<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Two-qubit_amplitude_damping_code" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-graded="True"> <h2 class="hd hd-2 unit-title">Two-qubit amplitude damping code</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk2-adamp-code" class="problems-wrapper" role="group" aria-labelledby="s12-wk2-adamp-code-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code/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-wk2-adamp-code-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Two-qubit amplitude damping code &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Amplitude damping is an important process in real physical systems; it models spontaneous emission, inelastic scattering, thermalization of spins to the lattice, and many other microscopic processes where energy is exchanged between the system and environment. In this problem, we study a quantum code adapted for this error mechanism. &lt;/p&gt; &lt;p&gt; Recall that the amplitude damping channel for a single qubit is described by [mathjaxinline]{\cal E}(\rho ) = \sum _ k E_ k \rho E_ k^\dagger[/mathjaxinline], where the operation elements are &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]E_0 = \left[ \begin{array}{cc} {1} &amp;amp; {0}\\ {0} &amp;amp; \sqrt{1-g} \end{array} \right] ~ ~ ~ ~ ~ ~ ~ ~ E_1 = \left[ \begin{array}{cc} {0} &amp;amp; {\sqrt{g}}\\ {0} &amp;amp; {0} \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; You may think of [mathjaxinline]g[/mathjaxinline] as being [mathjaxinline]g = 1-e^{-t/T_1}[/mathjaxinline], where [mathjaxinline]t[/mathjaxinline] is time and [mathjaxinline]T_1[/mathjaxinline] is the amplitude damping time constant. &lt;/p&gt; &lt;p&gt; Let [mathjaxinline]|0_ L\rangle = |01\rangle[/mathjaxinline] and [mathjaxinline]|1_ L\rangle = |10\rangle[/mathjaxinline] be a quantum code encoding one logical qubit using two physical qubits. Define [mathjaxinline]|\psi \rangle = a|0_ L\rangle + b |1_ L\rangle[/mathjaxinline]. Compute the output state &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]\rho &#39; = {\cal E}(|\psi \rangle ) = \sum _{j,k=\{ 0,1\} } (E_ j \otimes E_ k)\, |\psi \rangle \langle \psi | \, (E_ j\otimes E_ k)^\dagger[/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; which results when each physical qubit is subject to amplitude damping. &amp;#160;&lt;br/&gt;&lt;/p&gt; &lt;p&gt; Please enter your answer as a matrix, ie a list of lists, of this form &lt;tt class=&#34;tt&#34;&gt;[ [ 1,0,0,0 ], [0,1,0,0], [0,0,1,0], [0,0,0,1] ]&lt;/tt&gt;. Assume that [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] are real-valued. Use &lt;tt class=&#34;tt&#34;&gt;^&lt;/tt&gt; for exponents, and explicitly include the multiplication operator &lt;tt class=&#34;tt&#34;&gt;*&lt;/tt&gt;. You may use standard functions like &lt;tt class=&#34;tt&#34;&gt;sqrt&lt;/tt&gt;. Express the results in terms of [mathjaxinline]g[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], and [mathjaxinline]b[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\rho &#39; =[/mathjaxinline]&lt;/p&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-wk2-adamp-code_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk2-adamp-code_2_1&#34; id=&#34;input_s12-wk2-adamp-code_2_1&#34; aria-describedby=&#34;status_s12-wk2-adamp-code_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;80&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk2-adamp-code_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk2-adamp-code_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-wk2-adamp-code_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk2-adamp-code_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk2-adamp-code_2_1_dynamath&#34; name=&#34;input_s12-wk2-adamp-code_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_s12-wk2-adamp-code_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;Two-qubit amplitude damping code&#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-wk2-adamp-code&#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-wk2-adamp-code&#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-wk2-adamp-code-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-wk2-adamp-code-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-wk2-adamp-code-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-wk2-adamp-code2"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code2" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk2-adamp-code2" class="problems-wrapper" role="group" aria-labelledby="s12-wk2-adamp-code2-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code2" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code2/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-wk2-adamp-code2-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code2-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Two-qubit amplitude damping code II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code2-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; In class we saw that distance between states could be measured by the trace distance. An alternate is the &lt;em&gt;fidelity&lt;/em&gt;, which is equal to 1 for two states that are equal and 0 for two orthogonal states. In this way, it generalizes the inner product to density matrices. More information (not needed to complete this problem) on fidelity can be found &lt;a href=&#34;https://en.wikipedia.org/wiki/Fidelity_of_quantum_states&#34; target=&#34;_blank&#34;&gt;here&lt;/a&gt;. &lt;/p&gt; &lt;p&gt; Compute the fidelity [mathjaxinline]F(|\psi \rangle ,\rho &#39;) = \sqrt{\langle \psi |\rho &#39;|\psi \rangle }[/mathjaxinline] of [mathjaxinline]\rho &#39;[/mathjaxinline] with respect to [mathjaxinline]|\psi \rangle[/mathjaxinline]. Express the result as a function of [mathjaxinline]g[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]F(|\psi \rangle ,\rho &#39;) =[/mathjaxinline]&lt;/p&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-wk2-adamp-code2_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk2-adamp-code2_2_1&#34; id=&#34;input_s12-wk2-adamp-code2_2_1&#34; aria-describedby=&#34;status_s12-wk2-adamp-code2_2_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-wk2-adamp-code2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk2-adamp-code2_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-wk2-adamp-code2_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk2-adamp-code2_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk2-adamp-code2_2_1_dynamath&#34; name=&#34;input_s12-wk2-adamp-code2_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_s12-wk2-adamp-code2_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;Two-qubit amplitude damping code 