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<h2 class="hd hd-2 unit-title">Projective measurement</h2>
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Projective measurement
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In this problem, we will consider indirect measurements where the probe interacts with the system and projective measurement is made on the probe. </p>
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Let us first take CNOT as an interaction between the system and probe as shown in the figure below. You will see that it is the circuit that indirectly carries out the projective measurement. Let [mathjaxinline]|{\psi }\rangle =a|{0}\rangle +b|{1}\rangle[/mathjaxinline] where [mathjaxinline]a,\ b[/mathjaxinline] are real. What is the probability of getting the result 0? And what is the output state in the system when the measurement result is 0? Answer in the ket notation. </p>
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<p style="display:inline">probability:</p>
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Suppose we do not learn the measurement result. Then, the output state is not a pure state anymore. It is now a mixed state, which is a probabilistic mixture of the states that correspond to two measurement outcomes. What is the density matrix for the output state in the system? Answer in a matrix form e.g. <tt class="ttfamily">[[0,1],[0,0]]</tt> for [mathjaxinline]|{0}\rangle \langle {1}|[/mathjaxinline], and in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">Phase damping and phase flip</h2>
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Phase damping and phase flip
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What is the output mixed state in the system for the circuit shown below? [mathjaxinline]R_ y(\theta )[/mathjaxinline] is the rotation with [mathjaxinline]y[/mathjaxinline] axis by [mathjaxinline]\theta[/mathjaxinline]. Answer in a matrix form e.g. <tt class="ttfamily">[[0,1],[0,0]]</tt> for [mathjaxinline]|{0}\rangle \langle {1}|[/mathjaxinline], and in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. Do not forget to explicitly input the multiplication sign <tt class="ttfamily">*</tt>. It is known as the phase damping channel which models the effect of decoherence. </p>
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Let [mathjaxinline]\rho =|{\psi }\rangle \langle {\psi }|[/mathjaxinline] where [mathjaxinline]|{\psi }\rangle =a|{0}\rangle +b|{1}\rangle[/mathjaxinline]. Suppose we flip a biased coin which gives the head with probability [mathjaxinline]1-p[/mathjaxinline] and the tail with probability [mathjaxinline]p[/mathjaxinline]. We do nothing on [mathjaxinline]\rho[/mathjaxinline] if we get the head and apply [mathjaxinline]Z[/mathjaxinline] if we get the tail. What is the state after this process? Answer in a matrix form e.g. <tt class="ttfamily">[[0,1],[0,0]]</tt> for [mathjaxinline]|{0}\rangle \langle {1}|[/mathjaxinline], and in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p>
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Express [mathjaxinline]p[/mathjaxinline] in terms of [mathjaxinline]\cos (\theta /2)[/mathjaxinline] so that these two processes (the one with controlled [mathjaxinline]R_ y(\theta )[/mathjaxinline] and the one with coin flip) are equivalent (i.e. give the same output mixed state). It shows that the two different-looking quantum processes, phase damping and phase flip are actually equivalent. </p>
<p>
<p style="display:inline">[mathjaxinline]p=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">The 9-qubit code I</h2>
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The 9-qubit code
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<p>
The 9-qubit code is a quantum code that corrects any single qubit error. The codewords for the 9-qubit code are defined by </p>
<table id="a0000000007" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000008">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |{0_ L}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(\frac{|{000}\rangle +|{111}\rangle }{\sqrt{2}}\right)^{\otimes 3}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.5)</td>
</tr>
<tr id="a0000000009">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |{1_ L}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(\frac{|{000}\rangle -|{111}\rangle }{\sqrt{2}}\right)^{\otimes 3}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.6)</td>
</tr>
</table>
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<li>
<p>
What is the expectation value of the following observables for [mathjaxinline]|{1_ L}\rangle[/mathjaxinline]? Note that [mathjaxinline]Z_ i[/mathjaxinline] refers to the [mathjaxinline]Z[/mathjaxinline] operator acting on the [mathjaxinline]i[/mathjaxinline]th qubit etc. and identity operators are implicit e.g. [mathjaxinline]Z_1Z_2= Z\otimes Z \otimes I \otimes I\otimes I\otimes I\otimes I\otimes I\otimes I[/mathjaxinline]. </p>
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<p>
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<h2 class="hd hd-2 unit-title">The 9-qubit code</h2>
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<h3 class="hd hd-3 problem-header" id="The_9-qubit_code_II-problem-title" aria-describedby="block-v1:MITx+8.370.3x+1T2018+type@problem+block@The_9-qubit_code_II-problem-progress" tabindex="-1">
The 9-qubit code II
</h3>
<div class="problem-progress" id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@The_9-qubit_code_II-problem-progress"></div>
<div class="problem">
<div>
<p>
Continuing with our consideration of the 9-qubit code, a quantum code that corrects any single qubit error. The codewords for the 9-qubit code are defined by </p>
<table id="a0000000010" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000011">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |{0_ L}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(\frac{|{000}\rangle +|{111}\rangle }{\sqrt{2}}\right)^{\otimes 3}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.7)</td>
