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<h2 class="hd hd-2 unit-title">Hadamard gate by measurement</h2>
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<p><big class="xxlarge">Implementation of Clifford gates by measurement</big></p><p>
Clifford circuits may be composed from just [mathjaxinline]H[/mathjaxinline], [mathjaxinline]S[/mathjaxinline], and CNOT gates. Alternatively, on stabilizer states, any Clifford circuit operation can also be composed from conditional application of the single-qubit [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] Pauli operations, measurement of [mathjaxinline]{\tt X}[/mathjaxinline] and [mathjaxinline]{\tt Y}[/mathjaxinline], and measurement of [mathjaxinline]{\tt XX}[/mathjaxinline], [mathjaxinline]{\tt ZZ}[/mathjaxinline], and [mathjaxinline]{\tt ZX}[/mathjaxinline]. This is a very useful observation, even beyond FTQC. </p>
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<h3 class="hd hd-3 problem-header" id="s14-wk4-H-by-meas-problem-title" aria-describedby="block-v1:MITx+8.371.2x+2T2018+type@problem+block@s14-wk4-H-by-meas-problem-progress" tabindex="-1">
Hadamard gate by measurement
</h3>
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<p>
The Hadamard gate can be implemented using only stabilizer measurements and classically controlled single-qubit [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] Pauli operations. In this problem, we show that the output of the following circuit is [mathjaxinline]H|\psi \rangle[/mathjaxinline]: </p>
<p>
<center>
<img src="/assets/courseware/v1/fac4308c22b67e74f8e7f1cdfda9ebf2/asset-v1:MITx+8.371.2x+2T2018+type@asset+block/images_ps7-clifford-measure-hadamard.png" width="500"/>
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</p>
<p>
Let us show this by working out how the initial stabilizer and the normalizer operations transform. </p>
<p>
Note that there are many equivalent expressions for the logical qubit operators, since they are invariant under multiplication by any stabilizer element. However, there is a simple and natural procedure to follow when updating a normalizer after measurement of some Pauli operator [mathjaxinline]m[/mathjaxinline], with measurement result [mathjaxinline]a[/mathjaxinline]. Suppose [mathjaxinline]m[/mathjaxinline] anticommutes with only one stabilizer generator [mathjaxinline]g[/mathjaxinline]. Then the stabilizer is updated by replacing [mathjaxinline]g[/mathjaxinline] with [mathjaxinline](-1)^ a m[/mathjaxinline]. Correspondingly, any logical qubit operator [mathjaxinline]\bar L[/mathjaxinline] which also anticommutes with [mathjaxinline]m[/mathjaxinline] should be updated by replacing it with [mathjaxinline]g \bar L[/mathjaxinline], so that it then commutes with the new stabilizer. </p>
<p>
The initial state [mathjaxinline]S_0[/mathjaxinline] may be described the stabilizer generator [mathjaxinline]g[/mathjaxinline] and logical qubit operators [mathjaxinline]\bar{X}[/mathjaxinline], [mathjaxinline]\bar{Z}[/mathjaxinline]: </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
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<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZ}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
[mathjaxinline]\bar{X}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XI}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]\bar{Z}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]{\tt ZI}[/mathjaxinline] </td>
</tr>
</table>
</center>
<p>
Note that the right-most labels correspond to the bottom qubit wire (qubit [mathjaxinline]0[/mathjaxinline]) in the circuit shown above. </p>
<p>
Give the logical qubit operators for the states specified after each operation in the circuit. </p>
<p>
Input your answers using symbolic mathematical expressions, below. Be sure to include explicit [mathjaxinline]{\tt *}[/mathjaxinline] operators for multiplication. You may use <tt class="tt">^</tt> for exponentiation. Enter stabilizer operators as strings of letters, eg [mathjaxinline]{\tt IXI}[/mathjaxinline]. Expressions such as <tt class="tt">(-1)^c * XZZ</tt> are acceptable. </p>
<ol class="enumerate">
<li value="1">
<p>
Give the natural logical qubit operators for the state [mathjaxinline]S_1[/mathjaxinline] after the measurement of [mathjaxinline]ZX[/mathjaxinline], with measurement result [mathjaxinline]\color{red}{a}[/mathjaxinline]: </p>
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<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </td>
<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]S_1[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZ}[/mathjaxinline] </td>
<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]{\tt (-1)^{\color{red}{a}} * ZX}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XI}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<p style="display:inline">[mathjaxinline]\bar{Z}[/mathjaxinline]</p>
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<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]{\tt ZI}[/mathjaxinline] </td>
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<li value="2">
<p>
Give the natural logical qubit operators for the state [mathjaxinline]S_2[/mathjaxinline] after the measurement of [mathjaxinline]XI[/mathjaxinline], with measurement result [mathjaxinline]\color{red}{b}[/mathjaxinline]: </p>