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-wk2-adamp-code2&#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-wk2-adamp-code2&#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-wk2-adamp-code2-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-wk2-adamp-code2-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-wk2-adamp-code2-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-wk2-adamp-code3"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code3" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk2-adamp-code3" class="problems-wrapper" role="group" aria-labelledby="s12-wk2-adamp-code3-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code3" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code3/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-wk2-adamp-code3-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code3-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Two-qubit amplitude damping code III &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk2-adamp-code3-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose we project the output state into the space orthogonal to [mathjaxinline]|00\rangle[/mathjaxinline] (say by performing a measurement of [mathjaxinline]Z\otimes I + I \otimes Z[/mathjaxinline] to measure the total excitation number), and keep only the cases when we do not obtain [mathjaxinline]|00\rangle[/mathjaxinline]. What is the resulting state? &lt;/p&gt; &lt;p&gt; Enter your answer using &amp;#8220;ket&#34; notation, e.g. [mathjaxinline]|0\rangle[/mathjaxinline] is [mathjaxinline]{\tt |0&amp;gt;}[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;state =&lt;/p&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-wk2-adamp-code3_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/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk2-adamp-code3_2_1&#34; id=&#34;input_s12-wk2-adamp-code3_2_1&#34; aria-describedby=&#34;status_s12-wk2-adamp-code3_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk2-adamp-code3_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk2-adamp-code3_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-wk2-adamp-code3_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk2-adamp-code3_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk2-adamp-code3_2_1_dynamath&#34; name=&#34;input_s12-wk2-adamp-code3_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_s12-wk2-adamp-code3_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;Two-qubit amplitude damping code 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-wk2-adamp-code3&#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-wk2-adamp-code3&#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-wk2-adamp-code3-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-wk2-adamp-code3-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-wk2-adamp-code3-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-3" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box004113"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box004113" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <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>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Fun_with_stabilizers_I_Groups_and_generators" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-graded="True"> <h2 class="hd hd-2 unit-title">Fun with stabilizers I: Groups and generators</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-stab-group" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-stab-group-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group/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-wk3-stab-group-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Groups and generators &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Stabilizers are a powerful way to describe quantum states and transformations. In this problem we explore some basic properties of stabilizer sets, which arise from the fact that a stabilizer is an &lt;a href=&#34;http://en.wikipedia.org/wiki/Abelian_group&#34; target=&#34;_blank&#34;&gt;Abelian group&lt;/a&gt;. &lt;/p&gt; &lt;p&gt; Recall that if an [mathjaxinline]n[/mathjaxinline]-qubit state [mathjaxinline]|\psi \rangle[/mathjaxinline] is stabilized by [mathjaxinline]{\cal S} = \langle g_1, g_2, \ldots , g_ n \rangle[/mathjaxinline], then [mathjaxinline]g|\psi \rangle = |\psi \rangle[/mathjaxinline] for all [mathjaxinline]g\in S[/mathjaxinline]. To verify that [mathjaxinline]{\cal S}[/mathjaxinline] is an abelian group, we can observe that: &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]I^{\otimes n}[/mathjaxinline] is the identity element &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]g[/mathjaxinline] has an inverse: [mathjaxinline]gg^\dagger = I^{\otimes n}[/mathjaxinline] (up to factors of [mathjaxinline]\pm 1[/mathjaxinline] and [mathjaxinline]\pm i[/mathjaxinline], [mathjaxinline]g^\dagger = g[/mathjaxinline]) &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; Multiplication of stabilizer elements is commutative and associative &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; The set is closed under multiplication &lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;p&gt; In fact, [mathjaxinline]{\cal S}[/mathjaxinline] is a subgroup of the Pauli group, distinguished by an important fact: all elements of the stabilizer must commute with each other. It is because of this that all of the elements may share simultaneous eigenvectors. &lt;/p&gt; &lt;p&gt; You are given that [mathjaxinline]{\cal S} = \{ {\tt XXIZ}, {\tt YXIY}, {\tt IZIX}, {\tt ZZII}, {\tt -YYIZ}, {\tt XYIY}, {\tt ZIIX}, {\tt IIII}\}[/mathjaxinline]. Give a minimal list of elements of [mathjaxinline]{\cal S}[/mathjaxinline] which can be combined by multiplication to produce all of the elements in [mathjaxinline]{\cal S}[/mathjaxinline]. We say that such a list is a set of &lt;em&gt;generators&lt;/em&gt; of [mathjaxinline]{\cal S}[/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_s12-wk3-stab-group_2_1&#34; class=&#34; capa_inputtype textline&#34;&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-stab-group_2_1&#34; id=&#34;input_s12-wk3-stab-group_2_1&#34; aria-describedby=&#34;status_s12-wk3-stab-group_2_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab-group_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab-group_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-wk3-stab-group_2_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-wk3-stab-group_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;Groups and generators&#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-wk3-stab-group&#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-wk3-stab-group&#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-wk3-stab-group-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-wk3-stab-group-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-wk3-stab-group-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-wk3-stab-group2"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group2" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-stab-group2" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-stab-group2-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group2" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group2/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-wk3-stab-group2-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group2-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Groups