554f\u984c\u306b\u7f6e\u304d\u63db\u308f\u308a\u307e\u3059\u3002<\/p>\n<p>$$\u00a0 \\begin{split} \\underset{\\bm{x}} {\\rm{minimize}} \\quad\u00a0 &amp;f({\\bm x}) + \\alpha \\sum_{l=1}^{L}(\\bm{F}^{\\top}_l \\bm{x} &#8211; C_l)^2\u00a0 \\\\ {\\rm subject \\ to} \\quad &amp;\\bm{G}^{\\top}_{m} \\bm{x} \\leq C_{m} \\qquad(m=1,\\cdots, M) \\\\ \\end{split} \\tag{2&#8242;} $$<\/p>\n<p>\u5f0f <span>(2) <\/span>\u306b\u306f ${\\bm F}^{\\top}_{l} {\\bm x} = C_{l}$\u00a0 $(l=1,\\dots,L)$ \u3068\u3044\u3046\u5236\u7d04\u304c\u3042\u308a\u307e\u3057\u305f\u304c\u3001\u5f0f <span>(2\u2019) <\/span>\u3067\u306f\u305d\u308c\u304c\u7121\u304f\u306a\u308a\u3001\u4ee3\u308f\u308a\u306b\u76ee\u7684\u95a2\u6570\u306b\u5f0f<span>(3)<\/span>\u306e\u9805\u3092 $\\alpha$ \u500d\u3057\u305f\u3082\u306e\u304c\u8db3\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u5236\u7d04\u3092\u6e80\u305f\u3055\u306a\u304b\u3063\u305f\u5834\u5408<span>\u00a0 ( <\/span>\u3059\u306a\u308f\u3061 ${\\bm F}^{\\top}_l {\\bm x} &#8211; C_l \\ne 0$ \u306e\u5834\u5408 <span>)<\/span>\u3001\u305d\u306e\u8aa4\u5dee\u306e\u4e8c\u4e57\u3092 $\\alpha$ \u500d\u3057\u305f\u3082\u306e\u304c\u30b3\u30b9\u30c8\u95a2\u6570\u306b\u8db3\u3055\u308c\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u4fc2\u6570 $\\alpha$ \u306f\u7f70\u91d1\u4fc2\u6570\u3068\u547c\u3070\u308c\u3001\u5f0f (3) \u306e\u9805\u304c\u30b3\u30b9\u30c8\u95a2\u6570\u306b\u53ca\u307c\u3059\u5f71\u97ff\u306e\u5f37\u3055\u3092\u6c7a\u3081\u308b\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001$\\alpha$ \u306f $ f ( {\\bm x} ) $ \u3088\u308a\u3082\u76f8\u5bfe\u7684\u306b\u5927\u304d\u306a\u5024\u306b\u8a2d\u5b9a\u3057\u307e\u3059\u3002<\/p>\n<p>\u4ee5\u4e0a\u306e\u3088\u3046\u306b\u3057\u3066\u5236\u7d04\u3092\u76ee\u7684\u95a2\u6570\u306b\u542b\u3081\u308b\u624b\u6cd5\u3092\u7f70\u91d1\u6cd5\u3068\u8a00\u3044\u307e\u3059\u3002\u307e\u305f\u3001\u7f70\u91d1\u6cd5\u306b\u3088\u308a\u76ee\u7684\u95a2\u6570\u306b\u52a0\u308f\u308b\u9805\u3092\u300c\u7f70\u91d1\u9805\u300d\u3068\u547c\u3073\u3001\u5236\u7d04\u3092\u76ee\u7684\u95a2\u6570\u306b\u542b\u3081\u308b\u3053\u3068\u3092\u300c\u7de9\u548c\u300d\u3068\u547c\u3073\u307e\u3059\u3002\u540c\u3058\u3088\u3046\u306b\u3057\u3066\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04 ${\\bm G}^{\\top}_{m} {\\bm x} \\leq D_{m}$ \u306b\u3064\u3044\u3066\u3082\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u73fe\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>$$ \\displaystyle \\sum_{m=1}^{M} ({\\bm G}^{\\top}_{m} {\\bm x}-D_{m}+s_{m})^2 \\tag{4} $$<\/p>\n<p>\u305f\u3060\u3057\u3001$s_m \\in \\{ 0,1,\\cdots, D_m \\}$ \u3068\u3057\u307e\u3059\u3002\u3053\u306e\u65b0\u305f\u306b\u8ffd\u52a0\u3055\u308c\u305f\u5909\u6570 $s_m$ \u306e\u3053\u3068\u3092\u300c\u30b9\u30e9\u30c3\u30af\u5909\u6570\u300d\u3068\u547c\u3073\u307e\u3059\u3002\u3053\u306e\u30b9\u30e9\u30c3\u30af\u5909\u6570 $s_m$ \u306f\u5909\u6570 $\\bm{x}$ \u3068\u540c\u69d8\u3001QA \u306b\u3088\u3063\u3066\u6700\u9069\u89e3\u3092\u6c42\u3081\u308b\u5bfe\u8c61\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u5f0f (4) \u3092\u76ee\u7684\u95a2\u6570\u306b\u52a0\u3048\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u5f0f (2&#8242;) \u306f\u4ee5\u4e0b\u306e\u554f\u984c\u306b\u7de9\u548c\u3055\u308c\u307e\u3059\u3002<\/p>\n<p>$$ \\displaystyle \\underset{\\bm{x}} {\\rm{minimize}} \\quad\u00a0 f({\\bm x}) + \\alpha \\sum_{l=1}^{L}({\\bm F}^{\\top}_{l} {\\bm x} &#8211; C_l)^2 + \\beta \\sum_{m=1}^{M} ({\\bm G}^{\\top}_{m} \\bm{x} -D_m + s_m)^2 \\tag{2&#8221;}$$<\/p>\n<p>\u00a0\u3053\u308c\u306b\u3088\u308a\u3001\u5f0f <span>(2) <\/span>\u306f\u5236\u7d04\u7121\u3057\u306e\u6700\u9069\u5316\u554f\u984c\u306b\u306a\u308a\u307e\u3057\u305f\u3002\u3053\u3053\u3067\u3001$\\beta$ \u306f $\\alpha$ \u3068\u540c\u69d8\u306e\u7f70\u91d1\u4fc2\u6570\u3067\u3059\u3002\u3057\u304b\u3057\u3001$s_m \\in \\{ 0,1,\\cdots, D_m \\}$ \u3067\u3042\u308b\u305f\u3081\u3001 $D_m \\geq 2$ \u306e\u3068\u304d $s_m$ \u306f\u4e8c\u5024\u5909\u6570\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<span>QUBO <\/span>\u306f\u300c\u4e8c\u5024\u300d\u6700\u9069\u5316\u554f\u984c\u306a\u306e\u3067\u3001$s_m$ \u304c\u4e8c\u5024\u5909\u6570\u3067\u306a\u304b\u3063\u305f\u5834\u5408\u3001 $s_m = 2^{0} \\times y_1 + 2^1 \\times y_2 + 2^2 + y_3 + \\cdots $ \u3068\u3044\u3046\u3088\u3046\u306b\u88dc\u52a9\u7684\u306a\u4e8c\u5024\u5909\u6570\u3067\u8868\u73fe\u3057\u76f4\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u00a0<span> ( <\/span>\u8a73\u3057\u304f\u306f <strong>9.2. \u88dc\u8db32 : \u30b9\u30e9\u30c3\u30af\u5909\u6570\u306e\u4e8c\u5024\u5909\u6570\u306b\u3088\u308b\u5c55\u958b <\/strong>\u306b\u3066\u7c21\u5358\u306a\u4f8b\u3092\u7528\u3044\u3066\u8aac\u660e\u3057\u3066\u3044\u307e\u3059 <span>)<\/span>\u3002\u3055\u3089\u306b\u3001\u3053\u306e\u3088\u3046\u306a\u5c55\u958b\u3092\u5168\u3066\u306e\u4e0d\u7b49\u5f0f\u5236\u7d04 ${\\bm G}^{\\top}_m {\\bm x} \\leq D_m \\ (m=0,1,2,\\cdots,M)$ \u306b\u5bfe\u3057\u3066\u884c\u3046\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u5c55\u958b\u306b\u3088\u308a\u591a\u304f\u306e\u4e8c\u5024\u5909\u6570\u304c\u8ffd\u52a0\u3055\u308c\u3001\u7269\u7406\u91cf\u5b50\u30d3\u30c3\u30c8\u304c $y_1, y_2, y_3,\\cdots$ \u306e\u500b\u6570\u5206\u3060\u3051\u4f59\u8a08\u306b\u5fc5\u8981\u3068\u306a\u308a\u307e\u3059\u3002<span>D-Wave 2000Q <\/span>\u306e\u7269\u7406\u91cf\u5b50\u30d3\u30c3\u30c8\u6570\u306f 2048 \u500b\u3067\u3042\u308b\u305f\u3081\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u4f7f\u3063\u3066\u3057\u307e\u3046\u3068\u3001\u5909\u6570 ${\\bm x}$ \u306e\u6570\u304c\u5c11\u306a\u3044<span> ( = <\/span>\u30b5\u30a4\u30ba\u306e\u5c0f\u3055\u3044 <span>) <\/span>\u554f\u984c\u3057\u304b\u89e3\u3051\u306a\u304f\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n<p>\u4ee5\u4e0a\u306e\u7406\u7531\u304b\u3089\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04\u306b\u5bfe\u3057\u3066\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u7528\u3044\u308b\u3053\u3068\u306f\u5f97\u7b56\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u672c\u8ad6\u6587\u3067\u306f\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u4f7f\u308f\u305a\u306b\u89e3\u304f\u65b9\u6cd5\u3092\u958b\u767a\u3057\u3066\u3044\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u5f93\u6765\u3088\u308a\u3082\u30b5\u30a4\u30ba\u306e\u5927\u304d\u306a\u4e0d\u7b49\u5f0f\u5236\u7d04\u4ed8\u304d\u306e\u554f\u984c\u304c\u89e3\u3051\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E8%83%8C%E6%99%AF\"><\/span>\u80cc\u666f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"%E6%9C%80%E9%81%A9%E5%8C%96%E3%83%A2%E3%83%BC%E3%83%89%E3%81%A8%E3%82%B5%E3%83%B3%E3%83%97%E3%83%AA%E3%83%B3%E3%82%B0%E3%83%A2%E3%83%BC%E3%83%89\"><\/span>\u6700\u9069\u5316\u30e2\u30fc\u30c9\u3068\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u672c\u8ad6\u6587\u3067\u4f7f\u7528\u3059\u308b <span>D-Wave 2000Q <\/span>\u306b\u306f\u300c\u6700\u9069\u5316\u30e2\u30fc\u30c9\u300d\u3068\u300c\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9\u300d\u3068\u547c\u3070\u308c\u308b 2 \u3064\u306e\u5f8c\u51e6\u7406\u306e\u30e2\u30fc\u30c9\u304c\u642d\u8f09\u3055\u308c\u3066\u3044\u307e\u3059\u3002<strong>6. \u5b9f\u9a13 <\/strong>\u3067\u306f\u4e21\u30e2\u30fc\u30c9\u306e\u6bd4\u8f03\u3082\u884c\u3046\u305f\u3081\u3001\u3053\u3053\u3067\u306f\u305d\u306e 2 \u3064\u306e\u30e2\u30fc\u30c9\u306b\u3064\u3044\u3066\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n<p>\u307e\u305a\u3001\u6700\u9069\u5316\u30e2\u30fc\u30c9\u3067\u306f\u3001<span>QA <\/span>\u3067\u5f97\u3089\u308c\u305f\u30b5\u30f3\u30d7\u30eb ( \u89e3\u306e\u5019\u88dc ) \u306b\u5bfe\u3057\u3066\u5c40\u6240\u7684\u306a\u66f4\u65b0\u3092\u884c\u3046\u3053\u3068\u306b\u3088\u3063\u3066\u3001\u3088\u308a\u4f4e\u3044\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u5f97\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u30b3\u30b9\u30c8\u95a2\u6570\u3068\u306f\u3001\u5f0f <span>(2\u2019\u2019) <\/span>\u306e\u3088\u3046\u306a\u3001\u6700\u9069\u5316\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306b\u6700\u5c0f\u5316\u3059\u308b\u95a2\u6570\u306e\u3053\u3068\u3067\u3059\u3002\u3059\u306a\u308f\u3061\u3001\u6700\u9069\u5316\u30e2\u30fc\u30c9\u306f\u3001\u89e3\u304d\u305f\u3044\u6700\u9069\u5316\u554f\u984c\u306b\u5bfe\u3057\u3066\u3001\u306a\u308b\u3079\u304f\u826f\u3044\u89e3\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u30e2\u30fc\u30c9\u3067\u3059\u3002<\/p>\n<p>\u4e00\u65b9\u3001\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9\u3067\u306f\u3001<span>QA <\/span>\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u305f\u30b5\u30f3\u30d7\u30eb\u3092\u3001\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u78ba\u7387\u5206\u5e03\u306b\u4fee\u6b63\u3057\u307e\u3059\u3002<\/p>\n<p>$$ P({\\bm x}) = \\frac{1}{Z} \\exp [-\\beta E({\\bm x})]\u00a0 \\tag{5}$$<\/p>\n<p>\u5f0f (5) \u306e\u78ba\u7387\u5206\u5e03\u306f\u30dc\u30eb\u30c4\u30de\u30f3\u5206\u5e03\u3068\u547c\u3070\u308c\u307e\u3059\u3002\u3053\u3053\u3067\u3001$\\beta$ \u306f\u9006\u6e29\u5ea6\u3067\u3059\u3002$\\beta$ \u3092\u5909\u5316\u3055\u305b\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u78ba\u7387\u5206\u5e03 $P({\\bm x})$ \u304c\u5909\u5316\u3057\u307e\u3059\u3002\u4f8b\u3048\u3070\u3001$\\beta \\to \\infty $ \u3068\u3059\u308b\u3068\u3001\u30a8\u30cd\u30eb\u30ae\u30fc $E({\\bm x})$ \u3092\u6700\u3082\u5c0f\u3055\u304f\u3059\u308b\u30b5\u30f3\u30d7\u30eb\u306e\u307f\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u4e00\u65b9\u3001$\\beta \\to 0$ \u3068\u3059\u308b\u3068\u3001$P({\\bm x})$ \u304b\u3089\u591a\u69d8\u306a\u30b5\u30f3\u30d7\u30eb\u304c\u751f\u6210\u3055\u308c\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3001\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9\u306f\u30dc\u30eb\u30c4\u30de\u30f3\u5206\u5e03\u304b\u3089\u30b5\u30f3\u30d7\u30eb\u3092\u53d6\u5f97\u3057\u305f\u3044\u5834\u5408\u306b\u7528\u3044\u308b\u30e2\u30fc\u30c9\u3067\u3059[2]\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95\"><\/span>\u65b9\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"%E6%8B%A1%E5%BC%B5%E3%83%A9%E3%82%B0%E3%83%A9%E3%83%B3%E3%82%B8%E3%83%A5%E6%B3%95\"><\/span>\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u672c\u8ad6\u6587\u3067\u63d0\u6848\u3059\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u3001\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5\u3068 ADMM \u306b\u57fa\u3065\u3044\u3066\u3044\u307e\u3059\u3002\u3053\u3053\u304b\u3089\u306f\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u4f7f\u308f\u305a\u306b\u4e0d\u7b49\u5f0f\u5236\u7d04\u3092\u7de9\u548c\u3059\u308b\u65b9\u6cd5\u3092\u8b70\u8ad6\u3057\u306a\u304c\u3089\u3001\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5\u3068 ADMM \u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<p>\u4eca\u56de\u306f\u3001\u7c21\u5358\u306e\u305f\u3081\u5f0f (2) \u306b\u304a\u3044\u3066\u4e0d\u7b49\u5f0f\u5236\u7d04\u306e\u307f\u3092\u8003\u3048\u307e\u3059\u3002\u4e0d\u7b49\u5f0f\u5236\u7d04 ${\\bm G}^{\\top}_m {\\bm x} \\leq D_m \\ (m=0,1,2,\\cdots,M)$ \u3092\u7de9\u548c\u3059\u308b\u305f\u3081\u306b\u3001\u3053\u3053\u3067\u306f\u30d8\u30f4\u30a3\u30b5\u30a4\u30c9\u306e\u968e\u6bb5\u95a2\u6570 $\\Theta (x)$ \u3092\u7528\u3044\u307e\u3059\u3002<\/p>\n<p>$$ \\begin{equation} \\Theta (x) = \\left \\{\\begin{array}{l}1\u3000{\\rm if} \\ x &gt; 0 \\\\0\u3000{\\rm if}\\\u00a0 x \\leq 0 \\end{array}\\right.\\end{equation} $$<\/p>\n<p>$\\Theta (x)$ \u306e\u30b0\u30e9\u30d5\u3092\u4ee5\u4e0b\u306e\u56f3 1 \u306b\u793a\u3057\u307e\u3059\u3002<\/p>\n<div id=\"attachment_5884\" style=\"width: 310px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5884\" src=\"\/T-Wave\/wp-content\/uploads\/2023\/02\/d8d4b6242c10922efc82e1c97e9a03a4-300x247.png\" alt=\"\" width=\"300\" height=\"247\" class=\"wp-image-5884 size-medium\" \/><p id=\"caption-attachment-5884\" class=\"wp-caption-text\">\u56f31. \u30d8\u30f4\u30a3\u30b5\u30a4\u30c9\u306e\u968e\u6bb5\u95a2\u6570 $\\Theta({\\bm x})$ \u306e\u30b0\u30e9\u30d5\u3002<\/p><\/div>\n<p>\u3053\u306e\u968e\u6bb5\u95a2\u6570 $\\Theta (x)$ \u3068\u7f70\u91d1\u6cd5\u3092\u7d44\u307f\u5408\u308f\u305b\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04 ${\\bm G}^{\\top}_m {\\bm x} \\leq D_m \\ (m=0,1,2,\\cdots,M)$ \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7de9\u548c\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>$$ E_{{\\rm ineq}} ({\\bm x}) = f({\\bm x})+ \\gamma \\sum_{m=1}^{M} \\Theta ({\\bm G}^{\\top}_{m} {\\bm x} &#8211; D_m) \\tag{6} $$<\/p>\n<p>\u4efb\u610f\u306e $m$ \u306b\u3064\u3044\u3066 $\\Theta ({\\bm G}^{\\top}_{m} {\\bm x}^{\\ast} -D_m)$ \u304c\u30bc\u30ed\u306e\u3068\u304d\u3001 ${\\bm x}^{\\ast}$ \u306f\u5b9f\u884c\u53ef\u80fd\u89e3\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u308c\u306f\u3001${\\bm G}^{\\top}_m {\\bm x}^{\\ast} \\leq D_m \\Leftrightarrow {\\bm G}^{\\top}_m {\\bm x}^{\\ast}\u00a0 &#8211; D_m \\leq 0$\u00a0 \u00a0\u306e\u3068\u304d\u3001\u968e\u6bb5\u95a2\u6570\u306e\u5b9a\u7fa9\u3088\u308a $\\Theta ({\\bm G}^{\\top}_{m} {\\bm x}^{\\ast} &#8211; D_m)=0$ \u3068\u306a\u308b\u304b\u3089\u3067\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u4f7f\u308f\u306a\u304f\u3066\u3082\u4e0d\u7b49\u5f0f\u5236\u7d04\u3092\u7de9\u548c\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<p>\u3057\u304b\u3057\u3001\u5f0f <span>(6) <\/span>\u306f\u968e\u6bb5\u95a2\u6570\u306e\u4e2d\u306b\u5909\u6570 ${\\bm x}$ \u304c\u542b\u307e\u308c\u3066\u3044\u308b\u305f\u3081\u3001<span>QUBO <\/span>\u5f62\u5f0f\u306b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u3053\u306e\u307e\u307e\u3067\u306f <span>D-Wave 2000Q <\/span>\u3092\u7528\u3044\u305f\u6700\u5c0f\u5316\u306f\u4e0d\u53ef\u80fd\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u5f0f <span>(6) <\/span>\u306e\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u6b21\u306e\u6700\u9069\u5316\u554f\u984c\u306b\u66f8\u304d\u63db\u3048\u307e\u3059\u3002<\/p>\n<p>$$\u00a0 \\begin{split} \\underset{\\bm{x}} {\\rm{minimize}} \\quad\u00a0 &amp;f({\\bm x}) + \\gamma \\sum_{m=1}^{M} \\Theta(z_m) \\\\ {\\rm subject \\ to} \\quad &amp;\\bm{G}^{\\top}_{m} \\bm{x} -D_m = z_m \\qquad(m=1,\\cdots, M) \\\\ \\end{split} \\tag{7} $$<\/p>\n<p>\u3053\u3053\u3067\u3001 $\\{ z_m \\} \\in \\mathbb{Z}^N$ \u306f\u88dc\u52a9\u5909\u6570\u3067\u3059\u3002\u3059\u306a\u308f\u3061\u3001 $\\Theta({\\bm G}^{\\top}_m {\\bm x})$ \u304c $\\Theta(z_m)$ \u3068\u7b49\u4fa1\u306b\u306a\u308b\u3088\u3046\u306b\u3001\u5236\u7d04\u3068\u3057\u3066 ${\\bm G}^{\\top}_m {\\bm x} &#8211; D_m = z_m$ \u3092\u8ab2\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u968e\u6bb5\u95a2\u6570\u306e\u4e2d\u306b\u306f\u5909\u6570 ${\\bm x}$ \u304c\u7121\u304f\u306a\u308a\u307e\u3057\u305f\u3002\u305f\u3060\u3057\u3001\u65b0\u305f\u306a\u7b49\u5f0f\u5236\u7d04\u304c\u767b\u5834\u3057\u305f\u305f\u3081\u3001\u7de9\u548c\u3057\u76f4\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u306b\u7528\u3044\u308b\u306e\u304c\u300c\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5\u300d\u3067\u3059\u3002\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5\u3092\u7528\u3044\u3066\u7b49\u5f0f\u5236\u7d04\u3092\u7de9\u548c\u3057\u305f\u30b3\u30b9\u30c8\u95a2\u6570 $E_{\\rm{aug}}$ \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<p>$$\u00a0 \\displaystyle E_{\\rm aug}({\\bm x},{\\bm z},{\\bm \\lambda}) = f({\\bm x}) + \\gamma \\sum_{m=1}^{M} \\Theta(z_m)+ \\sum_{m=1}^{M} \\lambda_{m} ({\\bm G}^{\\top}_m {\\bm x} &#8211; D_m &#8211; z_m) + \\frac{\\rho}{2} \\sum_{m=1}^{M} ({\\bm G}^{\\top}_m {\\bm x} -D_m -z_m)^2\u00a0 \\tag{8}$$<\/p>\n<p>\u5f0f <span>(8) <\/span>\u306e\u7b2c <span>4 <\/span>\u9805\u306f\u3001\u5f0f <span>(3) <\/span>\u3068\u540c\u3058\u7f70\u91d1\u9805\u3067\u3059\u3002\u305d\u3057\u3066\u3001\u7b2c\uff13\u9805\u306f\u300c\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u672a\u5b9a\u4e57\u6570\u6cd5\u300d\u306b\u3088\u308b\u9805\u3067\u3059\u3002\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u672a\u5b9a\u4e57\u6570\u6cd5\u306f\u3001\u7f70\u91d1\u6cd5\u3068\u540c\u3058\u3088\u3046\u306b\u5236\u7d04\u3092\u7de9\u548c\u3059\u308b\u65b9\u6cd5\u306e\u4e00\u3064\u3067\u3059\u3002\u7f70\u91d1\u6cd5\u3068\u306e\u9055\u3044\u306f\u3001 $({\\bm G}^{\\top}_m {\\bm x} &#8211; D_m &#8211; z_m)$ \u304c <span>1 <\/span>\u6b21\u306b\u306a\u3063\u3066\u3044\u308b\u70b9\u3068\u3001\u4fc2\u6570 $\\lambda_m$ \u304c $m=1,2,\\cdots, M$ \u3068\u3044\u3046\u5236\u7d04\u306b\u4f9d\u5b58\u3057\u3066\u3044\u308b\u70b9\u3067\u3059 ( \u3053\u306e\u4fc2\u6570\u306f\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u672a\u5b9a\u5b9a\u6570\u3068\u547c\u3070\u308c\u307e\u3059 )\u3002<\/p>\n<p>\u3053\u306e\u3088\u3046\u306b\u3001\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5\u306f\u3001\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u672a\u5b9a\u4e57\u6570\u6cd5\u3068\u7f70\u91d1\u6cd5\u3092\u7d44\u307f\u5408\u308f\u305b\u3066\u5236\u7d04\u3092\u7de9\u548c\u3059\u308b\u65b9\u6cd5\u3067\u3059\u3002\u4ee5\u964d\u3067\u306f\u3001\u5f0f <span>(8) <\/span>\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u5bfe\u8c61\u306e\u6700\u9069\u5316\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u3092\u76ee\u6307\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"ADMM\"><\/span>ADMM<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>ADMM\u306f\u3001\u5f0f <span>(8) <\/span>\u306e\u3088\u3046\u306a\u3001\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5\u3092\u7528\u3044\u3066\u5b9a\u7fa9\u3057\u305f\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u6700\u5c0f\u5316\u3059\u308b\u305f\u3081\u306b\u5e83\u304f\u7528\u3044\u3089\u308c\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3059\u3002<span>ADMM <\/span>\u3067\u306f\u3001\u4ee5\u4e0b\u306e <span>3 <\/span>\u30b9\u30c6\u30c3\u30d7\u304b\u3089\u306a\u308b\u9010\u6b21\u6700\u9069\u5316\u3092\u7e70\u308a\u8fd4\u3057\u9069\u7528\u3057\u3066 ${\\bm x}$\u3001${\\bm z}$\u3001\u304a\u3088\u3073\u4e57\u6570 ${\\bm \\lambda}$ \u3092\u66f4\u65b0\u3057\u307e\u3059\u3002<\/p>\n<p>$$ {\\bm x}^{\\ast}[t+1]=\\underset{\\bm{x}} {\\rm{argmin}}\u00a0 \\ \\\u00a0 \u00a0E_{\\rm aug}({\\bm x}, {\\bm z}^{\\ast}[t+1], {\\bm \\lambda}[t]) \\tag{9a}$$<\/p>\n<p>$$ {\\bm z}^{\\ast}[t+1]=\\underset{\\bm{z}} {\\rm{argmin}} \\ \\ E_{\\rm aug}({\\bm x}^{\\ast}[t+1], {\\bm z}, {\\bm \\lambda}[t]) \\tag{9b}$$<\/p>\n<p>$$ \\lambda_{m}[t+1]=\\lambda_{m}[t]+\\rho ({\\bm G}^{\\top}_{m} {\\bm x}^{\\ast}[t+1] &#8211; D_m &#8211; z_m^{\\ast}[t+1] ) \\quad (m=1,\\cdots,M) \\tag{9c}$$<\/p>\n<p>\u3053\u3053\u3067\u3001 $t$ \u306f\u53cd\u5fa9\u56de\u6570\u306b\u76f8\u5f53\u3057\u307e\u3059\u3002\u53ce\u675f\u3059\u308b\u307e\u3067\u5f0f (<span>9a) <\/span>~ (<span>9c) <\/span>\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u3067\u3001$E_{\\rm aug}({\\bm x},{\\bm z},{\\bm \\lambda})$ \u3092\u6700\u5c0f\u306b\u3059\u308b\u89e3\u3092\u5f97\u307e\u3059\u3002<\/p>\n<p>\u3059\u306a\u308f\u3061\u3001<span>ADMM <\/span>\u3067\u306f\u3001\u306f\u3058\u3081\u306b ${\\bm z}$ \u3092\u56fa\u5b9a\u3057\u3066 ${\\bm x}$ \u306b\u3064\u3044\u3066\u6700\u9069\u5316\u3057 <span>( <\/span>\u5f0f <span>(9a) )<\/span>\u3001\u6b21\u306b ${\\bm x}$ \u3092\u56fa\u5b9a\u3057\u3066 ${\\bm z}$ \u306b\u3064\u3044\u3066\u6700\u9069\u5316\u3057<span> ( <\/span>\u5f0f <span>(9b) )<\/span>\u3001\u6700\u5f8c\u306b\u6700\u9069\u5316\u3057\u305f ${\\bm x}$ \u3068 ${\\bm z}$ \u3092\u7528\u3044\u3066\u00a0 $\\lambda_{m}$ \u3092\u66f4\u65b0\u3059\u308b <span>( <\/span>\u5f0f <span>(9c) )<\/span>\u3001\u3068\u3044\u3046 <span>3 <\/span>\u30b9\u30c6\u30c3\u30d7\u3092\u7e70\u308a\u8fd4\u3057\u307e\u3059\u3002\u30dd\u30a4\u30f3\u30c8\u306f\u3001${\\bm x}$ \u3068 ${\\bm z}$ \u306b\u3064\u3044\u3066\u306e\u6700\u9069\u5316\u3092\u5225\u3005\u306b\u884c\u3046\u70b9\u3067\u3059\u3002\u5f0f <span>(9a) <\/span>\u306e ${\\bm x}$ \u306b\u3064\u3044\u3066\u306e\u6700\u9069\u5316\u3067\u306f\u3001 $E_{\\rm aug}$ \u5185\u306e\u968e\u6bb5\u95a2\u6570\u306b\u95a2\u3059\u308b\u9805\u306f\u95a2\u4fc2\u3057\u306a\u3044\u305f\u3081 <span>QUBO <\/span>\u5f62\u5f0f\u306b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u672c\u8ad6\u6587\u3067\u306f\u5f0f <span>(9a) <\/span>\u306e ${\\bm x}$ \u306b\u3064\u3044\u3066\u306e\u6700\u9069\u5316\u306b <span>D-Wave 2000Q <\/span>\u3092\u7528\u3044\u3066\u3044\u307e\u3059\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E6%96%B0%E3%81%97%E3%81%84%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0\"><\/span>\u65b0\u3057\u3044\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u672c\u8ad6\u6587\u3067\u306f\u3001<span>ADMM <\/span>\u3068 <span>QA <\/span>\u3092\u7d44\u307f\u5408\u308f\u305b\u305f\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u63d0\u6848\u3057\u3066\u3044\u307e\u3059\u3002\u901a\u5e38\u306e <span>ADMM <\/span>\u3068\u306e\u4e3b\u306a\u9055\u3044\u306f\u3001\u4e0a\u8ff0\u306e\u901a\u308a\u3001\u5f0f <span>(9a) <\/span>\u3092\u89e3\u304f\u305f\u3081\u306b <span>D-Wave 2000Q <\/span>\u3092\u7528\u3044\u308b\u70b9\u3067\u3059\u3002<\/p>\n<p>\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u306f\u3001$E_{\\rm aug}({\\bm x},{\\bm z},{\\bm \\lambda})$ \u306b\u5bfe\u3057\u3066 <span>QA <\/span>\u3092\u9069\u7528\u3057\u3066\u591a\u6570\u306e\u30b5\u30f3\u30d7\u30eb $\\{ {\\bm x}_{\\nu} \\}$ ( $\\nu$ \u306f\u5404\u30b5\u30f3\u30d7\u30eb\u306e\u30a4\u30f3\u30c7\u30c3\u30af\u30b9 <span>) <\/span>\u3092\u53d6\u5f97\u3057\u307e\u3059\u3002\u305d\u3057\u3066\u3001$E_{\\rm aug}({\\bm x},{\\bm z},{\\bm \\lambda})$ \u3092\u6700\u5c0f\u5316\u3059\u308b\u89e3 ${\\bm x}^{\\ast}_{\\rm cost}$ \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<p>$$ {\\bm x}^{\\ast}_{\\rm cost}= \\underset{ \\{ {\\bm x}_{\\nu} \\} } {\\rm{argmin}} \\ \\ E_{\\rm aug}({\\bm x} = {\\bm x}_{\\nu},{\\bm z},{\\bm \\lambda}) \\tag{10}$$<\/p>\n<p>\u3059\u306a\u308f\u3061\u3001\u30b5\u30f3\u30d7\u30eb $\\{ {\\bm x}_{\\nu} \\}$ \u306e\u4e2d\u3067 $E_{\\rm aug}$ \u3092\u6700\u5c0f\u306b\u3059\u308b\u3082\u306e\u3092 ${\\bm x}^{\\ast}_{\\rm cost}$ \u3068\u304a\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u3001${\\bm x}^{\\ast}_{\\rm cost}$ \u306f\u5fc5\u305a\u3057\u3082\u5b9f\u884c\u53ef\u80fd\u89e3\u3067\u3042\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u305f\u3060\u3057\u3001$\\{ {\\bm x}_{\\nu} \\}$ \u306e\u4ed6\u306e\u30b5\u30f3\u30d7\u30eb\u306f\u5b9f\u884c\u53ef\u80fd\u89e3\u3068\u3057\u307e\u3059\u3002\u307e\u305f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9f\u884c\u53ef\u80fd\u89e3 ( \u4e0d\u7b49\u5f0f\u5236\u7d04\u3092\u5168\u3066\u6e80\u305f\u3059\u89e3 ) \u306e\u4e2d\u3067 $f({\\bm x})$ \u3092\u6700\u5c0f\u5316\u3059\u308b\u89e3 ${\\bm x}^{\\ast}_{\\rm feas}$ \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<p>$$ {\\bm x}^{\\ast}_{\\rm feas} = \\underset{ \\{ {\\bm x}_{\\nu} \\} } {\\rm{argmin}}\\ \\\u00a0 f({\\bm x})\u00a0 \\ \\ {\\rm s.t.} \\ \\ {\\bm G}^{\\top}_{m} {\\bm x} \\leq D_m\u00a0 \\quad(m=1,\\cdots, M) \\tag{11}$$<\/p>\n<p>${\\bm x}^{\\ast}_{\\rm cost}$ \u3068 ${\\bm x}^{\\ast}_{\\rm feas}$ \u3092\u5225\u306b\u5b9a\u7fa9\u3057\u305f\u306e\u306f\u3001${\\bm z}$ \u3068 ${\\bm \\lambda}$ \u306e\u66f4\u65b0\u306b\u306f ${\\bm x}^{\\ast}_{\\rm cost}$ \u3092\u4f7f\u7528\u3057\u3001\u5b9f\u884c\u53ef\u80fd\u89e3\u306e\u63a2\u7d22\u306b\u306f ${\\bm x}^{\\ast}_{\\rm feas}$ \u3092\u4f7f\u7528\u3059\u308b\u305f\u3081\u3067\u3059\u3002<\/p>\n<p>\u4ee5\u4e0b\u306b\u3001\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u8a73\u7d30\u3092\u8a18\u8f09\u3057\u307e\u3059\u3002\u306a\u304a\u3001\u672c\u8ad6\u6587\u3067\u306f <span>QUBO <\/span>\u3092\u89e3\u304f\u306e\u306b D-Wave 2000Q \u3092\u7528\u3044\u3066\u3044\u307e\u3059\u304c\u3001\u4ed6\u306e <span>QUBO <\/span>\u30bd\u30eb\u30d0\u3092\u9069\u7528\u3059\u308b\u3053\u3068\u3082\u53ef\u80fd\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001<span>D-Wave 2000Q <\/span>\u3088\u308a\u591a\u304f\u306e\u5909\u6570\u3092\u6271\u3046\u3053\u3068\u304c\u3067\u304d\u308b <span>QUBO <\/span>\u30bd\u30eb\u30d0\u3092\u7528\u3044\u308c\u3070\u3001\u3055\u3089\u306b\u30b5\u30a4\u30ba\u306e\u5927\u304d\u306a\u4e0d\u7b49\u5f0f\u5236\u7d04\u4ed8\u304d\u306e\u6700\u9069\u5316\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<h4><span class=\"ez-toc-section\" id=\"_%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0%E3%81%AE%E6%89%8B%E9%A0%86\"><\/span><strong><span style=\"font-size: 20px;\">[ \u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u624b\u9806 ]<\/span><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<ol>\n<li>\n<p>\u00a0\u5404\u30d1\u30e9\u30e1\u30fc\u30bf\u3092 $\\{ z_m \\}=0$, $\\{ \\lambda_m \\}=0$, $t=1$ \u3068\u521d\u671f\u5316\u3059\u308b\u3002<\/p>\n<\/li>\n<li>\n<p>\u00a0\u30b5\u30a4\u30ba $N$ \u306e\u5b8c\u5168\u30b0\u30e9\u30d5\u3092 <span>embedding <\/span>\u3059\u308b\u3002\u203b $N$ \u306f\u5909\u6570 ${\\bm x}$ \u306e\u8981\u7d20\u6570 ( ${\\bm x} \\in \\{ 0, 1 \\}^N$ )\u3002<\/p>\n<\/li>\n<li>\u00a0\u5f0f <span>(8) <\/span>\u3092\u7528\u3044\u3066 <span>QUBO <\/span>\u884c\u5217\u3092\u8a08\u7b97\u3059\u308b\u3002<\/li>\n<li>\u00a0\u8a08\u7b97\u3057\u305f <span>QUBO <\/span>\u884c\u5217\u3092 <span>D-Wave 2000Q <\/span>\u306b\u9069\u7528\u3057\u3001\u30b5\u30f3\u30d7\u30eb $\\{ {\\bm x}_{\\nu} \\}$ \u3092\u53d6\u5f97\u3059\u308b\u3002<\/li>\n<li>\n<p>\u00a0\u30b5\u30f3\u30d7\u30eb $\\{ {\\bm x}_{\\nu} \\}$ \u3092\u7528\u3044\u3066\u3001\u5f0f <span>(10) <\/span>\u3068\u5f0f <span>(11) <\/span>\u304b\u3089 ${\\bm x}^{\\ast}_{\\rm cost}$ \u3068 ${\\bm x}^{\\ast}_{\\rm feas}$ \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n<\/li>\n<li>\n<p>\u00a0 $z_{m}^{\\ast} = \\min \\ (0, {\\bm G}^{\\top}_m {\\bm x}^{\\ast}_{\\rm cost} -D_m) \\quad (m=1,\\cdots,M)$ \u3068\u3057\u3066 ${\\bm z}^{\\ast}$ \u3092\u66f4\u65b0\u3059\u308b\u3002<\/p>\n<\/li>\n<li>\n<p>\u00a0 $\\lambda_m = \\lambda_m + \\rho ({\\bm G}^{\\top}_m {\\bm x}^{\\ast}_{\\rm cost} -D_m-z_m^{\\ast})\u00a0 \\quad (m=1,\\cdots,M)$ \u3068\u3057\u3066 ${\\bm \\lambda}$ \u3092\u66f4\u65b0\u3059\u308b\u3002<\/p>\n<\/li>\n<li>\n<p>\u00a0\u53ce\u675f\u3057\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u3092\u78ba\u8a8d\u3059\u308b<span> ( <\/span>\u4ee5\u4e0b\u306e\u57fa\u6e96\u306e\u3044\u305a\u308c\u304b\u3092\u6e80\u305f\u3057\u305f\u6642\u306b\u8a08\u7b97\u3092\u7d42\u4e86\u3059\u308b <span>) <\/span>\u3002<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li style=\"text-align: left;\">\n<p>\u00a0$t&gt;t_{\\rm max}$<\/p>\n<\/li>\n<li>\u00a0$E_{\\rm ineq}({\\bm x}_{\\rm feas}^{\\ast})$ \u304c $t_{\\rm conv}$ \u56de\u306e\u30b9\u30c6\u30c3\u30d7\u3067\u6539\u5584\u3055\u308c\u306a\u3044<\/li>\n<li>\u00a0$\\sqrt{\\sum_{m}({\\bm G}^{\\top}_m {\\bm x}_{\\rm feas} -D_m-z_m)^2} &lt; \\epsilon \\qquad$ \u203b $ t_{\\rm max}, t_{\\rm conv}, \\epsilon$ \u306f\u4e88\u3081\u8a2d\u5b9a\u3057\u3066\u304a\u304f\u30d1\u30e9\u30e1\u30fc\u30bf\u3002<\/li>\n<\/ol>\n<\/li>\n<li>$t \\leftarrow t+1$<\/li>\n<li>\u624b\u9806 3. ~ 9. \u3092\u53ce\u675f\u3059\u308b\u307e\u3067\u7e70\u308a\u8fd4\u3059\u3002<\/li>\n<\/ol>\n<p>\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u304a\u3044\u3066\u3001\u624b\u9806 <span>3. ~ 5. <\/span>\u304c\u5f0f <span>(9a) <\/span>\u306e ${\\bm x}$ \u306b\u3064\u3044\u3066\u306e\u6700\u9069\u5316\u3001\u624b\u9806 <span>6. <\/span>\u304c\u5f0f <span>(9b) <\/span>\u306e ${\\bm z}$ \u306b\u3064\u3044\u3066\u306e\u6700\u9069\u5316\u3001\u624b\u9806 <span>7. <\/span>\u304c\u5f0f <span>(9c) <\/span>\u306e ${\\bm \\lambda}$\u00a0 \u306e\u66f4\u65b0\u306b\u3042\u305f\u308a\u307e\u3059\u3002\u624b\u9806 <span>3. ~ 