<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="VerticalStudentView" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Quantum_Fourier_Transform__1_of_3_" data-runtime-class="LmsRuntime" data-has-score="False"> <h2 class="hd hd-2 unit-title">Quantum Fourier Transform (1 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_I"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_I" data-runtime-class="LmsRuntime" data-has-score="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_Quantum_Fourier_Transform_I" class="problems-wrapper" role="group" aria-labelledby="Quantum_Fourier_Transform_I-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_I" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_I/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;Quantum_Fourier_Transform_I-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_I-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Quantum Fourier Transform I &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_I-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; The Quantum Fourier transform is a quantum analogue of the discrete Fourier transform. Let [mathjaxinline]j[/mathjaxinline] be a [mathjaxinline]n[/mathjaxinline] bit string. We define [mathjaxinline]U_{QFT}[/mathjaxinline] by &lt;/p&gt; &lt;table id=&#34;a0000000004&#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_{QFT} |{j}\rangle = \frac{1}{\sqrt{2^{n}}}\sum _{k=0}^{2^ n-1} e^{\frac{2\pi i jk}{2^ n}}|{k}\rangle[/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]jk[/mathjaxinline] in the exponent is the ordinary multiplication. QFT is a key submodule used in another important algorithm called the phase estimation algorithm, which we are going to see in the next problem. &lt;/p&gt; &lt;p&gt; Compute the following. Choose the global phase so that the coefficient for [mathjaxinline]|{00}\rangle[/mathjaxinline] becomes real. Input &lt;tt class=&#34;ttfamily&#34;&gt;i&lt;/tt&gt; for the imaginary number [mathjaxinline]i[/mathjaxinline]. &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]U_{QFT}|{01}\rangle[/mathjaxinline] &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Quantum_Fourier_Transform_I_2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Quantum_Fourier_Transform_I_2_1&#34; id=&#34;input_Quantum_Fourier_Transform_I_2_1&#34; aria-describedby=&#34;status_Quantum_Fourier_Transform_I_2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Quantum_Fourier_Transform_I_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Quantum_Fourier_Transform_I_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_Quantum_Fourier_Transform_I_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Quantum_Fourier_Transform_I_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Quantum_Fourier_Transform_I_2_1_dynamath&#34; name=&#34;input_Quantum_Fourier_Transform_I_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]U_{QFT}\left[\frac{1}{\sqrt{2}}\left(|{00}\rangle +|{11}\rangle \right)\right][/mathjaxinline] &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Quantum_Fourier_Transform_I_3_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Quantum_Fourier_Transform_I_3_1&#34; id=&#34;input_Quantum_Fourier_Transform_I_3_1&#34; aria-describedby=&#34;status_Quantum_Fourier_Transform_I_3_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Quantum_Fourier_Transform_I_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Quantum_Fourier_Transform_I_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_Quantum_Fourier_Transform_I_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Quantum_Fourier_Transform_I_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Quantum_Fourier_Transform_I_3_1_dynamath&#34; name=&#34;input_Quantum_Fourier_Transform_I_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]U_{QFT}^{\dagger }|{+++}\rangle[/mathjaxinline] &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Quantum_Fourier_Transform_I_4_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Quantum_Fourier_Transform_I_4_1&#34; id=&#34;input_Quantum_Fourier_Transform_I_4_1&#34; aria-describedby=&#34;status_Quantum_Fourier_Transform_I_4_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Quantum_Fourier_Transform_I_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Quantum_Fourier_Transform_I_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_Quantum_Fourier_Transform_I_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Quantum_Fourier_Transform_I_4_1&#34; 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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+8.370.2x+1T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.370.2x+1T2018/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-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="VerticalStudentView" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Quantum_Fourier_Transform__2_of_3_" data-runtime-class="LmsRuntime" data-has-score="False"> <h2 class="hd hd-2 unit-title">Quantum Fourier Transform (2 