and generators II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group2-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; You are given that [mathjaxinline]\{ {\tt XIX},{\tt ZIY}\}[/mathjaxinline] generate [mathjaxinline]{\cal S}[/mathjaxinline]. Give a list of additional operators which are also in [mathjaxinline]{\cal S}[/mathjaxinline]. Your input should be a list of operators, eg of the form &lt;tt class=&#34;tt&#34;&gt;[XXX,YYY]&lt;/tt&gt;: &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_s12-wk3-stab-group2_2_1&#34; class=&#34; capa_inputtype textline&#34;&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-stab-group2_2_1&#34; id=&#34;input_s12-wk3-stab-group2_2_1&#34; aria-describedby=&#34;status_s12-wk3-stab-group2_2_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab-group2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab-group2_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-wk3-stab-group2_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&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;Groups and generators 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-wk3-stab-group2&#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-wk3-stab-group2&#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-wk3-stab-group2-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-wk3-stab-group2-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-wk3-stab-group2-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-wk3-stab-group3"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group3" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-stab-group3" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-stab-group3-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group3" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group3/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-wk3-stab-group3-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group3-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Groups and generators III &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab-group3-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; If [mathjaxinline]{\cal S}[/mathjaxinline] acts on an [mathjaxinline]n[/mathjaxinline]-qubit space, and the minimal generator set for [mathjaxinline]{\cal S}[/mathjaxinline] has [mathjaxinline]d[/mathjaxinline] elements, then what is the dimension of the vector space which is stabilized by [mathjaxinline]{\cal S}[/mathjaxinline]? &amp;#160;&lt;br/&gt;&amp;#160;&lt;br/&gt;Give your answer in terms of [mathjaxinline]n[/mathjaxinline] and [mathjaxinline]d[/mathjaxinline]. If necessary, remember to use &lt;tt class=&#34;tt&#34;&gt;^&lt;/tt&gt; to indicate exponentiation. &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_s12-wk3-stab-group3_2_1&#34; class=&#34; capa_inputtype textline&#34;&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-stab-group3_2_1&#34; id=&#34;input_s12-wk3-stab-group3_2_1&#34; aria-describedby=&#34;status_s12-wk3-stab-group3_2_1&#34; value=&#34;&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab-group3_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab-group3_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-wk3-stab-group3_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&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;Groups and generators 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-wk3-stab-group3&#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-wk3-stab-group3&#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-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Fun_with_stabilizers_II_Elementary_stabilizers" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-graded="True"> <h2 class="hd hd-2 unit-title">Fun with stabilizers II: Elementary stabilizers</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab0"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab0" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-stab0" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-stab0-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab0" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab0/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="7" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;s12-wk3-stab0-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab0-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Elementary stabilizers &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab0-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Stabilizers are one of the most useful way to describe states and transformations in quantum information. In this problem we epxlore some descriptions of basic stabilizer states. &lt;/p&gt; &lt;p&gt; Recall that if an [mathjaxinline]n[/mathjaxinline]-qubit state [mathjaxinline]|\psi \rangle[/mathjaxinline] is stabilized by [mathjaxinline]S = \langle g_1, g_2, \ldots , g_ n \rangle[/mathjaxinline], then [mathjaxinline]g|\psi \rangle = |\psi \rangle[/mathjaxinline] for all [mathjaxinline]g\in S[/mathjaxinline]. Note that [mathjaxinline]g_1, g_2, \ldots , g_ n[/mathjaxinline] are the &lt;em&gt;generators&lt;/em&gt; of [mathjaxinline]S[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; Give stabilizer generator sets for the following states (normalizations suppressed). Enter each set as a list of stabilizers, eg &lt;tt class=&#34;tt&#34;&gt;[XI,IX]&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]|0\rangle +i|1\rangle ~[/mathjaxinline]: &lt;/p&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-wk3-stab0_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-wk3-stab0_2_1&#34; id=&#34;input_s12-wk3-stab0_2_1&#34; aria-describedby=&#34;status_s12-wk3-stab0_2_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab0_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab0_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-wk3-stab0_2_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-wk3-stab0_solution_1&#34;/&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]|1\rangle ~[/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-wk3-stab0_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-wk3-stab0_3_1&#34; id=&#34;input_s12-wk3-stab0_3_1&#34; aria-describedby=&#34;status_s12-wk3-stab0_3_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab0_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab0_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-wk3-stab0_3_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-wk3-stab0_solution_2&#34;/&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]|00\rangle +|11\rangle ~[/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-wk3-stab0_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-wk3-stab0_4_1&#34; id=&#34;input_s12-wk3-stab0_4_1&#34; aria-describedby=&#34;status_s12-wk3-stab0_4_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab0_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab0_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-wk3-stab0_4_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-wk3-stab0_solution_3&#34;/&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]|00\rangle -|11\rangle ~[/mathjaxinline]: &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-wk3-stab0_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-wk3-stab0_5_1&#34; id=&#34;input_s12-wk3-stab0_5_1&#34; aria-describedby=&#34;status_s12-wk3-stab0_5_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab0_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab0_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-wk3-stab0_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-wk3-stab0_solution_4&#34;/&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]|01\rangle +|10\rangle ~[/mathjaxinline]: &lt;/p&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-wk3-stab0_6_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-wk3-stab0_6_1&#34; id=&#34;input_s12-wk3-stab0_6_1&#34; aria-describedby=&#34;status_s12-wk3-stab0_6_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab0_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab0_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-wk3-stab0_6_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-wk3-stab0_solution_5&#34;/&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]|01\rangle -|10\rangle ~[/mathjaxinline]: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk3-stab0_7_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-wk3-stab0_7_1&#34; id=&#34;input_s12-wk3-stab0_7_1&#34; aria-describedby=&#34;status_s12-wk3-stab0_7_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab0_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab0_7_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-wk3-stab0_7_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-wk3-stab0_solution_6&#34;/&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]|000\rangle +|111\rangle ~[/mathjaxinline]: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 7&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_s12-wk3-stab0_8_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-wk3-stab0_8_1&#34; id=&#34;input_s12-wk3-stab0_8_1&#34; aria-describedby=&#34;status_s12-wk3-stab0_8_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab0_8_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab0_8_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-wk3-stab0_8_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-wk3-stab0_solution_7&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Elementary stabilizers&#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-wk3-stab0&#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-wk3-stab0&#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-wk3-stab0-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-wk3-stab0-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-wk3-stab0-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-wk3-stab02"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab02" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-stab02" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-stab02-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab02" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab02/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-wk3-stab02-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab02-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Elementary stabilizers II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab02-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Give stabilizer generator sets for the following &lt;em&gt;vector spaces&lt;/em&gt;, specified by the basis sets given. Enter each stabilizer generator set as a list of stabilizers, eg &lt;tt class=&#34;tt&#34;&gt;[XI,IX]&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]|001\rangle[/mathjaxinline], [mathjaxinline]|110\rangle[/mathjaxinline]}&lt;/p&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-wk3-stab02_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-wk3-stab02_2_1&#34; id=&#34;input_s12-wk3-stab02_2_1&#34; aria-describedby=&#34;status_s12-wk3-stab02_2_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab02_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab02_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-wk3-stab02_2_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-wk3-stab02_solution_1&#34;/&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]|00\rangle +|11\rangle[/mathjaxinline], [mathjaxinline]|01\rangle +|10\rangle[/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-wk3-stab02_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-wk3-stab02_3_1&#34; id=&#34;input_s12-wk3-stab02_3_1&#34; aria-describedby=&#34;status_s12-wk3-stab02_3_1&#34; value=&#34;&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab02_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab02_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-wk3-stab02_3_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-wk3-stab02_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Elementary stabilizers 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-wk3-stab02&#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-wk3-stab02&#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-wk3-stab02-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-wk3-stab02-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-wk3-stab02-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@html+block@site_search_box009135"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box009135" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <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>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Fun_with_stabilizers_III_Four-qubit_stabilizer_state" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-graded="True"> <h2 class="hd hd-2 unit-title">Fun with stabilizers III: Four-qubit stabilizer state</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab1"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab1" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-stab1" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-stab1-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab1" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab1/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-wk3-stab1-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab1-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Four-qubit stabilizer state &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-stab1-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider the stabilizer generator set &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;a0000000009&#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 g_1[/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 IIZZ}[/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;a0000000010&#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 g_2[/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 ZZII}[/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;tr id=&#34;a0000000011&#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 g_3[/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 XXXX} \, .