{"id":2325,"date":"2022-02-06T16:39:15","date_gmt":"2022-02-06T07:39:15","guid":{"rendered":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=2325"},"modified":"2022-02-06T16:39:15","modified_gmt":"2022-02-06T07:39:15","slug":"%e3%82%a4%e3%82%b8%e3%83%b3%e3%82%b0%e6%a8%a1%e5%9e%8b%e3%81%a8qubo","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2022\/02\/06\/%e3%82%a4%e3%82%b8%e3%83%b3%e3%82%b0%e6%a8%a1%e5%9e%8b%e3%81%a8qubo\/","title":{"rendered":"\u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u3068QUBO"},"content":{"rendered":"<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\/2022\/02\/06\/%e3%82%a4%e3%82%b8%e3%83%b3%e3%82%b0%e6%a8%a1%e5%9e%8b%e3%81%a8qubo\/#%E3%82%A4%E3%82%B8%E3%83%B3%E3%82%B0%E6%A8%A1%E5%9E%8B\" >\u30a4\u30b8\u30f3\u30b0\u6a21\u578b<\/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\/2022\/02\/06\/%e3%82%a4%e3%82%b8%e3%83%b3%e3%82%b0%e6%a8%a1%e5%9e%8b%e3%81%a8qubo\/#QUBO\" >QUBO<\/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\/2022\/02\/06\/%e3%82%a4%e3%82%b8%e3%83%b3%e3%82%b0%e6%a8%a1%e5%9e%8b%e3%81%a8qubo\/#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE\" >\u53c2\u8003\u6587\u732e<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"%E3%82%A4%E3%82%B8%E3%83%B3%E3%82%B0%E6%A8%A1%E5%9E%8B\"><\/span>\u30a4\u30b8\u30f3\u30b0\u6a21\u578b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u30a4\u30b8\u30f3\u30b0\u6a21\u578b<span style=\"font-weight: 400;\">\u306f\u3001\u6b21\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/span><\/p>\n<p>\\begin{equation}<br \/>H(\\mathbf{\\sigma}) ~:=~ &#8211; \\sum_{i&lt;j} J_{ij} \\sigma _i \\sigma _j ~\u00a0 &#8211; \\sum_{i}^{N} h_i \\sigma _i<br \/>\\end{equation}<\/p>\n<p><span style=\"font-weight: 400;\">[latex]\\sigma_{i} \\in\\{-1,+1\\}[\/latex]\u306f\u30a4\u30b8\u30f3\u30b0\u5909\u6570\u3067\u3059\u3002[latex]J_{ij}[\/latex]\u306f2\u3064\u306e\u30a4\u30b8\u30f3\u30b0\u5909\u6570\u9593\u306e\u76f8\u4e92\u4f5c\u7528\u30d1\u30e9\u30e1\u30fc\u30bf\u3001[latex]h_i[\/latex]\u306f\u5c40\u6240\u78c1\u5834\u3092\u8868\u3059\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3059\u3002\u76f8\u4e92\u4f5c\u7528\u306e\u4fc2\u6570\u304c[latex]J_{ij}&gt;0[\/latex]<\/span><span style=\"font-weight: 400;\">\u306e\u3068\u304d\u3001\u3053\u308c\u3092\u5f37\u78c1\u6027\u7684\u76f8\u4e92\u4f5c\u7528\u3068\u8a00\u3044\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30b9\u30d4\u30f3\u304c\u305d\u308c\u305e\u308c\u540c\u3058\u65b9\u5411\u3092\u5411\u304f\u3068\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u4f4e\u304f\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u3001[latex]J_{ij}&lt;0[\/latex]<\/span><span style=\"font-weight: 400;\">\u306e\u3068\u304d\u3001\u3053\u308c\u3092\u53cd\u5f37\u78c1\u6027\u7684\u76f8\u4e92\u4f5c\u7528\u3068\u8a00\u3044\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30b9\u30d4\u30f3\u304c\u305d\u308c\u305e\u308c\u53cd\u5bfe\u65b9\u5411\u3092\u5411\u304f\u3068\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u4f4e\u304f\u306a\u308a\u307e\u3059\u3002<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"QUBO\"><\/span>QUBO<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>QUBO (Quadratic Unconstrained Binary Optimization) \u3067\u306f\u3001\u6b21\u306e\u30b3\u30b9\u30c8\u95a2\u6570\u3092\u6700\u5c0f\u5316\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>f(\\mathbf{x}) ~:=~&amp; \\sum_{i}\u00a0 \\sum_{j} Q_{ij} x _i x _j \\\\<br \/>~=~&amp; \\mathbf{x}^T Q \\mathbf{x}<br \/>\\end{align}<\/p>\n<p><span style=\"font-weight: 400;\">[latex]x_{i} \\in\\{0,1\\}[\/latex]\u306f\u4e8c\u5024\u5909\u6570\u3067\u3059\u3002[latex]Q[\/latex]\u3092QUBO\u884c\u5217\u3068\u547c\u3073\u307e\u3059\u3002<\/span><span style=\"font-weight: 400;\">[latex]\\sigma=2x- 1[\/latex]\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u3068\u3001\u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u3068QUBO\u306f\u7b49\u4fa1\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/span><\/p>\n<p>\u6709\u540d\u306a\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u591a\u304f\u306f\u3001\u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u3084QUBO\u3092\u7528\u3044\u3066\u5b9a\u5f0f\u5316\u304c\u53ef\u80fd\u3067\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u307e\u3059 <a href=\"https:\/\/doi.org\/10.3389\/fphy.2014.00005\">[Lucas, 2014]<\/a>\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 class=\"csl-left-margin\">[Lucas, 2014] Lucas, A. Ising formulations of many NP problems. <i>Front. Physics<\/i>\u00a0<b>2<\/b>, 2014, <a href=\"https:\/\/doi.org\/10.3389\/fphy.2014.00005\">10.3389\/fphy.2014.00005<\/a>.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u30a4\u30b8\u30f3\u30b0\u6a21\u578b \u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u306f\u3001\u6b21\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3067\u8868\u3055\u308c\u307e\u3059\u3002 \\begin{equation}H(\\mathbf{\\sigma}) ~:=~ &#8211; \\sum_{i&lt;j} J_{ij} \\sigma _i [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-2325","post","type-post","status-publish","format-standard","hentry","category-glossary"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u3068QUBO - T-QARD Harbor<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2022\/02\/06\/\u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u3068qubo\/\" \/>\n<meta property=\"og:locale\" content=\"ja_JP\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u30a4\u30b8\u30f3\u30b0\u6a21\u578b\u3068QUBO - 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