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Operator </td>
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[mathjaxinline]S_0[/mathjaxinline] </td>
<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]S_2[/mathjaxinline] </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-left:1px solid black; border:none">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">&#160;</td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IZ}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt (-1)^{\color{red}{b}} * XI}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XI}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<p style="display:inline">[mathjaxinline]\bar{Z}[/mathjaxinline]</p>
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[mathjaxinline]{\tt ZI}[/mathjaxinline] </td>
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</li>
<li value="3">
<p>
Still referring to state [mathjaxinline]S_2[/mathjaxinline], give a valid equivalent expression for [mathjaxinline]\bar{X}[/mathjaxinline] which has [mathjaxinline]{\tt I}[/mathjaxinline] on qubit [mathjaxinline]1[/mathjaxinline]: </p>
<p>
<p style="display:inline">[mathjaxinline]\bar{X} =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">Phase gate by measurement</h2>
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Phase gate by measurement
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The phase gate </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]S=\left(\begin{array}{cc} 1 &amp; 0 \\ 0 &amp; i \end{array}\right)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.1)</td>
</tr>
</table>
<p>
can be implemented using only stabilizer measurements and classically controlled single-qubit [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] Pauli operations. In this problem, we show that the output of the following circuit is [mathjaxinline]S|\psi \rangle[/mathjaxinline]: </p>
<p>
<center>
<img src="/assets/courseware/v1/f3427d4ea09326a540969d7807ee36e3/asset-v1:MITx+8.371.2x+2T2018+type@asset+block/images_ps7-clifford-measure-phase.png" width="500"/>
</center>
</p>
<p>
Let us show this by working out how the initial stabilizer and the normalizer operations transform. </p>
<p>
The initial state [mathjaxinline]S_0[/mathjaxinline] may be described the stabilizer generator [mathjaxinline]g[/mathjaxinline] and logical qubit operators [mathjaxinline]\bar{X}[/mathjaxinline], [mathjaxinline]\bar{Z}[/mathjaxinline]: </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZ}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
[mathjaxinline]\bar{X}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XI}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]\bar{Z}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]{\tt ZI}[/mathjaxinline] </td>
</tr>
</table>
</center>
<p>
Note that the right-most labels correspond to the bottom qubit wire (qubit [mathjaxinline]0[/mathjaxinline]) in the circuit shown above. </p>
<p>
Give the logical qubit operators for the states specified after each operation in the circuit. </p>
<p>
Input your answers using symbolic mathematical expressions, below. Be sure to include explicit [mathjaxinline]{\tt *}[/mathjaxinline] operators for multiplication. You may use <tt class="tt">^</tt> for exponentiation. Enter stabilizer operators as strings of letters, eg [mathjaxinline]{\tt IXI}[/mathjaxinline]. Expressions such as <tt class="tt">(-1)^c * XZZ</tt> are acceptable. </p>
<ol class="enumerate">
<li value="1">
<p>
Give the natural logical qubit operators for the state [mathjaxinline]S_1[/mathjaxinline] after the measurement of [mathjaxinline]ZX[/mathjaxinline], with measurement result [mathjaxinline]\color{red}{a}[/mathjaxinline]: </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </td>
<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]S_1[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZ}[/mathjaxinline] </td>
<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]{\tt (-1)^{\color{red}{a}} * ZX}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XI}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_s14-wk4-S-by-meas_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
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<p style="display:inline">[mathjaxinline]\bar{Z}[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]{\tt ZI}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_s14-wk4-S-by-meas_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
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</td>
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</center>
</li>
<li value="2">
<p>
Give the natural logical qubit operators for the state [mathjaxinline]S_2[/mathjaxinline] after the measurement of [mathjaxinline]IY[/mathjaxinline], with measurement result [mathjaxinline]\color{red}{b}[/mathjaxinline]: </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </td>
<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]S_2[/mathjaxinline] </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-left:1px solid black; border:none">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border:none">&#160;</td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IZ}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt (-1)^{\color{red}{b}} * IY}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XI}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<p style="display:inline">[mathjaxinline]\bar{Z}[/mathjaxinline]</p>
