<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Two-qubit_Clifford_circuits" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Two-qubit Clifford circuits</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@The_Gottesman-Knill_theorem"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="html" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@The_Gottesman-Knill_theorem" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><big class="xxlarge">The Gottesman-Knill theorem</big></p><p> Fault-tolerant quantum computation is made possible by computing on quantum codes. For stabilizer codes, the class of allowed gates is defined by the Clifford group. The Gottesman-Knill theorem asserts that circuits from gates in the Clifford group may be simulated efficiently on a classical computer. </p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-2qubit-clifford"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-2qubit-clifford" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-2qubit-clifford" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-2qubit-clifford-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-2qubit-clifford" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-2qubit-clifford/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="2" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-2qubit-clifford-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-2qubit-clifford-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Two-qubit Clifford circuits &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-2qubit-clifford-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; We explore some properties of quantum circuits constructed from gates in the Clifford group. These are often called simply &amp;#8220;Clifford circuits,&#34; for short, and here we are interested in two-qubit circuits. &lt;/p&gt; &lt;ol class=&#34;enumerate&#34;&gt; &lt;li value=&#34;1&#34;&gt; &lt;p&gt; Recall that the CNOT gate acts on a two-qubit Pauli group element [mathjaxinline]g\in P_2[/mathjaxinline] in a simple way, according to this table (the control qubit is on the left, and the target on the right): &lt;/p&gt; &lt;style&gt; .datagrid table { border-collapse: collapse; text-align: center; } .datagrid {font: normal 20px/150%; font-family:&#34;Lucida Console&#34;, Monaco, monospace; background: #fff; overflow: hidden; border: none; -webkit-border-radius: 3px; -moz-border-radius: 3px; border-radius: 3px; }.datagrid table td, .datagrid table th { padding: 3px 10px; }.datagrid table thead th {background:-webkit-gradient( linear, left top, left bottom, color-stop(0.05, #006699), color-stop(1, #00557F) );background:-moz-linear-gradient( center top, #006699 5%, #00557F 100% );filter:progid:DXImageTransform.Microsoft.gradient(startColorstr=&#39;#006699&#39;, endColorstr=&#39;#00557F&#39;);background-color:#006699; color:#ffffff; font-size: 25px; font-weight: bold; border-left: 1px solid #0070A8; } .datagrid table thead th:first-child { border: none; }.datagrid table tbody td { color: #00496B; border-left: 1px solid #E1EEF4;font-size: 20px;font-weight: normal; }.datagrid table tbody .alt td { background: #E1EEF4; color: #00496B; }.datagrid table tbody td:first-child { border-left: none; }.datagrid table tbody tr:last-child td { border-bottom: none; }.datagrid table tfoot td div { border-top: 1px solid #006699;background: #E1EEF4;} .datagrid table tfoot td { padding: 0; font-size: 20px } .datagrid table tfoot td div{ padding: 2px; }.datagrid table tfoot td ul { margin: 0; padding:0; list-style: none; text-align: center; }.datagrid table tfoot li { display: inline; }.datagrid table tfoot li a { text-decoration: none; display: inline-block; padding: 2px 8px; margin: 1px;color: #FFFFFF;border: 1px solid #006699;-webkit-border-radius: 3px; -moz-border-radius: 3px; border-radius: 3px; background:-webkit-gradient( linear, left top, left bottom, color-stop(0.05, #006699), color-stop(1, #00557F) );background:-moz-linear-gradient( center top, #006699 5%, #00557F 100% );filter:progid:DXImageTransform.Microsoft.gradient(startColorstr=&#39;#006699&#39;, endColorstr=&#39;#00557F&#39;);background-color:#006699; }.datagrid table tfoot ul.active, .datagrid table tfoot ul a:hover { text-decoration: none;border-color: #006699; color: #FFFFFF; background: none; background-color:#00557F;}div.dhtmlx_window_active, div.dhx_modal_cover_dv { position: fixed !important; } &lt;/style&gt; &lt;center style=&#34;boder:none; padding:none&#34;&gt; &lt;div class=&#34;datagrid&#34;&gt; &lt;table&gt; &lt;thead&gt; &lt;tr&gt; &lt;th&gt;g&lt;/th&gt; &lt;th&gt;CNOT on g&lt;/th&gt; &lt;/tr&gt; &lt;/thead&gt; &lt;tbody&gt; &lt;tr&gt; &lt;td&gt;IX&lt;/td&gt; &lt;td style=&#34;text-align:center&#34;&gt;IX&lt;/td&gt; &lt;/tr&gt; &lt;tr class=&#34;alt&#34;&gt; &lt;td&gt;IZ&lt;/td&gt; &lt;td style=&#34;text-align:center&#34;&gt;ZZ&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;XI&lt;/td&gt; &lt;td style=&#34;text-align:center&#34;&gt;XX&lt;/td&gt; &lt;/tr&gt; &lt;tr class=&#34;alt&#34;&gt; &lt;td&gt;ZI&lt;/td&gt; &lt;td style=&#34;text-align:center&#34;&gt;ZI&lt;/td&gt; &lt;/tr&gt; &lt;/tbody&gt; &lt;/table&gt; &lt;/div&gt; &lt;/center&gt; &lt;p&gt; This table can be expressed as a map, in which we say that CNOT maps the inputs &lt;tt class=&#34;tt&#34;&gt;[IX,IZ,XI,ZI]&lt;/tt&gt; to the outputs &lt;tt class=&#34;tt&#34;&gt;[IX,ZZ,XX,ZI]&lt;/tt&gt;. Following this notation, give a list which represents the result of the CNOT map acting on the inputs &lt;tt class=&#34;tt&#34;&gt;[XY,YY,ZY]&lt;/tt&gt;: &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;&lt;tt class=&#34;tt&#34;&gt;[XY,YY,ZY]&lt;/tt&gt;[mathjaxinline]\rightarrow[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-2qubit-clifford_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-2qubit-clifford_2_1&#34; id=&#34;input_s12-wk6-2qubit-clifford_2_1&#34; aria-describedby=&#34;status_s12-wk6-2qubit-clifford_2_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-2qubit-clifford_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-2qubit-clifford_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-2qubit-clifford_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;2&#34;&gt; &lt;p&gt; Consider the circuit &lt;tt class=&#34;tt&#34;&gt;[ CNOT(0,1), CNOT(1,0), CNOT(0,1) ]&lt;/tt&gt;: &lt;/p&gt; &lt;p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/a345971905ec1ca0cb75839b12e87af6/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/images_threecnot.png&#34; width=&#34;300&#34;/&gt; &lt;/center&gt; &lt;/p&gt; &lt;p&gt; Give the list which represents the output of this Clifford circuit acting on the inputs &lt;tt class=&#34;tt&#34;&gt;[IX,IZ,XI,ZI]&lt;/tt&gt;: &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;&lt;tt class=&#34;tt&#34;&gt;[IX,IZ,XI,ZI]&lt;/tt&gt;[mathjaxinline]\rightarrow[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-2qubit-clifford_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-2qubit-clifford_3_1&#34; id=&#34;input_s12-wk6-2qubit-clifford_3_1&#34; aria-describedby=&#34;status_s12-wk6-2qubit-clifford_3_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-2qubit-clifford_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-2qubit-clifford_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-2qubit-clifford_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;/ol&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Two-qubit