[/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;(1.8)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Note that these stabilizers act on a Hilbert space of four qubits. &lt;/p&gt; &lt;p&gt; What is the dimension of the space stabilized by [mathjaxinline]S=\langle g_1,g_2,g_3\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_s12-wk3-stab1_2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-stab1_2_1&#34; id=&#34;input_s12-wk3-stab1_2_1&#34; aria-describedby=&#34;status_s12-wk3-stab1_2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-stab1_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab1_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-wk3-stab1_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk3-stab1_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk3-stab1_2_1_dynamath&#34; name=&#34;input_s12-wk3-stab1_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&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;Four-qubit stabilizer state&#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-wk3-stab1&#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-wk3-stab1&#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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Please input your answer using &amp;#8220;ket&#34; notation, eg &lt;tt class=&#34;tt&#34;&gt;(|0011&amp;gt;+|0000&amp;gt;)/sqrt(2)&lt;/tt&gt;. The two states should be orthogonal to each other. They need not be properly normalized. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;multi-inputs-group&#34; role=&#34;group&#34;&gt;&lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]|0_ L\rangle =[/mathjaxinline] &lt;div id=&#34;inputtype_s12-wk3-stab12_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/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-stab12_2_1&#34; id=&#34;input_s12-wk3-stab12_2_1&#34; aria-describedby=&#34;status_s12-wk3-stab12_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-wk3-stab12_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab12_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-wk3-stab12_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk3-stab12_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk3-stab12_2_1_dynamath&#34; name=&#34;input_s12-wk3-stab12_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&gt; &lt;br/&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]|1_ L\rangle =[/mathjaxinline] &lt;div id=&#34;inputtype_s12-wk3-stab12_2_2&#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/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-stab12_2_2&#34; id=&#34;input_s12-wk3-stab12_2_2&#34; aria-describedby=&#34;status_s12-wk3-stab12_2_2&#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-wk3-stab12_2_2&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-stab12_2_2&#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-wk3-stab12_2_2&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk3-stab12_2_2&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk3-stab12_2_2_dynamath&#34; name=&#34;input_s12-wk3-stab12_2_2_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&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-wk3-stab12_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;Four-qubit stabilizer state 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-wk3-stab12&#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-wk3-stab12&#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-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Fun_with_stabilizers_IV_Action_of_quantum_gates_on_stabilizer_states" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-graded="True"> <h2 class="hd hd-2 unit-title">Fun with stabilizers IV: Action of quantum gates on stabilizer states</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-heis-circ" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-heis-circ-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ/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-wk3-heis-circ-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Action of quantum gates on stabilizer states &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose [mathjaxinline]|\psi \rangle[/mathjaxinline] is stabilized by [mathjaxinline]{\cal S}[/mathjaxinline], such that [mathjaxinline]g|\psi \rangle = |\psi \rangle[/mathjaxinline] for all [mathjaxinline]g\in {\cal S}[/mathjaxinline]. Then [mathjaxinline]|\phi \rangle = U|\psi \rangle[/mathjaxinline] is stabilized by [mathjaxinline]{\cal S}&#39; = U{\cal S}U^\dagger = \{ UgU^\dagger | g\in {\cal S}\}[/mathjaxinline], since &lt;/p&gt; &lt;table id=&#34;a0000000015&#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]UgU^\dagger |\phi \rangle = UgU^\dagger U|\psi \rangle = U |\psi \rangle = |\phi \rangle \, .[/mathjax]&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;/table&gt; &lt;p&gt; Thus, we may often gain deep appreciation into the operation of a quantum circuit [mathjaxinline]U[/mathjaxinline] by studying how it transforms stabilizer sets (by conjugation), rather than how it transforms general quantum states (by multiplication). This is analogous to the difference between the Heisenberg and Schr&amp;#246;dinger pictures of quantum mechanics. &lt;/p&gt; &lt;p&gt; Consider the vector space [mathjaxinline]V = \{ |01\rangle +|10\rangle ,|00\rangle +|11\rangle \}[/mathjaxinline] stabilized by the generator set [mathjaxinline]{\cal S} = \langle XX\rangle[/mathjaxinline]. Give the new stabilizer [mathjaxinline]{\cal S}&#39;[/mathjaxinline] resulting from operating on this two-qubit vector space with the following unitary gates. If the result is &lt;em&gt;not&lt;/em&gt; a stabilizer space, enter &lt;tt class=&#34;tt&#34;&gt;None&lt;/tt&gt;. Otherwise, enter the stabilizer as a list of generators, eg &lt;tt class=&#34;tt&#34;&gt;[IX,XI]&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]U = H^{\otimes 2} = H\otimes H[/mathjaxinline]:&lt;/p&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-wk3-heis-circ_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-wk3-heis-circ_2_1&#34; id=&#34;input_s12-wk3-heis-circ_2_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ_2_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-heis-circ_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ_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-wk3-heis-circ_2_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]U = S^{\otimes 