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_II"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_II" data-runtime-class="LmsRuntime" data-has-score="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_Quantum_Fourier_Transform_II" class="problems-wrapper" role="group" aria-labelledby="Quantum_Fourier_Transform_II-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_II" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_II/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;Quantum_Fourier_Transform_II-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_II-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Quantum Fourier Transform II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_II-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Continuing in the scenario of Part I of this problem, recall that the Quantum Fourier transform is a quantum analogue of the discrete Fourier transform, where [mathjaxinline]j[/mathjaxinline] is a [mathjaxinline]n[/mathjaxinline] bit string and [mathjaxinline]U_{QFT}[/mathjaxinline] is defined as &lt;/p&gt; &lt;table id=&#34;a0000000008&#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_{QFT} |{j}\rangle = \frac{1}{\sqrt{2^{n}}}\sum _{k=0}^{2^ n-1} e^{\frac{2\pi i jk}{2^ n}}|{k}\rangle[/mathjax]&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;/table&gt; &lt;p&gt; where [mathjaxinline]jk[/mathjaxinline] in the exponent is the ordinary multiplication. &lt;/p&gt; &lt;p&gt; [mathjaxinline]k[/mathjaxinline] in the equation above can be written in the binary form as [mathjaxinline]k=k_1k_2\dots k_{n}[/mathjaxinline], which is not to be confused with multiplication. In other words, [mathjaxinline]k = \sum _{l=0}^{n-1} 2^{l} k_{n-l}[/mathjaxinline] where [mathjaxinline]k_ i[/mathjaxinline] corresponds to the [mathjaxinline]i[/mathjaxinline]th bit of [mathjaxinline]k[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; By separately considering the sum for each bit, [mathjaxinline]U_{QFT} |{j}\rangle[/mathjaxinline] can be written in the tensor product form [mathjaxinline]U_{QFT} |{j}\rangle = |{\phi _1}\rangle \otimes |{\phi _2}\rangle \otimes \dots |{\phi _ n}\rangle[/mathjaxinline]. Find the expression for [mathjaxinline]|{\phi _ m}\rangle[/mathjaxinline] in terms of [mathjaxinline]j[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], and [mathjaxinline]n[/mathjaxinline]. Answer using the ket notation. You can input e^(i*pi) for [mathjaxinline]e^{i\pi }[/mathjaxinline] etc. Do not forget to explicitly include &lt;tt class=&#34;ttfamily&#34;&gt;*&lt;/tt&gt; for multiplication e.g. &lt;tt class=&#34;ttfamily&#34;&gt;2*|0&amp;gt;&lt;/tt&gt; etc. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Quantum_Fourier_Transform_II_2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Quantum_Fourier_Transform_II_2_1&#34; id=&#34;input_Quantum_Fourier_Transform_II_2_1&#34; aria-describedby=&#34;status_Quantum_Fourier_Transform_II_2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Quantum_Fourier_Transform_II_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Quantum_Fourier_Transform_II_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_Quantum_Fourier_Transform_II_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Quantum_Fourier_Transform_II_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Quantum_Fourier_Transform_II_2_1_dynamath&#34; name=&#34;input_Quantum_Fourier_Transform_II_2_1_dynamath&#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_Quantum_Fourier_Transform_II_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;html url_name=&#34;site_search_box1&#34;/&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Quantum Fourier Transform 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_Quantum_Fourier_Transform_II&#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_Quantum_Fourier_Transform_II&#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-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="VerticalStudentView" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Quantum_Fourier_Transform__3_of_3_" data-runtime-class="LmsRuntime" data-has-score="False"> <h2 class="hd hd-2 unit-title">Quantum Fourier Transform (3 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_III"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_III" data-runtime-class="LmsRuntime" data-has-score="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_Quantum_Fourier_Transform_III" class="problems-wrapper" role="group" aria-labelledby="Quantum_Fourier_Transform_III-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_III" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_III/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;Quantum_Fourier_Transform_III-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_III-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Quantum Fourier Transform III &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Quantum_Fourier_Transform_III-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Continuing in the scenario of Part I of this problem, recall that the Quantum Fourier transform is a quantum analogue of the discrete Fourier transform, where [mathjaxinline]j[/mathjaxinline] is a [mathjaxinline]n[/mathjaxinline] bit string and [mathjaxinline]U_{QFT}[/mathjaxinline] is defined as &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]U_{QFT} |{j}\rangle = \frac{1}{\sqrt{2^{n}}}\sum _{k=0}^{2^ n-1} e^{\frac{2\pi i jk}{2^ n}}|{k}\rangle[/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; where [mathjaxinline]jk[/mathjaxinline] in the exponent is the ordinary multiplication. &lt;/p&gt; &lt;p&gt; Complete the following circuit, which does the inverse QFT [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline] for [mathjaxinline]n=2[/mathjaxinline]. 