Clifford circuits&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-2qubit-clifford&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-2qubit-clifford&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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Let [mathjaxinline]C(U)[/mathjaxinline] denote a two-qubit controlled [mathjaxinline]U[/mathjaxinline] operation, with qubit [mathjaxinline]0[/mathjaxinline] being the target and qubit [mathjaxinline]1[/mathjaxinline] being the control, which acts as &lt;/p&gt; &lt;table id=&#34;a0000000002&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34; style=&#34;width:80%; border:none&#34;&gt;[mathjax]C(U) = \left[ \begin{array}{cc}{I}&amp;amp; {0}\\ {0}&amp;amp; {U}\end{array}\right] \, ,[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.1)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; where [mathjaxinline]U[/mathjaxinline] is a single-qubit gate, and the entries in this matrix are each [mathjaxinline]2\times 2[/mathjaxinline] matrices, acting on qubit 0. Thus, for example, [mathjaxinline]C(X)[/mathjaxinline] is the CNOT gate &lt;tt class=&#34;tt&#34;&gt;CNOT(1,0)&lt;/tt&gt;. &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; Give a Clifford circuit for [mathjaxinline]C(Z)[/mathjaxinline]. Input your answer as a list of gates, eg &lt;tt class=&#34;tt&#34;&gt;[H(1),CNOT(1,0)]&lt;/tt&gt;, where the gates in the circuit are applied in the order listed, from left to right. You may use &lt;tt class=&#34;tt&#34;&gt;X&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;Y&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;Z&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;H&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;S&lt;/tt&gt;, and &lt;tt class=&#34;tt&#34;&gt;CNOT&lt;/tt&gt; gates. For &lt;tt class=&#34;tt&#34;&gt;CNOT(1,0)&lt;/tt&gt;, qubit [mathjaxinline]1[/mathjaxinline] is the control, and qubit [mathjaxinline]0[/mathjaxinline] is the target. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]C(Z) =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-2qubit-clifford2_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-2qubit-clifford2_2_1&#34; id=&#34;input_s12-wk6-2qubit-clifford2_2_1&#34; aria-describedby=&#34;status_s12-wk6-2qubit-clifford2_2_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-2qubit-clifford2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-2qubit-clifford2_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-2qubit-clifford2_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; Give a Clifford circuit for [mathjaxinline]C(Y)[/mathjaxinline]: &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]C(Y) =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-2qubit-clifford2_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-2qubit-clifford2_3_1&#34; id=&#34;input_s12-wk6-2qubit-clifford2_3_1&#34; aria-describedby=&#34;status_s12-wk6-2qubit-clifford2_3_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-2qubit-clifford2_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-2qubit-clifford2_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-2qubit-clifford2_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;/li&gt; &lt;/ol&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Two-qubit Clifford circuits II&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-2qubit-clifford2&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-2qubit-clifford2&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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String.prototype.startsWith = function (str){ return this.indexOf(str) === 0; }; } if(typeof(String.prototype.trim) === "undefined") { String.prototype.trim = function() { return String(this).replace(/^\s+|\s+$/g, ''); }; } var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var amp = String.fromCharCode(38); var rlb = rb + lb; var mke = function(x){ return lb + x + rb; } var search_module_url = ""; var get_search_module_ficus = function(){ var cid = $('div.xblock').data('course-id'); if (cid){ console.log("cid = ", cid); // search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten console.log("3. search_module_url = ", search_module_url); return; } var course_root_link = $('span.nav-item-course').find('a').attr('href'); if (course_root_link){ console.log("course_root_link = ", course_root_link); search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/"); console.log("2. search_module_url = ", search_module_url); return } console.log("cannot determine search module url"); } var get_search_module = function(){ // find search this module link if (!($('div.course-index').length)){ return get_search_module_ficus(); } $('div.course-index').find('nav').find('a').each(function(){ if ($(this).text().trim().startsWith("Search this course")){ search_module_url = $(this).attr('href'); console.log("search_module_url = ", search_module_url); } }); } var go_to_search = function(){ get_search_module(); var sterm = $('#site-search-box').val(); // new_url = jump_to_url + "/Search_this_module/?q=" + sterm; new_url = search_module_url + "?q=" + sterm; console.log("sterm = ", sterm, " ; going to ", new_url); window.location.href = new_url; } if (!$('#site-search-box').length){ $("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'" + " id='site-search-box'" + rlb + "img src='" + course_static_url + "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb); } $("#site-search-box").keypress(function(event) { if (event.which == 13) { event.preventDefault(); go_to_search(); } }); // $('#site-search-box').bind("enterKey", go_to_search); var get = function(x){ return eval(x); } return {'course_static_url': course_static_url, 'jump_to_url': jump_to_url, 'go_to_search': go_to_search, 'get_search_module': get_search_module, 'get_search_module_ficus': get_search_module_ficus, 'get': get, } } var the_site_search = add_site_search(); var add_fix_transcript = function(){ if ($('div.wrap-instructor-info').length==0){ return; } $('div.xblock-student_view-video').each(function(key, vblock_e){ var vblock = $(vblock_e); var vuid = vblock.data('usage-id').split('@'); var vid; if (vuid.length==1){ vuid = vblock.data('usage-id').split(';_') vid = vuid[5]; }else{ vid = vuid[2]; } var mfnpre = vid.split("_video",1)[0]; var mfnid = mfnpre; // no periods mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var mke = function(x){ return lb + x + rb; } var ftid = "fix_transcript_" + mfnid; if (!