9. <\/span>\u306e\u53cd\u5fa9\u306b\u304a\u3044\u3066\u6bce\u56de <span>embedding <\/span>\u3059\u308b\u3053\u3068\u3092\u907f\u3051\u308b\u305f\u3081\u3001\u624b\u9806 <span>2. <\/span>\u3067\u306f\u4e88\u3081\u554f\u984c\u30b5\u30a4\u30ba $N$ \u306e\u5b8c\u5168\u30b0\u30e9\u30d5 ( \u3059\u306a\u308f\u3061\u5168\u7d50\u5408\u30b0\u30e9\u30d5 ) \u3092 <span>embedding <\/span>\u3057\u3066\u3044\u307e\u3059\u3002\u4e00\u65b9\u3067\u3001<span>unembedding <\/span>\u306f\u624b\u9806 <span>4. <\/span>\u3067\u53cd\u5fa9\u3054\u3068\u306b\u884c\u308f\u308c\u307e\u3059 ( e<span>mbedding <\/span>\u3068 <span>unembedding <\/span>\u306b\u95a2\u3057\u3066\u306f <strong>9.1.1. \u88dc\u8db31 : embedding\u3068unembedding <\/strong>\u3067\u8aac\u660e\u3057\u3066\u3044\u307e\u3059 )\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E5%AE%9F%E9%A8%93\"><\/span>\u5b9f\u9a13<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"%E5%AE%9F%E9%A8%93%E6%96%B9%E6%B3%95\"><\/span>\u5b9f\u9a13\u65b9\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h4><span class=\"ez-toc-section\" id=\"QKP\"><\/span>QKP<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>\u672c\u8ad6\u6587\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b <span>2 <\/span>\u6b21\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c ( <span>QKP : Quadratic Knapsack Problem\u00a0 ) <\/span>\u3092\u7528\u3044\u3066\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u6027\u80fd\u3092\u691c\u8a3c\u3057\u307e\u3059\u3002\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u3067\u306f\u3001\u91cd\u91cf\u3068\u4fa1\u5024\u3092\u6301\u3064\u8377\u7269\u3092\u3001\u5bb9\u91cf\u5236\u9650\u306e\u3042\u308b\u4e00\u3064\u306e\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u5165\u308c\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u5165\u308c\u305f\u8377\u7269\u306e\u4fa1\u5024\u304c\u6700\u5927\u306b\u306a\u308b\u3088\u3046\u306a\u8377\u7269\u306e\u5165\u308c\u65b9\u3092\u6c42\u3081\u307e\u3059\u3002\u3059\u306a\u308f\u3061\u3001\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306e\u5bb9\u91cf\u5236\u9650\u3092\u4e0d\u7b49\u5f0f\u5236\u7d04\u3068\u3057\u3066\u542b\u3080\u6700\u5927\u5316\u554f\u984c\u3067\u3059\u3002<\/p>\n<p>$$ \\begin{split} \\underset{\\bm{x}} {\\rm{maximize}} \\quad\u00a0 &amp;{\\bm x}^{\\top} P {\\bm x} \\\\ {\\rm subject \\ to} \\quad &amp;{\\bm w}^{\\top} {\\bm x} \\leq c\\\\ \\end{split} \\tag{7} $$<\/p>\n<p>\u3053\u3053\u3067\u3001$P=\\{ p_{i, j} \\} \\in \\mathbb{Z}_{+}^{N \\times N}$ ( $\\mathbb{Z}_{+}$ \u306f\u6b63\u306e\u6574\u6570\u306e\u96c6\u5408<span>) <\/span>\u306f\u8377\u7269\u306e\u4fa1\u5024\u3092\u8868\u3059\u884c\u5217\u3001 ${\\bm w} = \\{ w_i \\} \\in \\mathbb{Z}_{+}^{N}$ \u306f\u8377\u7269\u306e\u91cd\u91cf\u3092\u8868\u3059\u30d9\u30af\u30c8\u30eb\u3001 $c \\in \\mathbb{Z}$ \u306f\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306e\u5bb9\u91cf\u3092\u8868\u3059\u30b9\u30ab\u30e9\u30fc\u3067\u3059\u3002\u5909\u6570 ${\\bm x}=\\{ x_i\\} \\\u00a0 (i=1,\\cdots, N)$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<\/p>\n<p>$$ \\begin{equation} \\left \\{\\begin{array}{l}x_i = 0\u3000{\\rm if} \\quad i \\ \u756a\u76ee\u306e\u8377\u7269\u3092\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u5165\u308c\u306a\u3044 \\\\x_i = 1\u3000{\\rm if}\\quad\u00a0 i \\ \u756a\u76ee\u306e\u8377\u7269\u3092\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u5165\u308c\u308b\u00a0 \\end{array}\\right.\\end{equation} $$<\/p>\n<p>\u884c\u5217 $P$ \u306e\u975e\u5bfe\u89d2\u8981\u7d20 $p_{i, j} \\ (i \\ne j)$ \u306f\u3001\u8377\u7269 $i$ \u3068\u8377\u7269 $j$ \u304c\u4e21\u65b9\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u5165\u308c\u3089\u308c\u305f\u5834\u5408\u306b\u306e\u307f\u8ab2\u3055\u308c\u308b\u5831\u916c\u3067\u3059 ( $x_i$ \u307e\u305f\u306f $x_j$ \u304c\u30bc\u30ed\u306e\u3068\u304d $x_i P_{i,j} x_j =0$ \u3068\u306a\u308b\u305f\u3081\uff09\u3002\u306a\u304a\u3001<span>D-Wave 2000Q <\/span>\u3067\u306f\u6700\u5c0f\u5316\u554f\u984c\u3092\u6271\u3046\u306e\u3067\u3001<span>QUBO <\/span>\u5f62\u5f0f\u306b\u3059\u308b\u969b\u306b\u306f\u3001\u76ee\u7684\u95a2\u6570\u306e\u7b26\u53f7\u3092\u53cd\u8ee2\u3055\u305b\u3066 $f({\\bm x}) = &#8211; {\\bm x}^{\\top} P {\\bm x}$ \u3068\u3057\u307e\u3059\u3002\u307e\u305f\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04\u306b\u3064\u3044\u3066\u306f <strong>5. \u65b9\u6cd5<\/strong> \u3067\u8ff0\u3079\u305f\u624b\u6cd5\u3067\u7de9\u548c\u3057\u307e\u3059\u3002<\/p>\n<p>\u672c\u5b9f\u9a13\u3067\u306f\u3001 $P$ \u3068 ${\\bm w}$ \u3092\u30e9\u30f3\u30c0\u30e0\u306b\u751f\u6210\u3057\u307e\u3059\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u30da\u30a2 $(i, j)$ \u306e\u4fa1\u5024 $p_{i, j}$ \u3092\u78ba\u7387 $(1-\\Delta)$ \u3067 <span>0 \u3068\u3057<\/span>\u3001\u78ba\u7387 $\\Delta$ \u3067 <span>1 ~ <\/span><span>100 <\/span>\u306e\u4e00\u69d8\u5206\u5e03\u304b\u3089\u9078\u3073\u307e\u3059\u3002\u3064\u307e\u308a\u3001 $\\Delta$ \u304c <span>1 <\/span>\u306b\u8fd1\u3044\u3068\u304d\u3001QUBO \u884c\u5217\u306f\u975e\u30bc\u30ed\u8981\u7d20\u304c\u591a\u304f\u306a\u308a\u3001$\\Delta$ \u304c <span>0 <\/span>\u306b\u8fd1\u3044\u3068\u304d\u3001QUBO \u884c\u5217\u306f\u975e\u30bc\u30ed\u8981\u7d20\u304c\u5c11\u306a\u304f\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u5909\u6570\u540c\u58eb\u306e\u76f8\u4e92\u4f5c\u7528\u3092\u30b0\u30e9\u30d5\u3067\u8868\u3059\u3068\u3001$\\Delta \\approx 1$ <span>\u306e\u3068\u304d\u305d\u306e\u30b0\u30e9\u30d5\u306f\u5bc6\u3068\u306a\u308a\u3001$\\Delta \\approx 0$ \u306e\u3068\u304d\u758e\u3068\u306a\u308a\u307e\u3059\u3002<\/span>\u307e\u305f\u3001\u91cd\u307f $\\{w_i\\}$ \u306f $[1, 50]$ \u304b\u3089\u30e9\u30f3\u30c0\u30e0\u306b\u9078\u3073\u3001\u5bb9\u91cf $c$ \u306f $[50, \\sum_i w_i]$ \u306e\u4e00\u69d8\u5206\u5e03\u304b\u3089\u9078\u3073\u307e\u3059\u3002<\/p>\n<h4><span class=\"ez-toc-section\" id=\"MAPE\"><\/span>MAPE<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>\u65b0\u3057\u3044\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u5178\u578b\u7684\u306a\u6027\u80fd\u3092\u691c\u8a3c\u3059\u308b\u305f\u3081\u306b\u3001<span>10 <\/span>\u500b\u306e\u30a4\u30f3\u30b9\u30bf\u30f3\u30b9\u3092\u751f\u6210\u3057\u307e\u3059\u3002\u30a4\u30f3\u30b9\u30bf\u30f3\u30b9\u3068\u306f\u3001\u4e0a\u8ff0\u306e\u65b9\u6cd5\u3067\u30e9\u30f3\u30c0\u30e0\u306b\u751f\u6210\u3057\u305f\u4e00\u3064\u3072\u3068\u3064\u306e <span>QKP <\/span>\u306e\u3053\u3068\u3067\u3059\u3002\u305d\u3057\u3066\u3001\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u7cbe\u5ea6\u3092\u691c\u8a3c\u3059\u308b\u305f\u3081\u306b\u3001\u4ee5\u4e0b\u306e\u5e73\u5747\u7d76\u5bfe\u8aa4\u5dee ( <span>MAPE : Mean Absolute Percentage error ) <\/span>\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<p>$$ {\\rm MAPE} = \\frac{1}{N_{\\rm inst}} \\sum_{k=1}^{N_{\\rm inst}} \\frac{\u00a0 |f_k ({\\bm x}_{\\rm opt}^{\\ast}) &#8211; f_k ({\\bm x}_{\\rm feas}^{\\ast})|\u00a0 \u00a0}{f_k ({\\bm x}_{\\rm opt}^{\\ast})}\u00a0 \\tag{12}$$<\/p>\n<p>\u3053\u3053\u3067\u3001$f_k({\\bm x})$ \u306f $k$ \u756a\u76ee\u306e\u30a4\u30f3\u30b9\u30bf\u30f3\u30b9\u306e\u76ee\u7684\u95a2\u6570\u3067\u3042\u308a\u3001$N_{\\rm inst}$ \u306f\u30a4\u30f3\u30b9\u30bf\u30f3\u30b9\u306e\u6570<span> ( \u3059\u306a\u308f\u3061\u3001<\/span>\u3053\u3053\u3067\u306f $N_{\\rm inst}=10$ ) \u3067\u3059\u3002\u307e\u305f\u3001${\\bm x}_{\\rm opt}^{\\ast}$ \u306f <span>Gurobi <\/span>\u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6 ( \u6700\u9069\u5316\u554f\u984c\u3092\u53b3\u5bc6\u306b\u89e3\u304f\u30bd\u30eb\u30d0 ) \u3067\u5f97\u3089\u308c\u305f\u6700\u9069\u89e3\u3001${\\bm x}_{\\rm feas}^{\\ast}$ \u306f\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u5f97\u3089\u308c\u305f\u5b9f\u884c\u53ef\u80fd\u89e3\u3067\u3059\u3002${\\rm MAPE} = 0$ \u306e\u3068\u304d\u3001\u5168\u3066\u306e\u30a4\u30f3\u30b9\u30bf\u30f3\u30b9\u3067\u6700\u9069\u89e3\u304c\u5f97\u3089\u308c\u305f\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u307e\u305f\u3001<span>D-Wave 2000Q <\/span>\u306e\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u6642\u9593\u306f $20 \\ {\\rm \\mu s}$ \u306b\u8a2d\u5b9a\u3057\u3001\u30b5\u30f3\u30d7\u30eb\u306f <span>2000 <\/span>\u500b\u751f\u6210\u3057\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u4e88\u3081\u6c7a\u3081\u3066\u304a\u304f\u30d1\u30e9\u30e1\u30fc\u30bf\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a2d\u5b9a\u3057\u307e\u3059\u3002<\/p>\n<p>$$ \\rho = 0.1 \\tag{13a} $$<\/p>\n<p>$$ t_{\\rm max}= 30 \\tag{13b} $$<\/p>\n<p>$$ t_{\\rm conv} = 10 \\tag{13c} $$<\/p>\n<p>$$ \\epsilon = 10^{-3} \\tag{13d} $$<\/p>\n<p>&nbsp;<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E5%AE%9F%E9%A8%93%E7%B5%90%E6%9E%9C\"><\/span>\u5b9f\u9a13\u7d50\u679c<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h4><span class=\"ez-toc-section\" id=\"QKP_%E3%81%AB%E5%AF%BE%E3%81%99%E3%82%8B%E8%A7%A3%E3%81%AE%E7%B2%BE%E5%BA%A6%E3%81%AE%E6%A4%9C%E8%A8%BC\"><\/span>QKP \u306b\u5bfe\u3059\u308b\u89e3\u306e\u7cbe\u5ea6\u306e\u691c\u8a3c<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>QKP \u306b\u5bfe\u3057\u3066 ADMM \u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u5f97\u3089\u308c\u305f\u89e3\u306e\u7cbe\u5ea6\u3092 