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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+8.370.2x+1T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.370.2x+1T2018/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-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="VerticalStudentView" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Phase_Estimation__1_of_3_" data-runtime-class="LmsRuntime" data-has-score="False"> <h2 class="hd hd-2 unit-title">Phase Estimation (1 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_I"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_I" data-runtime-class="LmsRuntime" data-has-score="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_Phase_Estimation_I" class="problems-wrapper" role="group" aria-labelledby="Phase_Estimation_I-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_I" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_I/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;Phase_Estimation_I-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_I-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Phase Estimation I &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_I-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Let [mathjaxinline]U[/mathjaxinline] be a unitary with [mathjaxinline]U|{u_ i}\rangle =e^{2\pi i \phi _ i}|{u_ i}\rangle[/mathjaxinline] where [mathjaxinline]0 \leq \phi _ i &amp;lt;1[/mathjaxinline]. The phase estimation algorithm allows one to find out [mathjaxinline]\phi _ i[/mathjaxinline]. This submodule is widely used in numbers of important quantum algorithms, such as factoring algorithm, period finding algorithm etc. &lt;/p&gt; &lt;p&gt; The following figure is a part of the circuit for the phase estimation algorithm. The controlled- [mathjaxinline]U[/mathjaxinline] are applied controlled on [mathjaxinline]t[/mathjaxinline] qubits, which we call first register, to the bottom qubits, which we call second register. After this part, [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline], the inverse QFT, is applied to the first register and piecewise measurements are made on the first register. &lt;/p&gt; &lt;div class=&#34;figure&#34;&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/69ac7a3b1b5277055e84101077f55eed/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_ps3b_phase_estimation.png&#34; width=&#34;400&#34; style=&#34;scale:0.6&#34;/&gt; &lt;/center&gt; &lt;/div&gt; &lt;p&gt; Let us suppose that [mathjaxinline]\phi _ i[/mathjaxinline] can be exactly specified with [mathjaxinline]t[/mathjaxinline] decimal digits in a way that [mathjaxinline]\phi _ i = 0.\phi _{i,1}\phi _{i,2}\dots \phi _{i,t}[/mathjaxinline]. Note that it is the decimal for the binary representation, e.g. [mathjaxinline]1/2 = 0.1, 1/4 = 0.01[/mathjaxinline] etc. &lt;/p&gt; &lt;p&gt; Suppose we take [mathjaxinline]|{u_ j}\rangle[/mathjaxinline] for [mathjaxinline]|{u}\rangle[/mathjaxinline] in the figure. The state right before the inverse QFT takes the tensor product form. We write [mathjaxinline]\phi _ j = \psi _ j/2^ t[/mathjaxinline] so that [mathjaxinline]\psi _ j[/mathjaxinline] is an integer. What is the state of the [mathjaxinline]m[/mathjaxinline]th qubit from the top in the first register right before the inverse QFT? Answer using the ket notation in terms of [mathjaxinline]\psi _ j[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], [mathjaxinline]m[/mathjaxinline]. Input psi_j for [mathjaxinline]\psi _ j[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Phase_Estimation_I_2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Phase_Estimation_I_2_1&#34; id=&#34;input_Phase_Estimation_I_2_1&#34; aria-describedby=&#34;status_Phase_Estimation_I_2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Phase_Estimation_I_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Phase_Estimation_I_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_Phase_Estimation_I_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Phase_Estimation_I_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Phase_Estimation_I_2_1_dynamath&#34; name=&#34;input_Phase_Estimation_I_2_1_dynamath&#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_Phase_Estimation_I_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;Phase Estimation