$('#' + ftid).length){ var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb; html += "contribute transcript fix" + mke("/a") + mke("/span"); console.log("html = ", html); vblock.after(html) } $('#' + ftid).click(function(){ var cst = $('ol.subtitles').find('li.current'); var cindex = Number(cst.data('index')); var gurl; if (mfnpre.endsWith('_cq_sol')){ gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/"; }else{ gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/"; } gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1); console.log("going to ", gurl); window.open(gurl, "MITx 8.370x subtitle source"); }); }); } try{ add_fix_transcript(); } catch(err){ console.log(err); } try{ var rb = String.fromCharCode(62); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000); } catch(err){ console.log(err); } </script></span> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Multi-qubit_Clifford_circuits" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Multi-qubit Clifford circuits</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-multi-clifford" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-multi-clifford-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="3" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-multi-clifford-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Multi-qubit Clifford circuits &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; In seeking a systematic construction for arbitrary multi-qubit Clifford circuits, it is helpful to study the action of multi-qubit generalizations of the controlled-NOT gate, the [mathjaxinline]C(Z)[/mathjaxinline] gate, and the [mathjaxinline]C(Y)[/mathjaxinline] gate. &lt;/p&gt; &lt;ol class=&#34;enumerate&#34;&gt; &lt;li value=&#34;1&#34;&gt; &lt;p&gt; Recall that the two-qubit controlled-[mathjaxinline]Z[/mathjaxinline] gate [mathjaxinline]C(Z)[/mathjaxinline] and controlled-[mathjaxinline]Y[/mathjaxinline] gate [mathjaxinline]C(Y)[/mathjaxinline] can be composed from [mathjaxinline]C(X)[/mathjaxinline] (CNOT), [mathjaxinline]H[/mathjaxinline] (Hadamard), and [mathjaxinline]S[/mathjaxinline] (phase) gates. &lt;/p&gt; &lt;p&gt; A [mathjaxinline]n[/mathjaxinline]-qubit gate [mathjaxinline]C(g)[/mathjaxinline] can similarly be constructed from CNOT, [mathjaxinline]H[/mathjaxinline], and [mathjaxinline]S[/mathjaxinline] gates, where [mathjaxinline]g[/mathjaxinline] is an [mathjaxinline]n-1[/mathjaxinline] qubit Pauli operation (ie [mathjaxinline]g\in P_{n-1}[/mathjaxinline]). &lt;/p&gt; &lt;p&gt; For example, Give a three-qubit Clifford circuit for [mathjaxinline]C(ZX)[/mathjaxinline], where [mathjaxinline]ZX = Z\otimes X[/mathjaxinline] is a two-qubit Pauli operation. Input your answer as a list of gates, eg &lt;tt class=&#34;tt&#34;&gt;[H(1),CNOT(1,0)]&lt;/tt&gt;, where the gates in the circuit are applied in the order listed, from left to right. You may use &lt;tt class=&#34;tt&#34;&gt;X&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;Y&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;Z&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;H&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;S&lt;/tt&gt;, and &lt;tt class=&#34;tt&#34;&gt;CNOT&lt;/tt&gt; gates. For &lt;tt class=&#34;tt&#34;&gt;CNOT(1,0)&lt;/tt&gt;, qubit [mathjaxinline]1[/mathjaxinline] is the control, and qubit [mathjaxinline]0[/mathjaxinline] is the target. Let qubit [mathjaxinline]2[/mathjaxinline] be the control qubit for your gate, and let [mathjaxinline]ZX[/mathjaxinline] act on qubits [mathjaxinline]1[/mathjaxinline] and [mathjaxinline]0[/mathjaxinline], in that order, from left to right. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]C(ZX) =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-multi-clifford_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-multi-clifford_2_1&#34; id=&#34;input_s12-wk6-multi-clifford_2_1&#34; aria-describedby=&#34;status_s12-wk6-multi-clifford_2_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-multi-clifford_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-multi-clifford_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-multi-clifford_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_s12-wk6-multi-clifford_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;2&#34;&gt; &lt;p&gt; What is the action of [mathjaxinline]C(g)[/mathjaxinline] on the Pauli group elements &lt;tt class=&#34;tt&#34;&gt;Xg&lt;/tt&gt; = [mathjaxinline]X\otimes g[/mathjaxinline] and &lt;tt class=&#34;tt&#34;&gt;Zh&lt;/tt&gt; = [mathjaxinline]Z\otimes h[/mathjaxinline], where [mathjaxinline]g,h\in P_{n-1}[/mathjaxinline], assuming that [mathjaxinline][g,h]=0[/mathjaxinline]? &lt;/p&gt; &lt;p&gt; Give each of your answers as a string of Pauli operators, eg &lt;tt class=&#34;tt&#34;&gt;XY&lt;/tt&gt;: &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]C(g) \left[ X\otimes g \right] C(g)^\dagger =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-multi-clifford_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-multi-clifford_3_1&#34; id=&#34;input_s12-wk6-multi-clifford_3_1&#34; aria-describedby=&#34;status_s12-wk6-multi-clifford_3_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-multi-clifford_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-multi-clifford_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-multi-clifford_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]C(g) \left[ Z\otimes h \right] C(g)^\dagger =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-multi-clifford_4_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-multi-clifford_4_1&#34; id=&#34;input_s12-wk6-multi-clifford_4_1&#34; aria-describedby=&#34;status_s12-wk6-multi-clifford_4_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-multi-clifford_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-multi-clifford_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-multi-clifford_4_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;/li&gt; &lt;/ol&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Multi-qubit Clifford circuits&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-multi-clifford&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-multi-clifford&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Multi-qubit_Clifford_circuits_II" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Multi-qubit Clifford circuits II</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford2"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford2" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-multi-clifford2" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-multi-clifford2-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford2" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford2/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="3" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-multi-clifford2-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford2-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Multi-qubit Clifford circuits II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-multi-clifford2-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;ol class=&#34;enumerate&#34;&gt; &lt;li value=&#34;1&#34;&gt; &lt;p&gt; Consider the [mathjaxinline]n[/mathjaxinline] qubit Clifford circuit &lt;/p&gt; &lt;table id=&#34;a0000000003&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34; style=&#34;width:80%; border:none&#34;&gt;[mathjax]V = C(h) \left[ H\otimes I_{n-1}\right] C(g) \, ,[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.2)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; where [mathjaxinline]h,g\in P_{n-1}[/mathjaxinline], [mathjaxinline]I_{n-1}[/mathjaxinline] is the identity operation in [mathjaxinline]P_{n-1}[/mathjaxinline], and [mathjaxinline]H[/mathjaxinline] acts on the control qubit of [mathjaxinline]C(h)[/mathjaxinline] and [mathjaxinline]C(g)[/mathjaxinline]. This is the circuit &lt;center&gt;&lt;img src=&#34;/assets/courseware/v1/663b4453ce3f5cde535a6c364e46acaf/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/images_clifford-circuit-reduction-v1a.png&#34; width=&#34;400&#34; style=&#34;size:300px&#34;/&gt;&lt;/center&gt; It is convenient to analyze this circuit by expressing each of its components as a [mathjaxinline]2\times 2[/mathjaxinline] block matrix. Specifically, &lt;/p&gt; &lt;table id=&#34;a0000000004&#34; class=&#34;eqnarray&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000005&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle H[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle \frac{1}{\sqrt{2}} \left[ \begin{array}{cc}{I}&amp;amp; {I}\\ {I}&amp;amp; {-I}\end{array}\right][/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.3)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;a0000000006&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle C(g)[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle \left[ \begin{array}{cc}{I}&amp;amp; {0}\\ {0}&amp;amp; {g}\end{array}\right][/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.4)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;a0000000007&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle C(h)[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle \left[ \begin{array}{cc}{I}&amp;amp; {0}\\ {0}&amp;amp; {h}\end{array}\right] \, .[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.5)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Give the [mathjaxinline]2\times 2[/mathjaxinline] matrix for [mathjaxinline]\sqrt{2} V[/mathjaxinline]. Enter your result as a list of lists, eg &lt;tt class=&#34;tt&#34;&gt;[[1,-1],[1,1]]&lt;/tt&gt;. &lt;/p&gt; &lt;p&gt; Remember to explicitly include multiplication with &lt;tt class=&#34;tt&#34;&gt;*&lt;/tt&gt;. Note that for convenience, the [mathjaxinline]\sqrt{2}[/mathjaxinline] factor from [mathjaxinline]H[/mathjaxinline] has already been taken out to the left hand side of the equation. Keep the proper ordering of symbols in your expressions, as if [mathjaxinline]g[/mathjaxinline] and [mathjaxinline]h[/mathjaxinline] were to not commute: &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\sqrt{2} V =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-multi-clifford2_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/dc8916af8ae257e4388dae38f570cb8b/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_subtext.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-multi-clifford2_2_1&#34; id=&#34;input_s12-wk6-multi-clifford2_2_1&#34; aria-describedby=&#34;status_s12-wk6-multi-clifford2_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-multi-clifford2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-multi-clifford2_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-multi-clifford2_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk6-multi-clifford2_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk6-multi-clifford2_2_1_dynamath&#34; name=&#34;input_s12-wk6-multi-clifford2_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;2&#34;&gt; &lt;p&gt; What is the action of [mathjaxinline]V[/mathjaxinline] on the Pauli group elements &lt;tt class=&#34;tt&#34;&gt;Xg&lt;/tt&gt; = [mathjaxinline]X\otimes g[/mathjaxinline] and &lt;tt class=&#34;tt&#34;&gt;Zh&lt;/tt&gt; = [mathjaxinline]Z\otimes h[/mathjaxinline], where [mathjaxinline]g,h\in P_{n-1}[/mathjaxinline], assuming that [mathjaxinline][g,h]=0[/mathjaxinline]? &lt;/p&gt; &lt;p&gt; Give each of your answers as a string of Pauli operators, eg &lt;tt class=&#34;tt&#34;&gt;XY&lt;/tt&gt;: &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]V \left[ X\otimes g \right] V^\dagger =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-multi-clifford2_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-multi-clifford2_3_1&#34; id=&#34;input_s12-wk6-multi-clifford2_3_1&#34; aria-describedby=&#34;status_s12-wk6-multi-clifford2_3_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-multi-clifford2_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-multi-clifford2_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-multi-clifford2_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]V \left[ Z\otimes h \right] V^\dagger =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-multi-clifford2_4_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-multi-clifford2_4_1&#34; id=&#34;input_s12-wk6-multi-clifford2_4_1&#34; aria-describedby=&#34;status_s12-wk6-multi-clifford2_4_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-multi-clifford2_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-multi-clifford2_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-multi-clifford2_4_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;/li&gt; &lt;/ol&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Multi-qubit Clifford circuits II&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-multi-clifford2&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-multi-clifford2&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Decomposing_Clifford_operations" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Decomposing Clifford operations</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-clifford-decomposition"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-clifford-decomposition" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-clifford-decomposition" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-clifford-decomposition-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-clifford-decomposition" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-clifford-decomposition/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="5" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-clifford-decomposition-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-clifford-decomposition-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Decomposing Clifford operations &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-clifford-decomposition-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Recall that [mathjaxinline]P_ n[/mathjaxinline] is the group of [mathjaxinline]n[/mathjaxinline]-qubit Pauli operations. The set of unitary transforms [mathjaxinline]U[/mathjaxinline] such that [mathjaxinline]U P_ n U^{\dagger } = P_ n[/mathjaxinline] is the &lt;em&gt;normalizer&lt;/em&gt; of [mathjaxinline]P_ n[/mathjaxinline] in [mathjaxinline]SU(2^ n)[/mathjaxinline], denoted by [mathjaxinline]N(P_ n)[/mathjaxinline]. [mathjaxinline]N(P_ n)[/mathjaxinline] is known as the Clifford group. &lt;/p&gt; &lt;p&gt; In this problem, we shall prove the following theorem about Clifford group operations: &lt;/p&gt; &lt;blockquote class=&#34;quote&#34;&gt; Suppose [mathjaxinline]U[/mathjaxinline] is any unitary operator on [mathjaxinline]n[/mathjaxinline] qubits with the property that if [mathjaxinline]g \in P_ n[/mathjaxinline] then [mathjaxinline]U g U^{\dagger } \in P_ n[/mathjaxinline]. Then up to a global phase [mathjaxinline]U[/mathjaxinline] may be composed from [mathjaxinline]O(n^2)[/mathjaxinline] Hadamard, phase and CNOT gates. &lt;/blockquote&gt; &lt;p&gt; Let us prove the theorem in a few steps, below. &lt;/p&gt; &lt;ol class=&#34;enumerate&#34;&gt; &lt;li value=&#34;1&#34;&gt; &lt;p&gt; Suppose that the [mathjaxinline]n[/mathjaxinline]-qubit Clifford operation [mathjaxinline]U[/mathjaxinline] satisfies these two equations: &lt;/p&gt; &lt;table id=&#34;a0000000008&#34; class=&#34;eqnarray&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;eqn1p6&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U \left[ Z\otimes I_{n-1} \right] U^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle X\otimes g [/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.6)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;eqn1p7&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U \left[ X\otimes I_{n-1} \right] U^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle Z\otimes h \, .