</td>
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[mathjaxinline]{\tt ZI}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">&#160;</td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">&#160;</td>
</tr>
</table>
</center>
</li>
<li value="3">
<p>
Still for state [mathjaxinline]S_2[/mathjaxinline], give a valid equivalent expression for [mathjaxinline]\bar{X}[/mathjaxinline] which has [mathjaxinline]{\tt I}[/mathjaxinline] on qubit [mathjaxinline]0[/mathjaxinline]: </p>
<p>
<p style="display:inline">[mathjaxinline]\bar{X} =[/mathjaxinline]</p>
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<span id="solution_s14-wk4-S-by-meas_solution_1"/>
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<h2 class="hd hd-2 unit-title">CNOT gate by measurement</h2>
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<h3 class="hd hd-3 problem-header" id="s14-wk4-cnot-by-meas-problem-title" aria-describedby="block-v1:MITx+8.371.2x+2T2018+type@problem+block@s14-wk4-cnot-by-meas-problem-progress" tabindex="-1">
CNOT gate by measurement
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.2x+2T2018+type@problem+block@s14-wk4-cnot-by-meas-problem-progress"></div>
<div class="problem">
<div>
<p>
The CNOT gate can be implemented using only stabilizer measurements and classically controlled single-qubit [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] Pauli operations. In this problem, we show that the following circuits are equivalent: </p>
<p>
<center>
<img src="/assets/courseware/v1/680ba6de70807724bf3e81389b246cc8/asset-v1:MITx+8.371.2x+2T2018+type@asset+block/images_ps7-clifford-measure-cnot.png" width="700"/>
</center>
</p>
<p>
Let us show the equivalence by working out how the initial stabilizer and the normalizer operations transform. </p>
<p>
The initial state [mathjaxinline]S_0[/mathjaxinline] may be described the stabilizer generator [mathjaxinline]g[/mathjaxinline] and logical qubit operators [mathjaxinline]\bar{X}_1[/mathjaxinline], [mathjaxinline]\bar{Z}_1[/mathjaxinline], [mathjaxinline]\bar{X}_2[/mathjaxinline], [mathjaxinline]\bar{Z}_2[/mathjaxinline]: </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZI}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
[mathjaxinline]\bar{X}_1[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IIX}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
[mathjaxinline]\bar{Z}_1[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IIZ}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
[mathjaxinline]\bar{X}_2[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XII}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]\bar{Z}_2[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]{\tt ZII}[/mathjaxinline] </td>
</tr>
</table>
</center>
<p>
Note that the right-most labels correspond to the bottom qubit wire (qubit [mathjaxinline]0[/mathjaxinline]) in the circuit shown above. </p>
<p>
Give the logical qubit operators for the states specified after each operation in the circuit. Refer back to the problem on the Hadamard gate for the rules on how to update the logical qubit operators after Pauli measurements. </p>
<p>
Input your answers using symbolic mathematical expressions, below. Be sure to include explicit [mathjaxinline]{\tt *}[/mathjaxinline] operators for multiplication. You may use <tt class="tt">^</tt> for exponentiation. Enter stabilizer operators as strings of letters, eg [mathjaxinline]{\tt IXI}[/mathjaxinline]. Expressions such as <tt class="tt">(-1)^ c * XZZ</tt> are acceptable. </p>
<p>
Give the natural logical qubit operators for the state [mathjaxinline]S_1[/mathjaxinline] after the measurement of [mathjaxinline]IXX[/mathjaxinline], with measurement result [mathjaxinline]\color{red}{a}[/mathjaxinline]: </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </td>
<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]S_1[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZI}[/mathjaxinline] </td>
<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]{\tt (-1)^{\color{red}{a}} * IXX}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}_1[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IIX}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<p style="display:inline">[mathjaxinline]\bar{Z}_1[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IIZ}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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</td>
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<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}_2[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XII}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<p style="display:inline">[mathjaxinline]\bar{Z}_2[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]{\tt ZII}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
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<h2 class="hd hd-2 unit-title">CNOT gate by measurement II</h2>
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<h3 class="hd hd-3 problem-header" id="s14-wk4-cnot-by-meas2-problem-title" aria-describedby="block-v1:MITx+8.371.2x+2T2018+type@problem+block@s14-wk4-cnot-by-meas2-problem-progress" tabindex="-1">
CNOT gate by measurement II
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.2x+2T2018+type@problem+block@s14-wk4-cnot-by-meas2-problem-progress"></div>