I&#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_Phase_Estimation_I&#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_Phase_Estimation_I&#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;Phase_Estimation_I-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;Phase_Estimation_I-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;Phase_Estimation_I-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+8.370.2x+1T2018+type@html+block@html_site_search_box3xxx"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@html+block@html_site_search_box3xxx" data-runtime-class="LmsRuntime" data-has-score="False"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <span><a href="/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.370.2x+1T2018/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+8.370.2x+1T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.370.2x+1T2018/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-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="VerticalStudentView" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Phase_Estimation__2_of_3_" data-runtime-class="LmsRuntime" data-has-score="False"> <h2 class="hd hd-2 unit-title">Phase Estimation (2 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_II"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_II" data-runtime-class="LmsRuntime" data-has-score="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_Phase_Estimation_II" class="problems-wrapper" role="group" aria-labelledby="Phase_Estimation_II-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_II" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_II/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;Phase_Estimation_II-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_II-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Phase Estimation II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_II-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Continuing in the same scenario as the previous part of this problem: &lt;/p&gt; &lt;div class=&#34;figure&#34;&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/69ac7a3b1b5277055e84101077f55eed/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_ps3b_phase_estimation.png&#34; width=&#34;400&#34; style=&#34;scale:0.6&#34;/&gt; &lt;/center&gt; &lt;/div&gt; &lt;p&gt; Let us suppose that [mathjaxinline]\phi _ i[/mathjaxinline] can be exactly specified with [mathjaxinline]t[/mathjaxinline] decimal digits in a way that [mathjaxinline]\phi _ i = 0.\phi _{i,1}\phi _{i,2}\dots \phi _{i,t}[/mathjaxinline]. Note that it is the decimal for the binary representation, e.g. [mathjaxinline]1/2 = 0.1, 1/4 = 0.01[/mathjaxinline] etc. &lt;/p&gt; &lt;p&gt; Comparing the output of the QFT algorithm and the result of the previous problem, you notice that output of the inverse QFT gives [mathjaxinline]\psi[/mathjaxinline] as output. (Notice that the QFT circuit ends with reordering the qubits). Now, let&#39;s consider specific examples. Let [mathjaxinline]U=X[/mathjaxinline]. What is the smallest [mathjaxinline]t[/mathjaxinline] needed to exactly specify [mathjaxinline]\phi[/mathjaxinline]? If we take such [mathjaxinline]t[/mathjaxinline], and take [mathjaxinline]|{-}\rangle[/mathjaxinline] for the [mathjaxinline]|{u}\rangle[/mathjaxinline] in the figure, what is the state of the first register after the inverse QFT? Answer in the ket notation. This state should be [mathjaxinline]t_{min}[/mathjaxinline]-qubit state where [mathjaxinline]t_{min}[/mathjaxinline] is the answer you got. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;minimum [mathjaxinline]t[/mathjaxinline]:&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Phase_Estimation_II_2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Phase_Estimation_II_2_1&#34; id=&#34;input_Phase_Estimation_II_2_1&#34; aria-describedby=&#34;status_Phase_Estimation_II_2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Phase_Estimation_II_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Phase_Estimation_II_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_Phase_Estimation_II_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Phase_Estimation_II_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Phase_Estimation_II_2_1_dynamath&#34; name=&#34;input_Phase_Estimation_II_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;state after [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Phase_Estimation_II_3_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Phase_Estimation_II_3_1&#34; id=&#34;input_Phase_Estimation_II_3_1&#34; aria-describedby=&#34;status_Phase_Estimation_II_3_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Phase_Estimation_II_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Phase_Estimation_II_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_Phase_Estimation_II_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Phase_Estimation_II_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Phase_Estimation_II_3_1_dynamath&#34; name=&#34;input_Phase_Estimation_II_3_1_dynamath&#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_Phase_Estimation_II_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;html