[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.7)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Note that one can always arrange for this to be true (except in trivial cases) by replacing [mathjaxinline]U[/mathjaxinline] with [mathjaxinline]U_ r U[/mathjaxinline], where [mathjaxinline]U_ r[/mathjaxinline] involves at most swap gates and local Clifford gates. This is because [mathjaxinline]U[/mathjaxinline] must map [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] on the [mathjaxinline]n^{th}[/mathjaxinline] qubit to some nontrivial Pauli operations on at least one other qubit, and these can be turned into [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] using [mathjaxinline]H[/mathjaxinline] and [mathjaxinline]S[/mathjaxinline] operations (which permute among [mathjaxinline]X[/mathjaxinline], [mathjaxinline]Y[/mathjaxinline], and [mathjaxinline]Z[/mathjaxinline]). &lt;/p&gt; &lt;p&gt; Let us express [mathjaxinline]U[/mathjaxinline] in block matrix form, as &lt;/p&gt; &lt;table id=&#34;a0000000011&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34; style=&#34;width:80%; border:none&#34;&gt;[mathjax]U = \frac{1}{\sqrt{2}} \left[ \begin{array}{cc}{A}&amp;amp; {B}\\ {C}&amp;amp; {D}\end{array}\right] \, ,[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.8)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; where [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and [mathjaxinline]D[/mathjaxinline] are operators on the right-most [mathjaxinline]n-1[/mathjaxinline] qubits. Using the constraints of Eq.(1.6&amp;#8211;1.7), and the unitarity of [mathjaxinline]U[/mathjaxinline], express [mathjaxinline]B[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and [mathjaxinline]D[/mathjaxinline] in terms of [mathjaxinline]g[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], and [mathjaxinline]A[/mathjaxinline]. Keep in mind that the order of symbols in your expressions matters: &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]B =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-clifford-decomposition_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-clifford-decomposition_2_1&#34; id=&#34;input_s12-wk6-clifford-decomposition_2_1&#34; aria-describedby=&#34;status_s12-wk6-clifford-decomposition_2_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-clifford-decomposition_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-clifford-decomposition_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-clifford-decomposition_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]C =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-clifford-decomposition_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-clifford-decomposition_3_1&#34; id=&#34;input_s12-wk6-clifford-decomposition_3_1&#34; aria-describedby=&#34;status_s12-wk6-clifford-decomposition_3_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-clifford-decomposition_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-clifford-decomposition_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-clifford-decomposition_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]D =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-clifford-decomposition_4_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-clifford-decomposition_4_1&#34; id=&#34;input_s12-wk6-clifford-decomposition_4_1&#34; aria-describedby=&#34;status_s12-wk6-clifford-decomposition_4_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-clifford-decomposition_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-clifford-decomposition_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-clifford-decomposition_4_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;/li&gt; &lt;li value=&#34;2&#34;&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;What is [mathjaxinline]A^\dagger A[/mathjaxinline]?&amp;#160;&amp;#160;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-clifford-decomposition_5_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-clifford-decomposition_5_1&#34; id=&#34;input_s12-wk6-clifford-decomposition_5_1&#34; aria-describedby=&#34;status_s12-wk6-clifford-decomposition_5_1&#34; value=&#34;&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-clifford-decomposition_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-clifford-decomposition_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-clifford-decomposition_5_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_s12-wk6-clifford-decomposition_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;3&#34;&gt; &lt;p&gt; Recall the [mathjaxinline]n[/mathjaxinline] qubit Clifford circuit &lt;/p&gt; &lt;table id=&#34;a0000000012&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34; style=&#34;width:80%; border:none&#34;&gt;[mathjax]V = C(h) \left[ H\otimes I_{n-1}\right] C(g) \, ,[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.9)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; where [mathjaxinline]h,g\in P_{n-1}[/mathjaxinline], [mathjaxinline]I_{n-1}[/mathjaxinline] is the identity operation in [mathjaxinline]P_{n-1}[/mathjaxinline], and [mathjaxinline]H[/mathjaxinline] acts on the control qubit of [mathjaxinline]C(h)[/mathjaxinline] and [mathjaxinline]C(g)[/mathjaxinline]. This is the circuit &lt;center&gt;&lt;img src=&#34;/assets/courseware/v1/663b4453ce3f5cde535a6c364e46acaf/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/images_clifford-circuit-reduction-v1a.png&#34; width=&#34;400&#34; style=&#34;size:300px&#34;/&gt;&lt;/center&gt; &lt;/p&gt; &lt;p&gt; Give the block matrix for [mathjaxinline]U&#39; = VU[/mathjaxinline] (in terms of [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], [mathjaxinline]D[/mathjaxinline] and any necessary constants): &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]U&#39; =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-clifford-decomposition_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/dc8916af8ae257e4388dae38f570cb8b/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_subtext.