MAPE \u306b\u3088\u3063\u3066\u8a55\u4fa1\u3057\u305f\u5b9f\u9a13\u306e\u7d50\u679c\u3092\u793a\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u56f3 <span>2 <\/span>\u306b\u3001$\\Delta=0.2, 0.6, 1.0$ \u306e\u3068\u304d\u306e <span>MAPE <\/span>\u306e $N$ &#8211; \u4f9d\u5b58\u6027\u3092\u793a\u3057\u307e\u3059\u3002<span>DW(opt) <\/span>\u306f <strong>4. \u80cc\u666f \u6700\u9069\u5316\u30e2\u30fc\u30c9\u3068\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9 <\/strong>\u3067\u7d39\u4ecb\u3057\u305f <span>D-Wave 2000Q <\/span>\u306e\u6700\u9069\u5316\u30e2\u30fc\u30c9\u3092\u4f7f\u7528\u3057\u305f\u5834\u5408\u3067\u3001<span>DW(<\/span>\u03b2 <span>= \u2026) <\/span>\u306f\u9006\u6e29\u5ea6 $\\beta$ \u304c <span>\u2026 <\/span>\u306e\u3068\u304d\u306e\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9\u3092\u4f7f\u7528\u3057\u305f\u5834\u5408\u3067\u3059\u3002<\/p>\n<div id=\"attachment_5984\" style=\"width: 826px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5984\" src=\"\/T-Wave\/wp-content\/uploads\/2023\/02\/ad2e4928cfa3869ab8d967a120143718.png\" alt=\"\" width=\"816\" height=\"654\" class=\"wp-image-5984 size-full\" \/><p id=\"caption-attachment-5984\" class=\"wp-caption-text\">\u56f32. MAPE \u306e $N$ &#8211; \u4f9d\u5b58\u6027 ( <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2012.06119\">https:\/\/doi.org\/10.48550\/arXiv.2012.06119<\/a> )\u3002<\/p><\/div>\n<p>\u56f3 <span>2 <\/span>\u304b\u3089\u3001$N=64$ \u307e\u3067\u306e\u5168\u3066\u306e\u30a4\u30f3\u30b9\u30bf\u30f3\u30b9\u3067\u5b9f\u884c\u53ef\u80fd\u89e3\u304c\u5f97\u3089\u308c\u3066\u3044\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u4f7f\u7528\u3057\u305f\u5834\u5408\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u306e\u8ffd\u52a0\u306b\u3088\u308a\u3001<span>D-Wave 2000Q <\/span>\u3067\u306f $N=64$ \u306e\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f <span>D-Wave 2000Q <\/span>\u306b\u304a\u3044\u3066\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u7528\u3044\u308b\u4ee5\u524d\u306e\u624b\u6cd5\u3088\u308a\u3082\u5927\u304d\u306a\u30b5\u30a4\u30ba\u306e\u554f\u984c\u3092\u6271\u3048\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n<p>\u307e\u305f\u3001$\\Delta=0.2$ \u306e\u3068\u304d\u3001$N$ \u304c\u5897\u52a0\u3059\u308b\u3068 <span>4 <\/span>\u3064\u306e\u30e2\u30fc\u30c9\u5168\u3066\u306b\u304a\u3044\u3066 <span>MAPE <\/span>\u304c\u5897\u52a0\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002\u4e00\u65b9\u3001$\\Delta=0.2$ \u3067\u306f\u3001<span>DW(\u03b2 = 0.1) <\/span>\u3092\u9664\u304d\u3001$N$ \u304c\u5897\u52a0\u3057\u3066\u3082 <span>MAPE <\/span>\u306f\u307b\u307c\u30bc\u30ed\u306b\u8fd1\u3044\u5024\u3092\u7dad\u6301\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001<span>QKP <\/span>\u306e\u30b0\u30e9\u30d5\u304c\u5bc6\u3067\u3042\u308b\u307b\u3069\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u3063\u3066\u5b9f\u884c\u53ef\u80fd\u89e3\u3092\u6b63\u78ba\u306b\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>4 \u3064\u306e\u30e2\u30fc\u30c9\u3092\u6bd4\u8f03\u3059\u308b\u3068\u3001<span>DW(opt) <\/span>\u304a\u3088\u3073 <span>DW(\u03b2 = 10.0) <\/span>\u304c $\\Delta$ \u306e\u5024\u306b\u95a2\u308f\u3089\u305a\u6b63\u78ba\u306a\u89e3\u3092\u51fa\u3057\u3066\u3044\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001<span>QKP <\/span>\u3067\u306f <span>DW(opt) <\/span>\u3082\u3057\u304f\u306f <span>DW(\u03b2 = 10.0) <\/span>\u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u304c\u826f\u3044\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002\u307e\u305f\u3001<span>DW(\u03b2 = 0.1)<\/span> \u3067\u306f\u4f8b\u5916\u7684\u306b <span>MAPE <\/span>\u304c\u5927\u304d\u304f\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304b\u3089\u3001\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9\u306b\u304a\u3051\u308b \u03b2 \u306e\u8abf\u6574\u306f\u91cd\u8981\u3067\u3042\u308b\u3068\u8a00\u3048\u307e\u3059\u3002<\/p>\n<h4><span class=\"ez-toc-section\" id=\"%E8%A8%88%E7%AE%97%E6%99%82%E9%96%93%E3%81%AE%E6%A4%9C%E8%A8%BC\"><\/span>\u8a08\u7b97\u6642\u9593\u306e\u691c\u8a3c<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068<span> Gurobi <\/span>\u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6\u306e\u8a08\u7b97\u6642\u9593\u3092\u6bd4\u8f03\u3057\u307e\u3059\u3002\u6bd4\u8f03\u5bfe\u8c61\u306e\u6642\u9593\u3092\u305d\u308c\u305e\u308c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>$t_{\\rm QA}$ : D-Wave 2000Q \u306b\u642d\u8f09\u3055\u308c\u305f\u91cf\u5b50\u30c7\u30d0\u30a4\u30b9\u3078\u306e\u7dcf\u30a2\u30af\u30bb\u30b9\u6642\u9593 ( \u3059\u306a\u308f\u3061 <span>QA <\/span>\u306e\u5408\u8a08\u6642\u9593 )<\/li>\n<li>$t_{\\rm sampling}$ : \u30a4\u30f3\u30bf\u30fc\u30cd\u30c3\u30c8\u306e\u9045\u5ef6\u3084 $t_{\\rm QA}$\u3001<span>D-Wave 2000Q <\/span>\u3067\u306e\u4ed6\u306e\u51e6\u7406\u3092\u542b\u3080\u7dcf\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u6642\u9593<\/li>\n<li>$t_{\\rm unemb}$ : unembedding ( \u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u624b\u9806 <span>4. ) <\/span>\u306b\u304b\u304b\u3063\u305f\u5408\u8a08\u6642\u9593<\/li>\n<li>$t_{\\rm ADMM}$ : \u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u308b\u7dcf\u8a08\u7b97\u6642\u9593 ( \u3059\u306a\u308f\u3061 $t_{\\rm QA}$\u3001$t_{\\rm sampling}$\u3001$t_{\\rm unemb}$ \u306a\u3069\u306e\u51e6\u7406\u306e\u7dcf\u548c )<\/li>\n<li>$t_{\\rm Gurobi}$ : Gurobi \u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6\u306b\u3088\u308b\u7dcf\u8a08\u7b97\u6642\u9593<\/li>\n<\/ul>\n<p>\u305d\u308c\u3067\u306f\u3001$\\Delta=0.2, 0.6, 1.0$ \u306b\u304a\u3051\u308b\u5404\u8a08\u7b97\u6642\u9593\u306e $N$ &#8211; \u4f9d\u5b58\u6027\u3092\u4ee5\u4e0b\u306e\u56f3 <span>3 <\/span>\u306b\u793a\u3057\u307e\u3059\u3002<\/p>\n<div id=\"attachment_5992\" style=\"width: 675px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5992\" src=\"\/T-Wave\/wp-content\/uploads\/2023\/02\/2effa90e82c79a5bb1b665949beba0cd-1024x833.png\" alt=\"\" width=\"665\" height=\"541\" class=\"wp-image-5992 size-large\" \/><p id=\"caption-attachment-5992\" class=\"wp-caption-text\">\u56f33. $\\Delta=0.2, 0.6, 1.0$ \u306b\u304a\u3051\u308b\u8a08\u7b97\u6642\u9593\u306e $N$ &#8211; \u4f9d\u5b58\u6027 ( <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2012.06119\">https:\/\/doi.org\/10.48550\/arXiv.2012.06119<\/a> )\u3002<\/p><\/div>\n<p>\u8d64\u4e38 ( $t_{\\rm ADMM}$ ) \u3068\u9752\u4e38 ( $t_{\\rm Gurobi}$ ) \u306f\u3001\u305d\u308c\u305e\u308c <span>DW(\u03b2 = 10.0) <\/span>\u3068 <span>Gurobi <\/span>\u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6\u3092\u7528\u3044\u3066\u5f97\u3089\u308c\u305f\u30a4\u30f3\u30b9\u30bf\u30f3\u30b9\u5e73\u5747\u306e\u8a08\u7b97\u6642\u9593\u3067\u3059\u3002\u3053\u306e\u56f3\u304b\u3089\u3001$ \\Delta = 0.2, 0.6$ \u3067\u306f\u3001<span>Gurobi <\/span>\u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6\u306f <span>ADMM <\/span>\u3088\u308a\u5927\u5e45\u306b\u9ad8\u901f\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002\u3057\u304b\u3057\u3001$t_{\\rm Gurobi}$ \u306f $\\Delta$ \u3068 $N$ \u306e\u5897\u52a0\u306b\u4f34\u3044\u5287\u7684\u306b\u5897\u52a0\u3057\u3001$\\Delta=1.0$\u3001$N=64$ \u3067\u306f $t_{\\rm ADMM} &lt; t_{\\rm Gurobi}$ \u3068\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f $\\Delta$ \u304a\u3088\u3073 $N$ \u306e\u5897\u52a0\u306b\u4f34\u3044 <span>Gurobi <\/span>\u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6\u3088\u308a\u3082\u9ad8\u901f\u306b\u306a\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>\u307e\u305f\u3001$t_{\\rm sampling}$ \u304a\u3088\u3073 $t_{\\rm unemb}$ \u306f \u304c\u5927\u304d\u304f\u306a\u308b\u3068\u5897\u52a0\u3057\u307e\u3059\u304c\u3001$t_{\\rm QA}$ \u306f\u307b\u307c\u4e00\u5b9a\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001$t_{\\rm sampling}$ \u3084 $t_{\\rm unemb}$ \u306e\u3088\u3046\u306a\u8a08\u7b97\u30aa\u30fc\u30d0\u30fc\u30d8\u30c3\u30c9\u3092\u524a\u6e1b\u3059\u308c\u3070 $t_{\\rm ADMM}$ \u306f\u3088\u308a\u9ad8\u901f\u306b\u306a\u308a\u5f97\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u3088\u308a\u30b5\u30a4\u30ba\u304c\u5927\u304d\u304f\u5bc6\u306a\u30b0\u30e9\u30d5\u3092\u5b9f\u88c5\u3057\u305f\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u30de\u30b7\u30f3\u304c\u958b\u767a\u3055\u308c\u3001\u3082\u306f\u3084 <span>embedding <\/span>\u3084 <span>unembedding \u304c\u5fc5\u8981\u7121\u304f<\/span>\u306a\u308c\u3070\u3001\u672c\u624b\u6cd5\u306f\u9ad8\u901f\u5316\u3059\u308b\u3068\u4e88\u60f3\u3055\u308c\u307e\u3059\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E7%B5%90%E8%AB%96\"><\/span>\u7d50\u8ad6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u672c\u8ad6\u6587\u3067\u306f\u3001<span>D-Wave 2000Q <\/span>\u3092\u7528\u3044\u305f\u4e0d\u7b49\u5f0f\u5236\u7d04\u4ed8\u304d\u4e8c\u5024\u6700\u9069\u5316\u554f\u984c\u306e\u65b0\u3057\u3044\u89e3\u6cd5\u3092\u63d0\u6848\u3057\u307e\u3057\u305f\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u62e1\u5f35\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u6cd5\u3092\u7528\u3044\u3066\u65b0\u305f\u306a\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u5b9a\u7fa9\u3057\u3001<span>QA <\/span>\u3068 <span>ADMM <\/span>\u3092\u7d44\u307f\u5408\u308f\u305b\u305f\u30cf\u30a4\u30d6\u30ea\u30c3\u30c9\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u958b\u767a\u3057\u307e\u3057\u305f\u3002<\/p>\n<p>\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u6027\u80fd\u3092 <span>QKP <\/span>\u306b\u5bfe\u3057\u3066\u691c\u8a3c\u3057\u305f\u7d50\u679c\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u7528\u3044\u308b\u5f93\u6765\u306e\u624b\u6cd5\u3067\u306f\u8a08\u7b97\u3067\u304d\u306a\u3044\u5927\u898f\u6a21\u306a\u554f\u984c\u306b\u5bfe\u3057\u3066\u3001\u5b9f\u884c\u53ef\u80fd\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u305d\u3057\u3066\u3001\u554f\u984c\u306e\u30b0\u30e9\u30d5\u304c\u5bc6\u3067\u3042\u308b\u307b\u3069\u3001\u3088\u308a\u6b63\u78ba\u306a\u5b9f\u884c\u53ef\u80fd\u89e3\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u307e\u305f\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u9069\u3057\u305f\u5f8c\u51e6\u7406\u306f\u3001\u6700\u9069\u5316\u30e2\u30fc\u30c9\u3042\u308b\u3044\u306f $\\beta = 10.0$ \u306e\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30e2\u30fc\u30c9\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002\u3055\u3089\u306b\u3001\u8a08\u7b97\u6642\u9593\u306b\u3064\u3044\u3066\u691c\u8a3c\u3057\u305f\u7d50\u679c\u3001<span>QKP <\/span>\u304c\u30b5\u30a4\u30ba\u306e\u5927\u304d\u306a\u5bc6\u30b0\u30e9\u30d5\u3067\u4e0e\u3048\u3089\u308c\u308b\u5834\u5408\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f <span>Gurobi <\/span>\u30aa\u30d7\u30c6\u30a3\u30de\u30a4\u30b6\u3088\u308a\u3082\u9ad8\u901f\u306b\u8a08\u7b97\u3067\u304d\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002\u4e00\u822c\u306b\u3001\u758e\u30b0\u30e9\u30d5\u306b\u6bd4\u3079\u3066\u5bc6\u30b0\u30e9\u30d5\u306e\u65b9\u304c\u89e3\u304f\u3053\u3068\u304c\u96e3\u3057\u3044\u305f\u3081\u3001\u5bc6\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u6b63\u78ba\u306a\u89e3\u3092\u9ad8\u901f\u306b\u5f97\u3089\u308c\u308b\u70b9\u3082\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u9577\u6240\u3067\u3042\u308b\u3068\u8a00\u3048\u307e\u3059\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E3%81%82%E3%81%A8%E6%9B%B8%E3%81%8D\"><\/span>\u3042\u3068\u66f8\u304d<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u30b9\u30e9\u30c3\u30af\u5909\u6570\u306a\u3057\u3067\u4e0d\u7b49\u5f0f\u5236\u7d04\u3092\u7de9\u548c\u3059\u308b\u305f\u3081\u306b\u968e\u6bb5\u95a2\u6570\u3092\u4f7f\u3044\u3001\u3055\u3089\u306b\u968e\u6bb5\u95a2\u6570\u304c <span>QUBO <\/span>\u5f62\u5f0f\u306b\u3067\u304d\u306a\u3044\u554f\u984c\u3092 <span>ADMM <\/span>\u3067\u89e3\u6d88\u3057\u3066\u3044\u308b\u70b9\u304c\u5de7\u5999\u3060\u306a\u3068\u611f\u3058\u307e\u3057\u305f\u3002<\/p>\n<p>\u672c\u8ad6\u6587\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u3063\u3066\u3001\u30b9\u30e9\u30c3\u30af\u5909\u6570\u3092\u7528\u3044\u308b\u5f93\u6765\u624b\u6cd5\u3067\u306f\u89e3\u3051\u306a\u304b\u3063\u305f\u554f\u984c\u30b5\u30a4\u30ba\u306b\u5bfe\u5fdc\u3067\u304d\u308b\u3053\u3068\u306f\u5206\u304b\u308a\u307e\u3057\u305f\u304c\u3001\u5f93\u6765\u624b\u6cd5\u3067\u89e3\u3051\u308b\u554f\u984c\u30b5\u30a4\u30ba\u306b\u304a\u3044\u3066\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068\u89e3\u306e\u7cbe\u5ea6\u3092\u691c\u8a3c\u3057\u305f\u5834\u5408\u306b\u306f\u3001\u3069\u306e\u3088\u3046\u306a\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b\u304b\u304c\u6c17\u306b\u306a\u308a\u307e\u3057\u305f\u3002<\/p>\n<p>\u307e\u305f\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f <span>D-Wave 2000Q \u304c\u78ba\u7387\u7684<\/span>\u306b\u89e3\u3092\u51fa\u529b\u3059\u308b\u6027\u8cea\u3092\u7528\u3044\u3066\u3044\u307e\u3059\u304c\u3001\u30ea\u30d0\u30fc\u30b9\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0<span>[4]<\/span>\u3092\u7528\u3044\u3066\u524d\u306e\u53cd\u5fa9\u3067\u306e\u89e3\u3092\u521d\u671f\u89e3\u3068\u3057\u305f\u5834\u5408\u306b\u3001\u672c\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u308b\u89e3\u306e\u7cbe\u5ea6\u3084\u53ce\u675f\u306e\u901f\u3055\u306b\u3069\u306e\u3088\u3046\u306a\u9055\u3044\u304c\u73fe\u308c\u308b\u306e\u304b\u306b\u3064\u3044\u3066\u8208\u5473\u3092\u6301\u3061\u307e\u3057\u305f\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E8%A3%9C%E8%B6%B3\"><\/span>\u88dc\u8db3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"%E8%A3%9C%E8%B6%B3_1_embedding_%E3%81%A8_unembedding\"><\/span>\u88dc\u8db3 1 : embedding \u3068 unembedding<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u3053\u3053\u3067\u306f\u3001<span>D-Wave <\/span>\u30de\u30b7\u30f3\u306b\u3088\u308b\u6700\u9069\u5316\u3067\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u3066\u3044\u308b\u00a0<span>embedding <\/span>\u3068 <span>unembedding\u00a0 <\/span>\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n<h4><span class=\"ez-toc-section\" id=\"%E5%AF%86%E3%82%B0%E3%83%A9%E3%83%95%E3%81%A8%E7%96%8E%E3%82%B0%E3%83%A9%E3%83%95\"><\/span>\u5bc6\u30b0\u30e9\u30d5\u3068\u758e\u30b0\u30e9\u30d5<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><span>D-Wave <\/span>\u30de\u30b7\u30f3\u306f\u3001<span>QUBO \u3042\u308b\u3044<\/span>\u306f\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306e\u6700\u5c0f\u5316\u554f\u984c\u3092\u8fd1\u4f3c\u7684\u306b\u89e3\u304d\u307e\u3059\u3002<span>QUBO <\/span>\u3068\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306e\u6700\u5c0f\u5316\u554f\u984c\u306f\u3001<span>QUBO <\/span>\u306e\u4e8c\u5024\u5909\u6570 $x_i \\in \\{ 0, 1\\}$ \u3068\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306e\u30b9\u30d4\u30f3\u5909\u6570\u00a0 $s_i \\in \\{ -1, +1\\}$ \u306b\u3064\u3044\u3066 $s_i = 2 x_i -1$ \u3068\u3044\u3046\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u3053\u3068\u306b\u3088\u308a\u3001\u7b49\u4fa1\u306a\u6700\u9069\u5316\u554f\u984c\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059<span>[1]<\/span>\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4ee5\u964d\u306f\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u3092\u4e2d\u5fc3\u306b\u8aac\u660e\u3092\u9032\u3081\u307e\u3059\u3002\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306f\u3001\u4ee5\u4e0b\u306e\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n<p>$$\u00a0 \\frac{1}{2} \\sum_{i \\ne j} J_{ij} s_i s_j + \\sum_{i=1}^{N} h_i s_i $$<\/p>\n<p>\u3053\u306e\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306f\u3001\u30b9\u30d4\u30f3\u5909\u6570 $s_i$ \u3092\u30ce\u30fc\u30c9\u3001\u76f8\u4e92\u4f5c\u7528 $J_{ij}$ \u3092\u30a8\u30c3\u30b8\u306e\u91cd\u307f\u3001\u78c1\u5834 $h_i$ \u3092\u30ce\u30fc\u30c9\u306e\u91cd\u307f\u3068\u3059\u308b\u30b0\u30e9\u30d5\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<span>D-Wave <\/span>\u30de\u30b7\u30f3\u4e0a\u3067\u6700\u9069\u5316\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306b\u306f\u3001\u3053\u306e\u30b0\u30e9\u30d5\u3092 <span>D-Wave <\/span>\u30de\u30b7\u30f3\u4e0a\u306b\u5b9f\u88c5\u3055\u308c\u3066\u3044\u308b\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u306b\u5f53\u3066\u306f\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u4e0a\u3067\u306f\u3001\u7269\u7406\u91cf\u5b50\u30d3\u30c3\u30c8\u304c\u30ce\u30fc\u30c9\u306b\u3042\u305f\u308a\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u91cf\u5b50\u30d3\u30c3\u30c8\u540c\u58eb\u304c\u56de\u8def\u4e0a\u3067\u7d50\u7dda\u3055\u308c\u305f\u6240\u304c\u30a8\u30c3\u30b8\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u5168\u3066\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u304c\u4e92\u3044\u306b\u7d50\u5408\u3057\u3066\u3044\u308b\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u7269\u7406\u7684\u306a\u56de\u8def\u3068\u3057\u3066\u5b9f\u88c5\u3059\u308b\u4ee5\u4e0a\u3001\u91cf\u5b50\u30d3\u30c3\u30c8\u540c\u58eb\u3092\u7d50\u7dda\u3067\u304d\u308b\u306e\u306f\u3001\u4e92\u3044\u306b\u8fd1\u63a5\u3057\u3066\u3044\u308b\u7b87\u6240\u306b\u9650\u3089\u308c\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u30ce\u30fc\u30c9\u540c\u58eb\u306e\u7d50\u5408\u304c\u5c11\u306a\u3044\u30b0\u30e9\u30d5\u306e\u3053\u3068\u3092\u300c\u758e\u300d\u30b0\u30e9\u30d5\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n<p>\u4e00\u65b9\u3067\u3001\u30ce\u30fc\u30c9\u540c\u58eb\u306e\u7d50\u5408\u304c\u591a\u3044\u30b0\u30e9\u30d5\u306e\u3053\u3068\u3092\u300c\u5bc6\u300d\u30b0\u30e9\u30d5\u3068\u547c\u3073\u307e\u3059\u3002\u6700\u3082\u5bc6\u306a\u30b0\u30e9\u30d5\u306f\u3001\u5168\u3066\u306e\u30ce\u30fc\u30c9\u540c\u58eb\u304c\u30a8\u30c3\u30b8\u3067\u3064\u306a\u304c\u3063\u3066\u3044\u308b\u30b0\u30e9\u30d5\u3067\u3059\u3002\u305d\u306e\u3088\u3046\u306a\u30b0\u30e9\u30d5\u306f\u300c\u5b8c\u5168\u30b0\u30e9\u30d5\u300d\u3068\u547c\u3070\u308c\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u56f3 <span>4 <\/span>\u306b\u3001\u758e\u30b0\u30e9\u30d5\u3068\u5bc6\u30b0\u30e9\u30d5 ( \u5b8c\u5168\u30b0\u30e9\u30d5 ) \u306e\u4f8b\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n<div id=\"attachment_6000\" style=\"width: 675px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-6000\" src=\"\/T-Wave\/wp-content\/uploads\/2023\/02\/7a498be583c420c7d4a8d7e71b8e0b43-1024x430.png\" alt=\"\" width=\"665\" height=\"279\" class=\"wp-image-6000 