url_name=&#34;site_search_box2&#34;/&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Phase Estimation 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_Phase_Estimation_II&#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_Phase_Estimation_II&#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;Phase_Estimation_II-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;Phase_Estimation_II-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;Phase_Estimation_II-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> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="VerticalStudentView" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Phase_Estimation__3_of_3_" data-runtime-class="LmsRuntime" data-has-score="False"> <h2 class="hd hd-2 unit-title">Phase Estimation (3 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_III"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="a35cabee1ee111f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_III" data-runtime-class="LmsRuntime" data-has-score="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_Phase_Estimation_III" class="problems-wrapper" role="group" aria-labelledby="Phase_Estimation_III-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_III" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_III/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;Phase_Estimation_III-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_III-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Phase Estimation III &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@Phase_Estimation_III-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Continuing in the same scenario as the previous part of this problem: &lt;/p&gt; &lt;div class=&#34;figure&#34;&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/69ac7a3b1b5277055e84101077f55eed/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_ps3b_phase_estimation.png&#34; width=&#34;400&#34; style=&#34;scale:0.6&#34;/&gt; &lt;/center&gt; &lt;/div&gt; &lt;p&gt; Let us suppose that [mathjaxinline]\phi _ i[/mathjaxinline] can be exactly specified with [mathjaxinline]t[/mathjaxinline] decimal digits in a way that [mathjaxinline]\phi _ i = 0.\phi _{i,1}\phi _{i,2}\dots \phi _{i,t}[/mathjaxinline]. Note that it is the decimal for the binary representation, e.g. [mathjaxinline]1/2 = 0.1, 1/4 = 0.01[/mathjaxinline] etc. &lt;/p&gt; &lt;p&gt; Let [mathjaxinline]U=S[/mathjaxinline]. What is the smallest [mathjaxinline]t[/mathjaxinline] needed to exactly specify [mathjaxinline]\phi[/mathjaxinline]? If we take such [mathjaxinline]t[/mathjaxinline], and take [mathjaxinline]|{1}\rangle[/mathjaxinline] for the [mathjaxinline]|{u}\rangle[/mathjaxinline] in the figure, what is the state of the first register after the inverse QFT? Answer in the ket notation. This state should be [mathjaxinline]t_{min}[/mathjaxinline]-qubit state where [mathjaxinline]t_{min}[/mathjaxinline] is the answer you got. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;minimum [mathjaxinline]t[/mathjaxinline]:&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Phase_Estimation_III_2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Phase_Estimation_III_2_1&#34; id=&#34;input_Phase_Estimation_III_2_1&#34; aria-describedby=&#34;status_Phase_Estimation_III_2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Phase_Estimation_III_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Phase_Estimation_III_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_Phase_Estimation_III_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Phase_Estimation_III_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Phase_Estimation_III_2_1_dynamath&#34; name=&#34;input_Phase_Estimation_III_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;state after [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_Phase_Estimation_III_3_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_Phase_Estimation_III_3_1&#34; id=&#34;input_Phase_Estimation_III_3_1&#34; aria-describedby=&#34;status_Phase_Estimation_III_3_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_Phase_Estimation_III_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_Phase_Estimation_III_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_Phase_Estimation_III_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_Phase_Estimation_III_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_Phase_Estimation_III_3_1_dynamath&#34; name=&#34;input_Phase_Estimation_III_3_1_dynamath&#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_Phase_Estimation_III_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;html url_name=&#34;site_search_box3&#34;/&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Phase Estimation 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_Phase_Estimation_III&#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_Phase_Estimation_III&#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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