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-clifford-decomposition_6_1&#34; id=&#34;input_s12-wk6-clifford-decomposition_6_1&#34; aria-describedby=&#34;status_s12-wk6-clifford-decomposition_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-clifford-decomposition_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-clifford-decomposition_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-clifford-decomposition_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk6-clifford-decomposition_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk6-clifford-decomposition_6_1_dynamath&#34; name=&#34;input_s12-wk6-clifford-decomposition_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_s12-wk6-clifford-decomposition_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/li&gt; &lt;/ol&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Decomposing Clifford operations&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-clifford-decomposition&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-clifford-decomposition&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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String.prototype.startsWith = function (str){ return this.indexOf(str) === 0; }; } if(typeof(String.prototype.trim) === "undefined") { String.prototype.trim = function() { return String(this).replace(/^\s+|\s+$/g, ''); }; } var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var amp = String.fromCharCode(38); var rlb = rb + lb; var mke = function(x){ return lb + x + rb; } var search_module_url = ""; var get_search_module_ficus = function(){ var cid = $('div.xblock').data('course-id'); if (cid){ console.log("cid = ", cid); // search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten console.log("3. search_module_url = ", search_module_url); return; } var course_root_link = $('span.nav-item-course').find('a').attr('href'); if (course_root_link){ console.log("course_root_link = ", course_root_link); search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/"); console.log("2. search_module_url = ", search_module_url); return } console.log("cannot determine search module url"); } var get_search_module = function(){ // find search this module link if (!($('div.course-index').length)){ return get_search_module_ficus(); } $('div.course-index').find('nav').find('a').each(function(){ if ($(this).text().trim().startsWith("Search this course")){ search_module_url = $(this).attr('href'); console.log("search_module_url = ", search_module_url); } }); } var go_to_search = function(){ get_search_module(); var sterm = $('#site-search-box').val(); // new_url = jump_to_url + "/Search_this_module/?q=" + sterm; new_url = search_module_url + "?q=" + sterm; console.log("sterm = ", sterm, " ; going to ", new_url); window.location.href = new_url; } if (!$('#site-search-box').length){ $("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'" + " id='site-search-box'" + rlb + "img src='" + course_static_url + "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb); } $("#site-search-box").keypress(function(event) { if (event.which == 13) { event.preventDefault(); go_to_search(); } }); // $('#site-search-box').bind("enterKey", go_to_search); var get = function(x){ return eval(x); } return {'course_static_url': course_static_url, 'jump_to_url': jump_to_url, 'go_to_search': go_to_search, 'get_search_module': get_search_module, 'get_search_module_ficus': get_search_module_ficus, 'get': get, } } var the_site_search = add_site_search(); var add_fix_transcript = function(){ if ($('div.wrap-instructor-info').length==0){ return; } $('div.xblock-student_view-video').each(function(key, vblock_e){ var vblock = $(vblock_e); var vuid = vblock.data('usage-id').split('@'); var vid; if (vuid.length==1){ vuid = vblock.data('usage-id').split(';_') vid = vuid[5]; }else{ vid = vuid[2]; } var mfnpre = vid.split("_video",1)[0]; var mfnid = mfnpre; // no periods mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var mke = function(x){ return lb + x + rb; } var ftid = "fix_transcript_" + mfnid; if (!$('#' + ftid).length){ var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb; html += "contribute transcript fix" + mke("/a") + mke("/span"); console.log("html = ", html); vblock.after(html) } $('#' + ftid).click(function(){ var cst = $('ol.subtitles').find('li.current'); var cindex = Number(cst.data('index')); var gurl; if (mfnpre.endsWith('_cq_sol')){ gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/"; }else{ gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/"; } gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1); console.log("going to ", gurl); window.open(gurl, "MITx 8.370x subtitle source"); }); }); } try{ add_fix_transcript(); } catch(err){ console.log(err); } try{ var rb = String.fromCharCode(62); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000); } catch(err){ console.log(err); } </script></span> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Decomposition_exercise_2-qubit_Clifford_circuit" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Decomposition exercise: 2-qubit Clifford circuit</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-cg-decomp1" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-cg-decomp1-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-cg-decomp1-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Decomposition exercise: 2-qubit Clifford circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; [mathjaxinline]U[/mathjaxinline] is a two-qubit Clifford circuit which maps these Pauli operators: &lt;/p&gt; &lt;table id=&#34;a0000000013&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34; style=&#34;width:80%; border:none&#34;&gt;[mathjax]{\tt [IX,IZ,XI,ZI]} \longrightarrow {\tt [-YY,ZI,ZZ,ZX]} \, .[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.10)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Let us systematically construct a circuit for [mathjaxinline]U[/mathjaxinline] using CNOT, SWAP, [mathjaxinline]H[/mathjaxinline], and [mathjaxinline]S[/mathjaxinline], using the procedure from &lt;a href=&#34;/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/s12-wk6-clifford-decomposition&#34; target=&#34;_blank&#34;&gt;the previous problem&lt;/a&gt;. &lt;/p&gt; &lt;p&gt; Start with qubit [mathjaxinline]0[/mathjaxinline] (the right-most label). Give a Clifford circuit [mathjaxinline]W_1[/mathjaxinline], using just SWAP, [mathjaxinline]H[/mathjaxinline], and [mathjaxinline]S[/mathjaxinline] gates, to bring the set of Pauli operators into a form such that for [mathjaxinline]U_1 = W_1 U[/mathjaxinline], &lt;/p&gt; &lt;table id=&#34;a0000000014&#34; class=&#34;eqnarray&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000015&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U_1 {\tt IX}\, U_1^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle h \otimes {\tt Z}[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.11)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;a0000000016&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U_1 {\tt IZ}\, U_1^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle g \otimes {\tt X} \, .