2} = S\otimes S[/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-wk3-heis-circ_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-wk3-heis-circ_3_1&#34; id=&#34;input_s12-wk3-heis-circ_3_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ_3_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-heis-circ_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ_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-wk3-heis-circ_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]U = T^{\otimes 2} = T\otimes T[/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-wk3-heis-circ_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-wk3-heis-circ_4_1&#34; id=&#34;input_s12-wk3-heis-circ_4_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ_4_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-heis-circ_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ_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-wk3-heis-circ_4_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]U = Z^{\otimes 2} = Z\otimes Z[/mathjaxinline]:&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-wk3-heis-circ_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-wk3-heis-circ_5_1&#34; id=&#34;input_s12-wk3-heis-circ_5_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ_5_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-heis-circ_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ_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-wk3-heis-circ_5_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]U = {\rm cnot} = ((I+Z)\otimes I + (I-Z)\otimes X)/2[/mathjaxinline]:&lt;/p&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-wk3-heis-circ_6_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-wk3-heis-circ_6_1&#34; id=&#34;input_s12-wk3-heis-circ_6_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ_6_1&#34; 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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-wk3-heis-circ-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; 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&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-wk3-heis-circ2"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ2" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-heis-circ2" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-heis-circ2-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ2" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ2/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-wk3-heis-circ2-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ2-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Action of quantum gates on stabilizer states II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-heis-circ2-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider the vector space [mathjaxinline]V[/mathjaxinline] stabilized by [mathjaxinline]{\cal S} = \langle g_1,g_2,g_3\rangle[/mathjaxinline], where &lt;/p&gt; &lt;table id=&#34;a0000000016&#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;a0000000017&#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 g_1[/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 IIZZ}[/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;tr id=&#34;a0000000018&#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 g_2[/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 ZZII}[/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;(1.13)&lt;/td&gt; &lt;/tr&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 g_3[/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 XXXX} \, .[/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;/table&gt; &lt;p&gt; Give the new stabilizer [mathjaxinline]{\cal S}&#39;[/mathjaxinline] resulting from operating on this four qubit vector space with the following unitary gates. If the result is &lt;em&gt;not&lt;/em&gt; a stabilizer space, enter &lt;tt class=&#34;tt&#34;&gt;None&lt;/tt&gt;. Otherwise, enter the stabilizer as a list of generators, eg &lt;tt class=&#34;tt&#34;&gt;[IX,XI]&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]U_1 = H^{\otimes 4}[/mathjaxinline]:&lt;/p&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-wk3-heis-circ2_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-wk3-heis-circ2_2_1&#34; id=&#34;input_s12-wk3-heis-circ2_2_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ2_2_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-heis-circ2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ2_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-wk3-heis-circ2_2_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]U_2 = S^{\otimes 4}[/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-wk3-heis-circ2_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-wk3-heis-circ2_3_1&#34; id=&#34;input_s12-wk3-heis-circ2_3_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ2_3_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-heis-circ2_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ2_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-wk3-heis-circ2_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]U_3 = Z^{\otimes 4} = Z\otimes Z \otimes Z \otimes Z[/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-wk3-heis-circ2_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-wk3-heis-circ2_4_1&#34; id=&#34;input_s12-wk3-heis-circ2_4_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ2_4_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-heis-circ2_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ2_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-wk3-heis-circ2_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;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Action of quantum gates on stabilizer states 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-wk3-heis-circ2&#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-wk3-heis-circ2&#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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&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]U_1 {\cal S} U_1^\dagger = {\cal S}[/mathjaxinline]: True?&amp;#160;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;inputtype option-input inline&#34;&gt; &lt;select name=&#34;input_s12-wk3-heis-circ3_2_1&#34; id=&#34;input_s12-wk3-heis-circ3_2_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ3_2_1&#34;&gt; &lt;option value=&#34;option_s12-wk3-heis-circ3_2_1_dummy_default&#34;&gt;Select an option&lt;/option&gt; &lt;option value=&#34;true&#34;&gt; true&lt;/option&gt; &lt;option value=&#34;false&#34;&gt; false&lt;/option&gt; &lt;/select&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ3_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;/div&gt; &lt;p class=&#34;answer&#34; id=&#34;answer_s12-wk3-heis-circ3_2_1&#34;/&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]U_2 {\cal S} U_2^\dagger = {\cal S}[/mathjaxinline]: True?