<div class="problem">
<div>
<p>
Give the natural logical qubit operators for the state [mathjaxinline]S_2[/mathjaxinline] after the measurement of [mathjaxinline]ZZI[/mathjaxinline], with measurement result [mathjaxinline]\color{red}{b}[/mathjaxinline]: </p>
<p>
Remember to use [mathjaxinline]{\tt *}[/mathjaxinline] to explicitly indicate multiplication! </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </td>
<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]S_2[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZI}[/mathjaxinline] </td>
<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]{\tt (-1)^{\color{red}{b}} * ZZI}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}_1[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IIX}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<p style="display:inline">[mathjaxinline]\bar{Z}_1[/mathjaxinline]</p>
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[mathjaxinline]{\tt IIZ}[/mathjaxinline] </td>
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<p style="display:inline">[mathjaxinline]\bar{X}_2[/mathjaxinline]</p>
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[mathjaxinline]{\tt XII}[/mathjaxinline] </td>
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<p style="display:inline">[mathjaxinline]\bar{Z}_2[/mathjaxinline]</p>
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[mathjaxinline]{\tt ZII}[/mathjaxinline] </td>
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<h2 class="hd hd-2 unit-title">CNOT gate by measurement III</h2>
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<h3 class="hd hd-3 problem-header" id="s14-wk4-cnot-by-meas3-problem-title" aria-describedby="block-v1:MITx+8.371.2x+2T2018+type@problem+block@s14-wk4-cnot-by-meas3-problem-progress" tabindex="-1">
CNOT gate by measurement III
</h3>
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<div class="problem">
<div>
<p>
Give the natural logical qubit operators for the state [mathjaxinline]S_3[/mathjaxinline] after the measurement of [mathjaxinline]IXI[/mathjaxinline], with measurement result [mathjaxinline]\color{red}{c}[/mathjaxinline]: </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
Operator </td>
<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]S_0[/mathjaxinline] </td>
<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]S_3[/mathjaxinline] </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-left:1px solid black; border:none">
[mathjaxinline]g[/mathjaxinline] </td>
<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]{\tt IZI}[/mathjaxinline] </td>
<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]{\tt (-1)^{\color{red}{c}} * IXI}[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}_1[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IIX}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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</td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{Z}_1[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt IIZ}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<div class="unanswered inline">
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<span class="trailing_text" id="trailing_text_s14-wk4-cnot-by-meas3_3_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_s14-wk4-cnot-by-meas3_3_1" class="answer"/>
</div>
</div></div>
</td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border:none">
<p style="display:inline">[mathjaxinline]\bar{X}_2[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border:none">
[mathjaxinline]{\tt XII}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border:none">
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<span class="trailing_text" id="trailing_text_s14-wk4-cnot-by-meas3_4_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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</div>
</div></div>
</td>
</tr>
<tr>
<td style="text-align:center; border-right:1px solid black; border-left:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
<p style="display:inline">[mathjaxinline]\bar{Z}_2[/mathjaxinline]</p>
</td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
[mathjaxinline]{\tt ZII}[/mathjaxinline] </td>
<td style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; border:none">
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_s14-wk4-cnot-by-meas3_5_1" class=" capa_inputtype inline textline">
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<span class="trailing_text" id="trailing_text_s14-wk4-cnot-by-meas3_5_1"/>
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</div>
</div></div>
</td>
</tr>
</table>
</center>
<p>
Give a valid equivalent expression for [mathjaxinline]\bar{X}_2[/mathjaxinline] which has [mathjaxinline]{\tt I}[/mathjaxinline] on qubit [mathjaxinline]1[/mathjaxinline]: </p>
<p>
<p style="display:inline">[mathjaxinline]\bar{X}_2 =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_s14-wk4-cnot-by-meas3_6_1" class=" capa_inputtype inline textline">
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<span class="trailing_text" id="trailing_text_s14-wk4-cnot-by-meas3_6_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_s14-wk4-cnot-by-meas3_6_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_s14-wk4-cnot-by-meas3_solution_1"/>
</div></p>
<p>