size-large\" \/><p id=\"caption-attachment-6000\" class=\"wp-caption-text\">\u56f34. (a) \u758e\u30b0\u30e9\u30d5\u3068 (b) \u5bc6\u30b0\u30e9\u30d5 (\u5b8c\u5168\u30b0\u30e9\u30d5) \u306e\u4f8b\u3002<\/p><\/div>\n<h4><span class=\"ez-toc-section\" id=\"embedding\"><\/span>embedding<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>\u89e3\u304d\u305f\u3044\u554f\u984c\u306b\u5bfe\u5fdc\u3059\u308b\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u304c\u758e\u30b0\u30e9\u30d5\u3067\u3042\u308c\u3070\u3001\u305d\u306e\u307e\u307e\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u306b\u5165\u529b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u304c\u3001\u5bc6\u30b0\u30e9\u30d5\u3067\u3042\u3063\u305f\u5834\u5408\u306b\u306f\u3001\u4f55\u3089\u304b\u306e\u5de5\u592b\u3092\u3057\u3066\u758e\u306a\u7d50\u5408\u306e\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u306b\u57cb\u3081\u8fbc\u3080\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<span>D-Wave<\/span>\u30de\u30b7\u30f3\u306b\u306f\u3001\u3053\u306e\u300c\u57cb\u3081\u8fbc\u307f\u300d\u3092\u5b9f\u73fe\u3059\u308b\u6280\u8853\u304c\u5099\u308f\u3063\u3066\u304a\u308a\u3001<span>embedding<\/span>\u3068\u547c\u3070\u308c\u3066\u3044\u307e\u3059\u3002\u3059\u306a\u308f\u3061\u3001<span>embedding <\/span>\u306f\u300c\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306e\u30b9\u30d4\u30f3\u5909\u6570 \u2192 \u91cf\u5b50\u30d3\u30c3\u30c8 \u3078\u306e\u7f6e\u63db\u300d\u3092\u884c\u3046\u51e6\u7406\u306b\u3042\u305f\u308a\u307e\u3059\u3002<\/p>\n<p>embedding \u306e\u4e00\u4f8b\u3092\u793a\u3057\u307e\u3059\u3002\u89e3\u304d\u305f\u3044\u554f\u984c\u306b\u5bfe\u5fdc\u3059\u308b\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u304c\u4e0a\u56f3 <span>4(b) <\/span>\u306e\u30b0\u30e9\u30d5 ( <span>5 <\/span>\u30ce\u30fc\u30c9\u306e\u5b8c\u5168\u30b0\u30e9\u30d5 ) \u3067\u4e0e\u3048\u3089\u308c\u3001\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u304c\u4ee5\u4e0b\u306e\u56f3 <span>6 <\/span>\u3067\u4e0e\u3048\u3089\u308c\u3066\u3044\u305f\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u30b0\u30e9\u30d5\u306f\u30ad\u30e1\u30e9\u30b0\u30e9\u30d5\u3068\u547c\u3070\u308c\u3001<span>D-Wave 2000Q <\/span>\u306e\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u306e\u5358\u4f4d\u69cb\u9020 ( \u30e6\u30cb\u30c3\u30c8\u30fb\u30bb\u30eb ) \u3068\u3057\u3066\u63a1\u7528\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n<div id=\"attachment_6001\" style=\"width: 310px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-6001\" src=\"\/T-Wave\/wp-content\/uploads\/2023\/02\/022e1d1d32ae047015f90ca2ea8e9367-300x283.png\" alt=\"\" width=\"300\" height=\"283\" class=\"wp-image-6001 size-medium\" \/><p id=\"caption-attachment-6001\" class=\"wp-caption-text\">\u56f35. D-Wave \u30de\u30b7\u30f3\u4e0a\u306e\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u306e\u4e00\u4f8b (\u30ad\u30e1\u30e9\u30b0\u30e9\u30d5)\u3002<\/p><\/div>\n<p>\u3057\u304b\u3057\u3001\u56f3 <span>4(b) <\/span>\u306e\u30b0\u30e9\u30d5\u306f\u5168\u7d50\u5408\u30b0\u30e9\u30d5\u306a\u306e\u306b\u5bfe\u3057\u3001\u56f3 <span>5 <\/span>\u306e\u30ad\u30e1\u30e9\u30b0\u30e9\u30d5\u306f\u758e\u30b0\u30e9\u30d5\u3067\u3042\u308b\u305f\u3081\u3001\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306e\u30ce\u30fc\u30c9 ( \u30b9\u30d4\u30f3\u5909\u6570 ) \u3092\u30ad\u30e1\u30e9\u30b0\u30e9\u30d5\u306e\u30ce\u30fc\u30c9 ( \u91cf\u5b50\u30d3\u30c3\u30c8 ) \u306b\u4e00\u5bfe\u4e00\u3067\u5272\u308a\u5f53\u3066\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u3053\u306e\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b\u3001<span>embedding <\/span>\u3067\u306f 1 \u3064\u306e\u30b9\u30d4\u30f3\u5909\u6570\u3092\u8907\u6570\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u5272\u308a\u5f53\u3066\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u305d\u308c\u3089\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306f\u540c\u4e00\u306e\u30b9\u30d4\u30f3\u5909\u6570\u3092\u8868\u3059\u305f\u3081\u3001\u540c\u3058\u5411\u304d\u3092\u53d6\u308b\u3088\u3046\u306b\u5236\u5fa1\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u3001\u91cf\u5b50\u30d3\u30c3\u30c8\u304b\u3089\u306a\u308b\u30d6\u30ed\u30c3\u30af\u3092\u30c1\u30a7\u30fc\u30f3 ( <span>Chain ) <\/span>\u3068\u547c\u3073\u307e\u3059\u3002\u30c1\u30a7\u30fc\u30f3\u306e\u5c0e\u5165\u306b\u3088\u308a\u3001<span>5 <\/span>\u30ce\u30fc\u30c9\u306e\u5b8c\u5168\u30b0\u30e9\u30d5\u306f\u30ad\u30e1\u30e9\u30b0\u30e9\u30d5\u4e0a\u306b\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u57cb\u3081\u8fbc\u3080\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<div id=\"attachment_6002\" style=\"width: 675px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-6002\" src=\"\/T-Wave\/wp-content\/uploads\/2023\/02\/d1a0727a90aee040cbd3002d5ef2c20f-1024x605.png\" alt=\"\" width=\"665\" height=\"393\" class=\"wp-image-6002 size-large\" \/><p id=\"caption-attachment-6002\" class=\"wp-caption-text\">\u56f36. \u30c1\u30a7\u30fc\u30f3\u3092\u7528\u3044\u305f\u57cb\u3081\u8fbc\u307f\u306e\u4e00\u4f8b ( \u70b9\u7dda\u306f\u4f7f\u308f\u308c\u3066\u3044\u306a\u3044\u30a8\u30c3\u30b8 )\u3002<\/p><\/div>\n<p>\u56f3 <span>6 <\/span>\u306b\u793a\u3057\u305f\u57cb\u3081\u8fbc\u307f\u3067\u306f\u3001\u30ce\u30fc\u30c9 <span>1, 2, 5 <\/span>\u3092\u30c1\u30a7\u30fc\u30f3\u306b\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001<span>5 <\/span>\u7a2e\u985e\u306e\u30ce\u30fc\u30c9\u5168\u3066\u3092\u4ed6\u306e <span>4 <\/span>\u7a2e\u985e\u306e\u30ce\u30fc\u30c9\u3068\u7d50\u5408\u3055\u305b\u3001\u64ec\u4f3c\u7684\u306b\u5168\u7d50\u5408\u3092\u518d\u73fe\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<span>[3]<\/span><\/p>\n<h4><span class=\"ez-toc-section\" id=\"unembedding\"><\/span>unembedding<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3092\u5b9f\u884c\u3057\u305f\u5f8c\u3001\u30cf\u30fc\u30c9\u30a6\u30a7\u30a2\u30b0\u30e9\u30d5\u4e0a\u306e\u5404\u91cf\u5b50\u30d3\u30c3\u30c8\u306f <span>+1 <\/span>\u304b -1 \u306b\u5bfe\u5fdc\u3059\u308b\u3044\u305a\u308c\u304b\u306e\u72b6\u614b\u3092\u53d6\u308a\u307e\u3059\u304c\u3001\u305d\u308c\u3092\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306e\u30b9\u30d4\u30f3\u5909\u6570\u306b\u623b\u3059\u3053\u3068\u306b\u3088\u3063\u3066\u3001\u89e3\u304d\u305f\u3044\u6700\u9069\u5316\u554f\u984c\u306e\u89e3\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3053\u306e\u300c\u91cf\u5b50\u30d3\u30c3\u30c8 \u2192 \u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u306e\u30b9\u30d4\u30f3\u5909\u6570\u3078\u306e\u7f6e\u63db\u300d\u3068\u3044\u3046 <span>embedding <\/span>\u3068\u9006\u306e\u51e6\u7406\u3092 <span>unembedding <\/span>\u3068\u547c\u3073\u307e\u3059\u3002<span>unembedding <\/span>\u306b\u306f\u3001\u30c1\u30a7\u30fc\u30f3\u306b\u304a\u3044\u3066\u672c\u6765\u306f\u540c\u3058\u5024\u3092\u53d6\u308b\u3079\u304d\u91cf\u5b50\u30d3\u30c3\u30c8\u304c\u7570\u306a\u308b\u5024\u3092\u53d6\u3063\u3066\u3057\u307e\u3063\u305f\u5834\u5408 ( \u3053\u308c\u3092\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30fc\u30af ( <span>chain break ) <\/span>\u3068\u8a00\u3044\u307e\u3059 ) \u306a\u3069\u306b\u3001\u3069\u3061\u3089\u306e\u5024\u3092\u63a1\u7528\u3059\u308b\u304b\u3092\u6c7a\u5b9a\u3059\u308b\u51e6\u7406\u304c\u542b\u307e\u308c\u307e\u3059\u3002\u30c7\u30d5\u30a9\u30eb\u30c8\u3067\u306f\u3001\u30c1\u30a7\u30fc\u30f3\u306e\u4e2d\u3067\u3001\u305d\u306e\u5411\u304d\u3092\u53d6\u3063\u3066\u3044\u308b\u91cf\u5b50\u30d3\u30c3\u30c8\u304c\u591a\u3044\u65b9\u3092\u63a1\u7528\u3059\u308b\u300c\u591a\u6570\u6c7a\u6cd5\u300d\u304c\u8a2d\u5b9a\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u307e\u305f\u3001\u4ed6\u306b\u306f\u300c\u30a8\u30cd\u30eb\u30ae\u30fc\u6700\u5c0f\u5316\u6cd5\u300d\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u65b9\u6cd5\u3067\u306f\u3001\u5c40\u6240\u7684\u306a\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u3001\u3088\u308a\u4f4e\u3044\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u30b5\u30f3\u30d7\u30eb\u304c\u5f97\u3089\u308c\u308b\u3088\u3046\u306b\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E8%A3%9C%E8%B6%B3_2_%E3%82%B9%E3%83%A9%E3%83%83%E3%82%AF%E5%A4%89%E6%95%B0%E3%81%AE%E4%BA%8C%E5%80%A4%E5%A4%89%E6%95%B0%E3%81%AB%E3%82%88%E3%82%8B%E5%B1%95%E9%96%8B\"><\/span>\u88dc\u8db3 2 : \u30b9\u30e9\u30c3\u30af\u5909\u6570\u306e\u4e8c\u5024\u5909\u6570\u306b\u3088\u308b\u5c55\u958b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>3. \u8ab2\u984c <\/strong>\u306b\u3066\u3001\u5f0f <span>(4) <\/span>\u306e\u30b9\u30e9\u30c3\u30af\u5909\u6570 $s_m$ \u306f $s_m= 2^{0} \\times y_1 + 2^1 \\times y_2 + 2^2 + y_3 + \\cdots $ \u3068\u3044\u3046\u3088\u3046\u306b\u88dc\u52a9\u7684\u306a\u4e8c\u5024\u5909\u6570 $y_1,\u00a0 y_2, y_3, \\cdots$ \u3067\u8868\u73fe\u3057\u76f4\u3059\u5fc5\u8981\u304c\u3042\u308b\u3068\u8ff0\u3079\u307e\u3057\u305f\u3002\u3053\u306e\u3053\u3068\u3092\u7c21\u5358\u306a\u4f8b\u3067\u78ba\u8a8d\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<p>\u4eca\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04\u306e\u6570\u304c\u4e00\u3064 ( \u3059\u306a\u308f\u3061 $M=1$ ) \u3067\u3042\u308b\u3068\u3057\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04 $ {\\bm G}^{\\top} _m {\\bm x} \\leq D_m \\ (m=1,\\cdots,M)$ \u306e\u6dfb\u3048\u5b57 $m$ \u306f\u7701\u7565\u3057\u307e\u3059\u3002\u305d\u3057\u3066\u3001$N=3$\u3001${\\bm G}\u00a0= [1,2,2]^{\\top}$\u3001${\\bm x}=[x_1, x_2, x_3]^{\\top}$\u3001$D=3$ \u3067\u3042\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u4e0d\u7b49\u5f0f\u5236\u7d04 ${\\bm G}^{\\top} {\\bm x} \\leq D$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>$$ x_1 + 2 x_2 + 2 x_3 \\leq 3 \\tag{*} $$<\/p>\n<p>\u3053\u306e\u4e0d\u7b49\u5f0f\u3092\u5f0f <span>(4) <\/span>\u306e\u7f70\u91d1\u9805\u306b\u3057\u3066\u307f\u308b\u3068\u3001<\/p>\n<p>$$ (x_1 + 2 x_2 + 2 x_3 &#8211; 3 + s)^2 \\tag{**} $$<\/p>\n<p>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d $s$ \u304c\u3068\u308a\u5f97\u308b\u5024\u306f $\\{ 0,1,2,3\\}$ \u306b\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u4ee5\u4e0b\u306e\u8868\u304b\u3089\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002\u8868\u4e2d\u306e $s^{\\ast}$ \u306f\u3001\u5f0f <span>(**) <\/span>\u3092 0 \u306b\u3059\u308b $s$ \u306e\u5024\u3001\u3059\u306a\u308f\u3061 $s^{\\ast} = &#8211; (x_1 + 2 x_2 + 2 x_3 &#8211; 3)$ \u3067\u3059\u3002\u3053\u306e\u8868\u304b\u3089\u3001$s^{\\ast}$ \u304c\u8ca0\u306e\u3068\u304d\u306b\u306e\u307f\u4e0d\u7b49\u5f0f <span>(*) <\/span>\u306b\u53cd\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002\u5f93\u3063\u3066\u3001$s^{\\ast}$ \u304c $0,1,2,3$ \u306e\u5024\u3092\u53d6\u308c\u308b\u3088\u3046\u306b\u3059\u308c\u3070\u3001\u4e0d\u7b49\u5f0f <span>(*) <\/span>\u3092\u6e80\u305f\u3059\u6642\u306b\u306e\u307f\u7f70\u91d1\u9805 <span>(**) <\/span>\u304c 0 \u306b\u306a\u308b\u3088\u3046\u306b $s$ \u3067\u30ab\u30d0\u30fc\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<table style=\"height: 194px; width: 92.7575%; border-collapse: collapse; border-color: #000000;\">\n<tbody>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">$x_1$<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">$x_2$<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">$x_3$<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">$x_1 + 2 x_2 + 2 x_3 &#8211; 3$<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">$s^{\\ast}$<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 10px; text-align: center;\">0<\/td>\n<td style=\"width: 16.6202%; height: 10px; text-align: center;\">0<\/td>\n<td style=\"width: 18.013%; height: 10px; text-align: center;\">0<\/td>\n<td style=\"width: 26.1839%; height: 10px; text-align: center;\">-3<\/td>\n<td style=\"width: 17.8779%; height: 10px; text-align: center;\">3<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">-1<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">1<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">-1<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">1<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">-2<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">2<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">0<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">0<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 13.7418%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 16.6202%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 18.013%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 26.1839%; height: 23px; text-align: center;\">2<\/td>\n<td style=\"width: 17.8779%; height: 23px; text-align: center;\">-2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u4ee5\u4e0a\u3088\u308a\u3001\u4e0d\u7b49\u5f0f <span>(*) <\/span>\u3092\u6e80\u305f\u3059\u306b\u306f $s \\in \\{ 0,1,2,3\\}$ \u3068\u3059\u308c\u3070\u826f\u3044\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002$s$ \u306e\u3068\u308a\u5f97\u308b\u5024\u306e\u4e0a\u9650 ( \u3053\u3053\u3067\u306f 3 ) \u306f $D$ \u306b\u5bfe\u5fdc\u3059\u308b\u5024\u3067\u3042\u308b\u305f\u3081\u3001\u4e00\u822c\u306b\u306f $s \\in \\{ 0,1,\\cdots,D \\}$ \u3068\u3057\u307e\u3059\u3002<\/p>\n<p>\u3055\u3066\u3001\u3053\u3053\u3067\u306f $s \\in \\{ 0,1,2,3\\}$ \u3068\u3059\u308c\u3070\u826f\u3044\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u304c\u3001<span>QUBO <\/span>\u554f\u984c\u306f\u300c\u4e8c\u5024\u300d\u6700\u9069\u5316\u554f\u984c\u3067\u3042\u308b\u305f\u3081\u3001$s$ \u3092\u4e8c\u5024\u5909\u6570\u3067\u8868\u3057\u76f4\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u65b0\u305f\u306a\u5909\u6570 $y_1,y_2 \\in \\{\u00a0 0, 1\\}$ \u3092\u7528\u3044\u3066 $s=2^0 \\times y_1 + 2^1 \\times y_2$ \u3068\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001$s$ \u306f\u3061\u3087\u3046\u3069 $0,1,2,3$ \u306e\u5024\u3092\u3068\u308c\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 115px;\">\n<tbody>\n<tr style=\"height: 23px;\">\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">$y_1$<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">$y_2$<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">$s$<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">0<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">2<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">0<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">1<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">1<\/td>\n<td style=\"width: 33.3333%; height: 23px; text-align: center;\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u307e\u305f\u3001$D=2$ \u306e\u6642\u306b\u306f\u3001\u5148\u307b\u3069\u306e\u8b70\u8ad6\u304b\u3089 $s \\in \\{ 0,1,2 \\}$ \u3068\u3059\u308c\u3070\u826f\u3044\u308f\u3051\u3067\u3059\u304c\u3001\u4e8c\u5024\u5909\u6570 $ y_1, y_2 \\in \\{ 0, 1\\}$ \u3067 $s_m=2^0 \\times y_1 + 2^1 \\times y_2$ \u3068\u3059\u308b\u3068\u3001\u4e0a\u306e\u8868\u304c\u793a\u3059\u3088\u3046\u306b $y_1=y_2=1$ \u306e\u3068\u304d\u306b $s=3$ \u3092\u3068\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u5834\u5408\u306b\u306f $y_1=y_2=1$ \u306b\u306a\u3089\u306a\u3044\u3088\u3046\u306b $y_1 \\times y_2$ \u3068\u3044\u3046\u7f70\u91d1\u9805\u3092\u3001\u9069\u5207\u306a\u7f70\u91d1\u4fc2\u6570\u3092\u639b\u3051\u305f\u4e0a\u3067\u76ee\u7684\u95a2\u6570\u306b\u8db3\u3059\u3053\u3068\u3067\u5bfe\u51e6\u3055\u308c\u307e\u3059\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE\"><\/span>\u53c2\u8003\u6587\u732e<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol>\n<li>\n<p><span>T-Wave<\/span>\u7528\u8a9e\u96c6 \u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u3068<span>QUBO : <a href=\"\/T-Wave\/?p=2325\">\/T-Wave\/?p=2325<\/a><\/span><\/p>\n<\/li>\n<li>\n<p><span>D-Wave <\/span><span>\u30c9\u30ad\u30e5\u30e1\u30f3\u30c8 \u5f8c\u51e6\u7406\u30e2\u30fc\u30c9 D-Wave\u30b7\u30b9\u30c6\u30e0\u306e\u5f8c\u51e6\u7406\u65b9\u6cd5 : <\/span><\/p>\n<p><span><a href=\"https:\/\/dwavejapan.com\/app\/uploads\/2020\/02\/09-1105A-H_J-DeveloperGuidePostProcessing.pdf\">https:\/\/dwavejapan.com\/app\/uploads\/2020\/02\/09-1105A-H_J-DeveloperGuidePostProcessing.pdf<\/a><\/span><\/p>\n<\/li>\n<li>\n<p><span>NTT<\/span><span>\u30c7\u30fc\u30bf\u91cf\u5b50\u30b3\u30f3\u30d4\u30e5\u30fc\u30c6\u30a3\u30f3\u30b0\u30ac\u30a4\u30c9\u30e9\u30a4\u30f3\u3000\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u30de\u30b7\u30f3\/\u30a4\u30b8\u30f3\u30b0\u30de\u30b7\u30f3\u7de8 2021\u5e74 1\u6708\uff1a<\/span><\/p>\n<span>\u00a0<\/span><a href=\"https:\/\/www.nttdata.com\/jp\/ja\/-\/media\/nttdatajapan\/files\/news\/services_info\/2021\/012800\/012800-01.pdf\">https:\/\/www.nttdata.com\/jp\/ja\/-\/media\/nttdatajapan\/files\/news\/services_info\/2021\/012800\/012800-01.pdf<\/a><span>\u3000<\/span><\/li>\n<li>\n<p><span>T-Wave<\/span>\u7528\u8a9e\u96c6 <span>\u30ea\u30d0\u30fc\u30b9\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\uff1a<\/span><span><a href=\"\/T-Wave\/?p=2306\">\/T-Wave\/?p=2306<\/a> <\/span><\/p>\n<\/li>\n<\/ol>\n\n\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u9069\u5316\u554f\u984c\u306e\u4e2d\u306b\u306f\u3001\u4e0d\u7b49\u5f0f\u306e\u5f62\u3067\u8868\u3055\u308c\u308b\u5236\u7d04 ( = \u4e0d\u7b49\u5f0f\u5236\u7d04 ) \u3092\u542b\u3080\u554f\u984c\u304c\u591a\u304f\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u969b\u306b\u5f93\u6765\u306e\u624b\u6cd5\u3067\u306f\u88dc\u52a9\u5909\u6570\u304c\u7528\u3044\u307e\u3059\u304c\u3001\u591a\u6570\u306e\u7269\u7406\u91cf\u5b50\u30d3\u30c3\u30c8\u304c\u8ffd\u52a0\u3067\u5fc5\u8981\u3068\u306a\u308b\u305f\u3081\u554f\u984c\u306e\u898f\u6a21\u304c\u5236\u9650\u3055\u308c\u3066\u3057\u307e\u3044\u307e\u3059\u3002<br \/>\n\u305d\u3053\u3067\u3001\u672c\u8ad6\u6587\u3067\u306f\u3001\u4ea4\u4e92\u65b9\u5411\u4e57\u6570\u6cd5 ( ADMM ) \u3068\u3044\u3046\u65e2\u5b58\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3092\u7d44\u307f\u5408\u308f\u305b\u305f\u65b0\u305f\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u63d0\u6848\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u88dc\u52a9\u5909\u6570\u3092\u7528\u3044\u306a\u3044\u305f\u3081\u5f93\u6765\u3088\u308a\u3082\u5927\u898f\u6a21\u306a\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u53ef\u80fd\u3068\u306a\u308a\u307e\u3059\u3002<br \/>\n2 \u6b21\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c ( QKP ) 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