</tr>
<tr id="a0000000012">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |{1_ L}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(\frac{|{000}\rangle -|{111}\rangle }{\sqrt{2}}\right)^{\otimes 3}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.8)</td>
</tr>
</table>
<ul class="itemize">
<li>
<p>
Suppose there was a single qubit error and you get the expectation value as shown in the table. What error actually happens? Input <tt class="ttfamily">Z_4</tt> for [mathjaxinline]Z_4[/mathjaxinline] etc. </p>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="text-align:center; border-right:1px solid black; border:none">
observables </td>
<td style="text-align:center; border:none">
expectation value </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]Z_1Z_2[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
+1 </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]Z_2Z_3[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
+1 </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]Z_4Z_5[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
-1 </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]Z_5Z_6[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
-1 </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]Z_7Z_8[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
+1 </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]Z_8Z_9[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
+1 </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]X_1X_2X_3X_4X_5X_6[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
+1 </td>
</tr>
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]X_4X_5X_6X_7X_8X_9[/mathjaxinline] </td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border:none">
+1 </td>
</tr>
</table>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_The_9-qubit_code_II_2_1" class="text-input-dynamath capa_inputtype textline">
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<div class="unanswered ">
<input type="text" name="input_The_9-qubit_code_II_2_1" id="input_The_9-qubit_code_II_2_1" aria-describedby="status_The_9-qubit_code_II_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_The_9-qubit_code_II_2_1"/>
<span class="status unanswered" id="status_The_9-qubit_code_II_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_The_9-qubit_code_II_2_1" class="answer"/>
<div id="display_The_9-qubit_code_II_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_The_9-qubit_code_II_2_1_dynamath" name="input_The_9-qubit_code_II_2_1_dynamath"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
Suppose you have an encoded state [mathjaxinline]|{\psi _ L}\rangle =a|{0_ L}\rangle +b|{1_ L}\rangle[/mathjaxinline] and get the same syndrome above. But suppose this time that there are actually errors happening on two qubits without you knowing it. If you try to correct the error using the syndrome above assuming there was only single qubit error, what state will you end up obtaining? Answer in terms of [mathjaxinline]|{0_ L}\rangle[/mathjaxinline], [mathjaxinline]|{1_ L}\rangle[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline]. Input <tt class="ttfamily">|0&gt;,|1&gt;</tt> for [mathjaxinline]|{0_ L}\rangle ,|{1_ L}\rangle[/mathjaxinline]. </p>
<p><p style="display:inline">state</p>: <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_The_9-qubit_code_II_3_1" class="text-input-dynamath capa_inputtype textline">
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<div class="unanswered ">
<input type="text" name="input_The_9-qubit_code_II_3_1" id="input_The_9-qubit_code_II_3_1" aria-describedby="status_The_9-qubit_code_II_3_1" value="" class="math"/>
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<span class="status unanswered" id="status_The_9-qubit_code_II_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_The_9-qubit_code_II_3_1" class="answer"/>
<div id="display_The_9-qubit_code_II_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_The_9-qubit_code_II_3_1_dynamath" name="input_The_9-qubit_code_II_3_1_dynamath"/>
</div>
</div></div></p>
</li>
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<span id="solution_The_9-qubit_code_II_solution_1"/>
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// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
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if(typeof(String.prototype.trim) === "undefined")
{
String.prototype.trim = function()
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return String(this).replace(/^\s+|\s+$/g, '');
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var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
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console.log("cid = ", cid);
// search_module_url = "/courses/course-v1:MITx+8.370.3x+1T2018/" + cid + "/courseware/welcome/Search_this_course/";
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console.log("3. search_module_url = ", search_module_url);
return;
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console.log("course_root_link = ", course_root_link);
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console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('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")){
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console.log("search_module_url = ", search_module_url);
}
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var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
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if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
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'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
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if ($('div.wrap-instructor-info').length==0){
return;
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$('div.xblock-student_view-video').each(function(key, vblock_e){