[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.12)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; where [mathjaxinline]h[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are single qubit Pauli operators acting on qubit [mathjaxinline]1[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; Enter your answer as a list of gates, in the form &lt;big class=&#34;xlarge&#34;&gt;&lt;tt class=&#34;tt&#34;&gt;[H(0),SWAP(0,1)]&lt;/tt&gt;&lt;/big&gt;: &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]W_1 =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-cg-decomp1_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-cg-decomp1_2_1&#34; id=&#34;input_s12-wk6-cg-decomp1_2_1&#34; aria-describedby=&#34;status_s12-wk6-cg-decomp1_2_1&#34; value=&#34;&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-cg-decomp1_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-cg-decomp1_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-cg-decomp1_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_s12-wk6-cg-decomp1_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Decomposition exercise: 2-qubit Clifford circuit&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-cg-decomp1&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-cg-decomp1&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp1-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp1-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp1-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-2"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-2" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-cg-decomp1-2" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-cg-decomp1-2-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-2" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-2/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-cg-decomp1-2-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-2-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Decomposition exercise: 2-qubit Clifford circuit II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-2-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Give a Clifford circuit for [mathjaxinline]V[/mathjaxinline], such that for [mathjaxinline]U_2 = V U_1[/mathjaxinline], &lt;/p&gt; &lt;table id=&#34;a0000000018&#34; class=&#34;eqnarray&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000019&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U_2 {\tt IX}\, U_2^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle {\tt IX}[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.14)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;a0000000020&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U_2 {\tt IZ}\, U_2^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle {\tt IZ} \, .[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.15)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Enter your answer as a list of gates, in the form &lt;big class=&#34;xlarge&#34;&gt;&lt;tt class=&#34;tt&#34;&gt;[H(0),SWAP(0,1)]&lt;/tt&gt;&lt;/big&gt;. You may use CNOT, SWAP, [mathjaxinline]H[/mathjaxinline], and [mathjaxinline]S[/mathjaxinline]: &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]V =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-cg-decomp1-2_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-cg-decomp1-2_2_1&#34; id=&#34;input_s12-wk6-cg-decomp1-2_2_1&#34; aria-describedby=&#34;status_s12-wk6-cg-decomp1-2_2_1&#34; value=&#34;&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-cg-decomp1-2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-cg-decomp1-2_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-cg-decomp1-2_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_s12-wk6-cg-decomp1-2_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Decomposition exercise: 2-qubit Clifford circuit II&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-cg-decomp1-2&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-cg-decomp1-2&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp1-2-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp1-2-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp1-2-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-3"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-3" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-cg-decomp1-3" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-cg-decomp1-3-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-3" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-3/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-cg-decomp1-3-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-3-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Decomposition exercise: 2-qubit Clifford circuit III &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp1-3-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; There remains at most a single qubit Clifford circuit operation [mathjaxinline]W_2[/mathjaxinline], which is needed to complete the transformation such that for [mathjaxinline]U_3 = W_2 U_2[/mathjaxinline], &lt;/p&gt; &lt;table id=&#34;a0000000022&#34; class=&#34;eqnarray&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000023&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U_3 {\tt XI}\, U_3^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle {\tt XI}[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.17)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;a0000000024&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle U_3 {\tt ZI}\, U_3^\dagger[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle {\tt ZI} \, .[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.18)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Enter your answer as a list of gates, in the form &lt;big class=&#34;xlarge&#34;&gt;&lt;tt class=&#34;tt&#34;&gt;[H(0),SWAP(0,1)]&lt;/tt&gt;&lt;/big&gt;. You may use CNOT, SWAP, [mathjaxinline]H[/mathjaxinline], and [mathjaxinline]S[/mathjaxinline]: &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]W_2 =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-cg-decomp1-3_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-cg-decomp1-3_2_1&#34; id=&#34;input_s12-wk6-cg-decomp1-3_2_1&#34; aria-describedby=&#34;status_s12-wk6-cg-decomp1-3_2_1&#34; value=&#34;&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-cg-decomp1-3_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-cg-decomp1-3_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-cg-decomp1-3_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_s12-wk6-cg-decomp1-3_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; As a result, [mathjaxinline]W_2 V W_1 U = I[/mathjaxinline], (you may wish to prove for yourself that any unitary operation which maps [mathjaxinline]{\tt [IX,IZ,XI,ZI]} \longrightarrow {\tt [IX,IZ,XI,ZI]}[/mathjaxinline] must be identity). &lt;/p&gt; &lt;p&gt; And thus you now have a Clifford circuit for [mathjaxinline]U[/mathjaxinline], namely [mathjaxinline]W_1^\dagger V^\dagger W_2^\dagger = U[/mathjaxinline]. &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Decomposition exercise: 2-qubit Clifford circuit III&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-cg-decomp1-3&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-cg-decomp1-3&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Decomposition_exercise_3-qubit_Clifford_circuit" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Decomposition exercise: 3-qubit Clifford