&amp;#160;&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 class=&#34;inputtype option-input inline&#34;&gt; &lt;select name=&#34;input_s12-wk3-heis-circ3_3_1&#34; id=&#34;input_s12-wk3-heis-circ3_3_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ3_3_1&#34;&gt; &lt;option value=&#34;option_s12-wk3-heis-circ3_3_1_dummy_default&#34;&gt;Select an option&lt;/option&gt; &lt;option value=&#34;true&#34;&gt; true&lt;/option&gt; &lt;option value=&#34;false&#34;&gt; false&lt;/option&gt; &lt;/select&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ3_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;/div&gt; &lt;p class=&#34;answer&#34; id=&#34;answer_s12-wk3-heis-circ3_3_1&#34;/&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]U_3 {\cal S} U_3^\dagger = {\cal S}[/mathjaxinline]: True?&amp;#160;&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 class=&#34;inputtype option-input inline&#34;&gt; &lt;select name=&#34;input_s12-wk3-heis-circ3_4_1&#34; id=&#34;input_s12-wk3-heis-circ3_4_1&#34; aria-describedby=&#34;status_s12-wk3-heis-circ3_4_1&#34;&gt; &lt;option value=&#34;option_s12-wk3-heis-circ3_4_1_dummy_default&#34;&gt;Select an option&lt;/option&gt; &lt;option value=&#34;true&#34;&gt; true&lt;/option&gt; &lt;option value=&#34;false&#34;&gt; false&lt;/option&gt; &lt;/select&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-heis-circ3_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;/div&gt; &lt;p class=&#34;answer&#34; id=&#34;answer_s12-wk3-heis-circ3_4_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;/ul&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Action of quantum gates on stabilizer states 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-wk3-heis-circ3&#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-wk3-heis-circ3&#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-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Fun_with_stabilizers_V_Stabilizer_description_of_quantum_circuit_transformations" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-graded="True"> <h2 class="hd hd-2 unit-title">Fun with stabilizers V: Stabilizer description of quantum circuit transformations</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-circ-stab"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-circ-stab" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-circ-stab" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-circ-stab-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-circ-stab" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-circ-stab/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-wk3-circ-stab-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-circ-stab-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Stabilizer description of quantum circuit transformations &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-circ-stab-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider the following simple quantum circuit: &lt;/p&gt; &lt;p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/7b4bf266c4621c3b33a8acefa278c7dd/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/images_bell-circuit-stab-v2.png&#34; width=&#34;300&#34;/&gt; &lt;/center&gt; &lt;/p&gt; &lt;p&gt; The input state is [mathjaxinline]|00\rangle[/mathjaxinline], which is stabilized by [mathjaxinline]{\cal S}_0 = \langle {\tt IZ},{\tt ZI}\rangle[/mathjaxinline]. Give the stabilizers describing the state after the Hadamard and after the controlled-NOT gate. &lt;/p&gt; &lt;p&gt; Work this out by using the fact that [mathjaxinline]U[/mathjaxinline] acting on a state stabilized by [mathjaxinline]{\cal S}[/mathjaxinline] produces a state stabilized by [mathjaxinline]U{\cal S}U^\dagger[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; You can also double-check yourself by working out the explicit quantum states produced in the circuit, and figuring out the stabilizers for each of the states. However, for circuits composed of &lt;em&gt;Clifford group&lt;/em&gt; gates, namely circuits generated by [mathjaxinline]H[/mathjaxinline], [mathjaxinline]S[/mathjaxinline], and the controlled-NOT gate, it is much faster to work directly with the stabilizer. &lt;/p&gt; &lt;p&gt; Enter each stabilizer as a list of generators, eg &lt;tt class=&#34;tt&#34;&gt;[IX,XI]&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]{\cal S_1} = ~[/mathjaxinline]: &lt;/p&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-wk3-circ-stab_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-wk3-circ-stab_2_1&#34; id=&#34;input_s12-wk3-circ-stab_2_1&#34; aria-describedby=&#34;status_s12-wk3-circ-stab_2_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-circ-stab_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-circ-stab_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-wk3-circ-stab_2_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]{\cal S_2} = ~[/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-wk3-circ-stab_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-wk3-circ-stab_3_1&#34; id=&#34;input_s12-wk3-circ-stab_3_1&#34; aria-describedby=&#34;status_s12-wk3-circ-stab_3_1&#34; value=&#34;&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-circ-stab_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-circ-stab_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-wk3-circ-stab_3_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;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Stabilizer description of quantum circuit transformations&#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-wk3-circ-stab&#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-wk3-circ-stab&#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-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Topological_QEC_-_A_Projective_Plane_Code" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="VerticalStudentView" data-runtime-class="LmsRuntime" data-graded="True"> <h2 class="hd hd-2 unit-title">Topological QEC - A Projective Plane Code</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-ppcode1" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-ppcode1-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1/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-wk3-ppcode1-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Topological QEC - A Projective Plane Code &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Quantum error correction codes can be constructed using graphs on topological surfaces, where vertices and edges correspond to certain stabilizer operations, and the distance of the code is given by the shortest non-trivial topological chain of errors. In this problem, we explore a specific instance of such a construction, on a &lt;a href=&#34;http://en.wikipedia.org/wiki/Projective_plane&#34; target=&#34;_blank&#34;&gt;projective plane&lt;/a&gt;. &lt;/p&gt; &lt;p&gt; The projective plane in two dimensions, [mathjaxinline]\Re P^2[/mathjaxinline], can be drawn as a disc in which antipodal points on the boundary are identified. Recall that a cellulation [mathjaxinline]{\cal C}[/mathjaxinline] of a surface defines sets [mathjaxinline]F[/mathjaxinline], [mathjaxinline]E[/mathjaxinline], and [mathjaxinline]V[/mathjaxinline] of faces, edges, and vertices. For each [mathjaxinline]e\in E[/mathjaxinline], there corresponds a qubit on which [mathjaxinline]X_ e[/mathjaxinline] and [mathjaxinline]Z_ e[/mathjaxinline] are the Pauli [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] operators. Let [mathjaxinline]E_ f \subset E[/mathjaxinline] be the set of edges around face [mathjaxinline]f\in F[/mathjaxinline], and let [mathjaxinline]E_ v \subset E[/mathjaxinline] be the set of edges attached to vertex [mathjaxinline]v\in V[/mathjaxinline]. For the set of all [mathjaxinline]f[/mathjaxinline] and [mathjaxinline]v[/mathjaxinline], we define &lt;/p&gt; &lt;table id=&#34;a0000000020&#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]A_ f = \bigotimes _{e\in E_ f} Z_ e[/mathjax]&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; and &lt;/p&gt; &lt;table id=&#34;a0000000021&#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]B_ v = \bigotimes _{e\in E_ v} X_ e \, .[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.16)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; The set of all [mathjaxinline]A_ f[/mathjaxinline] and [mathjaxinline]B_ v[/mathjaxinline] is the stabilizer for the code. &lt;/p&gt; &lt;p&gt; Give a list of stabilizers (three [mathjaxinline]A_ f[/mathjaxinline] and the single unique [mathjaxinline]B_ v[/mathjaxinline]) for the cellulation: &lt;/p&gt; &lt;p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/389270d9a44cf14a27c6e1a67176c0e3/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/images_pplane1.png&#34; width=&#34;400&#34;/&gt; &lt;/center&gt; &lt;/p&gt; &lt;p&gt; This code acts on four qubits ([mathjaxinline]|E|=4[/mathjaxinline]), corresponding to the four edges, labeled [mathjaxinline]1[/mathjaxinline] through [mathjaxinline]4[/mathjaxinline]. Note there are only three distinct faces, labeled [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]. Be sure to verify that the stabilizers you give commute with each other. &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_s12-wk3-ppcode1_2_1&#34; class=&#34; capa_inputtype textline&#34;&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-ppcode1_2_1&#34; id=&#34;input_s12-wk3-ppcode1_2_1&#34; aria-describedby=&#34;status_s12-wk3-ppcode1_2_1&#34; value=&#34;&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_s12-wk3-ppcode1_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-ppcode1_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-wk3-ppcode1_2_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-wk3-ppcode1_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;Topological QEC - A Projective Plane Code&#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-wk3-ppcode1&#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-wk3-ppcode1&#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-wk3-ppcode1-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-wk3-ppcode1-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-wk3-ppcode1-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-wk3-ppcode1-2"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1-2" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_s12-wk3-ppcode1-2" class="problems-wrapper" role="group" aria-labelledby="s12-wk3-ppcode1-2-problem-title" data-problem-id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1-2" data-url="/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1-2/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-wk3-ppcode1-2-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1-2-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Topological QEC - A Projective Plane Code II &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk3-ppcode1-2-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; What state(s) do they stabilize? &lt;/p&gt; &lt;p&gt; You may input your answer using &amp;#8220;ket&#34; notation, eg &lt;tt class=&#34;tt&#34;&gt;(|0011&amp;gt;+|0000&amp;gt;)/sqrt(2)&lt;/tt&gt;. Input boxes for two possible states are provided; enter &lt;tt class=&#34;tt&#34;&gt;None&lt;/tt&gt; if the state is not needed. If you enter more than one state, make sure they are orthogonal to each other. States need not be normalized. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;multi-inputs-group&#34; role=&#34;group&#34;&gt;&lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]|0_ L\rangle =[/mathjaxinline] &lt;div id=&#34;inputtype_s12-wk3-ppcode1-2_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/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-ppcode1-2_2_1&#34; id=&#34;input_s12-wk3-ppcode1-2_2_1&#34; aria-describedby=&#34;status_s12-wk3-ppcode1-2_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-wk3-ppcode1-2_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-ppcode1-2_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-wk3-ppcode1-2_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk3-ppcode1-2_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk3-ppcode1-2_2_1_dynamath&#34; name=&#34;input_s12-wk3-ppcode1-2_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&gt; &lt;br/&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]|1_ L\rangle =[/mathjaxinline] &lt;div id=&#34;inputtype_s12-wk3-ppcode1-2_2_2&#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/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_s12-wk3-ppcode1-2_2_2&#34; id=&#34;input_s12-wk3-ppcode1-2_2_2&#34; aria-describedby=&#34;status_s12-wk3-ppcode1-2_2_2&#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-wk3-ppcode1-2_2_2&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_s12-wk3-ppcode1-2_2_2&#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-wk3-ppcode1-2_2_2&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_s12-wk3-ppcode1-2_2_2&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_s12-wk3-ppcode1-2_2_2_dynamath&#34; name=&#34;input_s12-wk3-ppcode1-2_2_2_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&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-wk3-ppcode1-2_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;Topological QEC - A Projective Plane Code 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-wk3-ppcode1-2&#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-wk3-ppcode1-2&#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-wk3-ppcode1-2-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-wk3-ppcode1-2-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-wk3-ppcode1-2-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@html+block@site_search_box011142"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box011142" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-request-token="8dce2890abb511f180ff12541d743da1" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-graded="True"> <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>