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var vuid = vblock.data('usage-id').split('@');
var vid;
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<h2 class="hd hd-2 unit-title">Quantum error correction condition</h2>
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<h3 class="hd hd-3 problem-header" id="problem_Quantum_error_correction_condition-problem-title" aria-describedby="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Quantum_error_correction_condition-problem-progress" tabindex="-1">
Quantum error correction condition
</h3>
<div class="problem-progress" id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Quantum_error_correction_condition-problem-progress"></div>
<div class="problem">
<div>
<p>
The quantum error correcting conditions are the necessary and sufficient conditions for a code to be able to correct a certain set of errors. Let [mathjaxinline]P[/mathjaxinline] be the projector onto the codeword space of the code and [mathjaxinline]\{ E_ i\}[/mathjaxinline] be some set of error operators. [mathjaxinline]\{ E_ i\}[/mathjaxinline] are correctable if and only if for any [mathjaxinline]i,\ j[/mathjaxinline] </p>
<table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]PE^{\dagger }_ iE_ jP = \alpha _{ij} P[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.9)</td>
</tr>
</table>
<p>
where [mathjaxinline]\alpha _{ij}[/mathjaxinline] is a Hermitian matrix. </p>
<p>
In this problem, we will explicitly see how it holds for the 9-qubit code. Using the codewords [mathjaxinline]|{0_ L}\rangle[/mathjaxinline] and [mathjaxinline]|{1_ L}\rangle[/mathjaxinline], the projector onto the codeword space can be written by [mathjaxinline]P=|{0_ L}\rangle \langle {0_ L}|+|{1_ L}\rangle \langle {1_ L}|[/mathjaxinline]. We will write it in a matrix form [mathjaxinline]P=\begin{bmatrix} 1 &amp; 0\\ 0 &amp; 1 \end{bmatrix}[/mathjaxinline] by taking [mathjaxinline]\{ |{0_ L}\rangle , |{1_ L}\rangle \}[/mathjaxinline] as a basis. </p>
<ul class="itemize">
<li>
<p>
Let [mathjaxinline]E_1=X_1[/mathjaxinline], [mathjaxinline]E_2=Z_1[/mathjaxinline], [mathjaxinline]E_3=Y_3[/mathjaxinline]. Compute a a 3[mathjaxinline]\times[/mathjaxinline]3 matrix [mathjaxinline]\alpha[/mathjaxinline]. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_problem_Quantum_error_correction_condition_2_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Quantum_error_correction_condition_2_1" id="input_problem_Quantum_error_correction_condition_2_1" aria-describedby="status_problem_Quantum_error_correction_condition_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Quantum_error_correction_condition_2_1"/>
<span class="status unanswered" id="status_problem_Quantum_error_correction_condition_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_problem_Quantum_error_correction_condition_2_1" class="answer"/>
<div id="display_problem_Quantum_error_correction_condition_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Quantum_error_correction_condition_2_1_dynamath" name="input_problem_Quantum_error_correction_condition_2_1_dynamath"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
Let [mathjaxinline]E_1=X_1[/mathjaxinline], [mathjaxinline]E_2=X_2X_3[/mathjaxinline], [mathjaxinline]E_3=Z_1[/mathjaxinline]. What is [mathjaxinline]PE^{\dagger }_1E_2P[/mathjaxinline]? Express in the matrix form with [mathjaxinline]\{ |{0_ L}\rangle ,|{1_ L}\rangle \}[/mathjaxinline] basis. Is this set of errors correctable? </p>
<p>
<p style="display:inline">Matrix:</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_problem_Quantum_error_correction_condition_3_1" class="text-input-dynamath capa_inputtype textline">
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<div class="unanswered ">
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<span class="status unanswered" id="status_problem_Quantum_error_correction_condition_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_problem_Quantum_error_correction_condition_3_1" class="equation">`{::}`</div>
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</p>
<p>
<p style="display:inline">Correctable?:</p>
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<option value="No"> No</option>
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<span class="status unanswered" id="status_problem_Quantum_error_correction_condition_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p class="answer" id="answer_problem_Quantum_error_correction_condition_4_1"/>
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</p>
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<li>
<p>
Let [mathjaxinline]E_1=Z_1[/mathjaxinline], [mathjaxinline]E_2=Z_2Z_3[/mathjaxinline], [mathjaxinline]E_3=X_1[/mathjaxinline]. What is [mathjaxinline]PE^{\dagger }_1E_2P[/mathjaxinline]? Express in terms of [mathjaxinline]|{0_ L}\rangle[/mathjaxinline] and [mathjaxinline]|{1_ L}\rangle[/mathjaxinline]. Is this set of errors correctable? </p>
<p>
<p style="display:inline">Matrix:</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_problem_Quantum_error_correction_condition_5_1" class="text-input-dynamath capa_inputtype textline">
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<div class="unanswered ">
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<span class="status unanswered" id="status_problem_Quantum_error_correction_condition_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_problem_Quantum_error_correction_condition_5_1" class="answer"/>
<div id="display_problem_Quantum_error_correction_condition_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Quantum_error_correction_condition_5_1_dynamath" name="input_problem_Quantum_error_correction_condition_5_1_dynamath"/>
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</div></div>
</p>
<p>
<p style="display:inline">Correctable?:</p>
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<p class="answer" id="answer_problem_Quantum_error_correction_condition_6_1"/>
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