circuit</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp2"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="problem" data-has-score="True" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp2" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk6-cg-decomp2" class="problems-wrapper" role="group" aria-labelledby="s12-wk6-cg-decomp2-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp2" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp2/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk6-cg-decomp2-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp2-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Decomposition exercise: 3-qubit Clifford circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk6-cg-decomp2-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; [mathjaxinline]U[/mathjaxinline] is a three-qubit Clifford circuit which maps these Pauli operators: &lt;/p&gt; &lt;table id=&#34;a0000000026&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34; style=&#34;width:80%; border:none&#34;&gt;[mathjax]{\tt [IIX, IIZ, IXI, IZI, XII, ZII]} \longrightarrow {\tt [IZX, IIZ, ZYZ, IZI, ZII, XZI]} \, .[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.20)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Construct a circuit for [mathjaxinline]U[/mathjaxinline] using CNOT, SWAP, [mathjaxinline]H[/mathjaxinline], and [mathjaxinline]S[/mathjaxinline]. You may use the procedure from &lt;a href=&#34;/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/s12-wk6-clifford-decomposition&#34; target=&#34;_blank&#34;&gt;the previous problem on decomposing Clifford operations&lt;/a&gt;, or any other method you wish. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]U =[/mathjaxinline]&lt;/p&gt; &lt;span&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk6-cg-decomp2_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk6-cg-decomp2_2_1&#34; id=&#34;input_s12-wk6-cg-decomp2_2_1&#34; aria-describedby=&#34;status_s12-wk6-cg-decomp2_2_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk6-cg-decomp2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk6-cg-decomp2_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_s12-wk6-cg-decomp2_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/span&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_s12-wk6-cg-decomp2_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Decomposition exercise: 3-qubit Clifford circuit&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_s12-wk6-cg-decomp2&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_s12-wk6-cg-decomp2&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp2-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp2-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;s12-wk6-cg-decomp2-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box018163"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="1e7bee46abba11f199130ebf0becc8e7" data-graded="True" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-block-type="html" data-has-score="False" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box018163" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <span><a href="/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript"> var add_site_search = function(){ course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', ''); jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', ''); if (typeof String.prototype.startsWith != 'function') { // see below for better implementation! String.prototype.startsWith = function (str){ return this.indexOf(str) === 0; }; } if(typeof(String.prototype.trim) === "undefined") { String.prototype.trim = function() { return String(this).replace(/^\s+|\s+$/g, ''); }; } var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var amp = String.fromCharCode(38); var rlb = rb + lb; var mke = function(x){ return lb + x + rb; } var search_module_url = ""; var get_search_module_ficus = function(){ var cid = $('div.xblock').data('course-id'); if (cid){ console.log("cid = ", cid); // search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten console.log("3. search_module_url = ", search_module_url); return; } var course_root_link = $('span.nav-item-course').find('a').attr('href'); if (course_root_link){ console.log("course_root_link = ", course_root_link); search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/"); console.log("2. search_module_url = ", search_module_url); return } console.log("cannot determine search module url"); } var get_search_module = function(){ // find search this module link if (!($('div.course-index').length)){ return get_search_module_ficus(); } $('div.course-index').find('nav').find('a').each(function(){ if ($(this).text().trim().startsWith("Search this course")){ search_module_url = $(this).attr('href'); console.log("search_module_url = ", search_module_url); } }); } var go_to_search = function(){ get_search_module(); var sterm = $('#site-search-box').val(); // new_url = jump_to_url + "/Search_this_module/?q=" + sterm; new_url = search_module_url + "?q=" + sterm; console.log("sterm = ", sterm, " ; going to ", new_url); window.location.href = new_url; } if (!$('#site-search-box').length){ $("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'" + " id='site-search-box'" + rlb + "img src='" + course_static_url + "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb); } $("#site-search-box").keypress(function(event) { if (event.which == 13) { event.preventDefault(); go_to_search(); } }); // $('#site-search-box').bind("enterKey", go_to_search); var get = function(x){ return eval(x); } return {'course_static_url': course_static_url, 'jump_to_url': jump_to_url, 'go_to_search': go_to_search, 'get_search_module': get_search_module, 'get_search_module_ficus': get_search_module_ficus, 'get': get, } } var the_site_search = add_site_search(); var add_fix_transcript = function(){ if ($('div.wrap-instructor-info').length==0){ return; } $('div.xblock-student_view-video').each(function(key, vblock_e){ var vblock = $(vblock_e); var vuid = vblock.data('usage-id').split('@'); var vid; if (vuid.length==1){ vuid = vblock.data('usage-id').split(';_') vid = vuid[5]; }else{ vid = vuid[2]; } var mfnpre = vid.split("_video",1)[0]; var mfnid = mfnpre; // no periods mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var mke = function(x){ return lb + x + rb; } var ftid = "fix_transcript_" + mfnid; if (!$('#' + ftid).length){ var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb; html += "contribute transcript fix" + mke("/a") + mke("/span"); console.log("html = ", html); vblock.after(html) } $('#' + ftid).click(function(){ var cst = $('ol.subtitles').find('li.current'); var cindex = Number(cst.data('index')); var gurl; if (mfnpre.endsWith('_cq_sol')){ gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/"; }else{ gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/"; } gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1); console.log("going to ", gurl); window.open(gurl, "MITx 8.370x subtitle source"); }); }); } try{ add_fix_transcript(); } catch(err){ console.log(err); } try{ var rb = String.fromCharCode(62); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000); } catch(err){ console.log(err); } </script></span> </div> </div> </div> </div>