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<span id="solution_s14-wk4-cnot-by-meas3_solution_2"/>
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console.log("3. search_module_url = ", search_module_url);
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if (course_root_link){
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// find search this module link
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$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
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new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
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if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
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event.preventDefault();
go_to_search();
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});
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var get = function(x){
return eval(x);
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'go_to_search': go_to_search,
'get_search_module': get_search_module,
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'get': get,
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mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
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var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
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console.log("html = ", html);
vblock.after(html)
}
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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/";
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gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
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window.open(gurl, "MITx 8.370x subtitle source");
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<h2 class="hd hd-2 unit-title">A cluster QC circuit</h2>
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<h3 class="hd hd-3 problem-header" id="s14-wk4-mbc-problem-title" aria-describedby="block-v1:MITx+8.371.2x+2T2018+type@problem+block@s14-wk4-mbc-problem-progress" tabindex="-1">
A cluster QC circuit
</h3>
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<p>
This problem works through an example of a brief measurement-based computation, using a cluster state. </p>
<p>
The figure below shows a cluster-model quantum computation, in which each qubit is represented as a circle, and links indicate qubits which have been connected in the cluster state. The letter in each circle indicates the eigenbasis of the measurement; thus, [mathjaxinline]X[/mathjaxinline] is equivalent to performing [mathjaxinline]H[/mathjaxinline] then measuring in the computational basis, while [mathjaxinline]Z[/mathjaxinline] denotes measurement in the computational basis. All the measurements may be done simultaneously; no conditional change of measurement basis is necessary in this specific computation: </p>
<p>
<center>
<img src="/assets/courseware/v1/8919302b6e260a3d2858509fa447f3bf/asset-v1:MITx+8.371.2x+2T2018+type@asset+block/images_cluster.png" width="500"/>
</center>
</p>
<ol class="enumerate">
<li value="1">
<p>
Which qubits are the input qubits and which are the output qubits? </p>
<p>
Specify each answer as a list of coordinates (x,y) where (0,0) is the bottom-left qubit, (3,0) is the bottom-right qubit, and (3,3) is the top-right qubit. For example, [mathjaxinline]{\tt [(0,0),(2,1)]}[/mathjaxinline] specifies the two qubits at (0,0) and (2,1). </p>
<p>
Hint: recall that the computation proceeds from left to right. </p>
<ul class="itemize">
<li>
<p>
<p style="display:inline">Input qubits:</p>
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</p>
</li>
<li>
<p>
<p style="display:inline">Output qubits:</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_s14-wk4-mbc_3_1" class=" capa_inputtype inline textline">
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</p>
</li>
</ul>
</li>
<li value="2">
<p>
What quantum circuit [mathjaxinline]U[/mathjaxinline] is simulated by this cluster computation? </p>
<p>
Input your answer as a quantum circuit, ie a list of gates, eg of the form [mathjaxinline]{\tt [X(0),X(2),CNOT(1,2),CPHASE(0,2)]}[/mathjaxinline]. The gates are applied in order from left to right, and qubits are numbered starting with [mathjaxinline]0[/mathjaxinline], from the bottom. You may number your qubits in any order, as long as you are consistent. </p>
<p>
<p style="display:inline">[mathjaxinline]U=[/mathjaxinline]</p>
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</p>
<p>
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<span id="solution_s14-wk4-mbc_solution_1"/>
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</li>
<li value="3">
<p>
What is the state [mathjaxinline]|{\rm out}\rangle[/mathjaxinline] of the unmeasured qubits after the appropriate Pauli corrections? </p>
<p>
Input your answer using explicit [mathjaxinline]{\tt *}[/mathjaxinline] for multiplication, using [mathjaxinline]{\tt sqrt(x)}[/mathjaxinline] if necessary. For example, [mathjaxinline]{\tt (|0&gt;+ |1&gt;)/sqrt(2)}[/mathjaxinline] would be a legal input. </p>
<p>
<p style="display:inline">[mathjaxinline]|{\rm out}\rangle =[/mathjaxinline]</p>
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<div id="display_s14-wk4-mbc_5_1" class="equation">`{::}`</div>
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