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<h2 class="hd hd-2 unit-title">CQ: Equivalent quantum states</h2>
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Equivalent quantum states
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Which of the following are equivalent quantum states? </p>
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<text>[mathjaxinline]\displaystyle \frac{|0\rangle +|1\rangle }{\sqrt{2}}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle 2\sin (\pi /4) |0\rangle + \sin (\pi /4)|1\rangle + \cos (3\pi /4)|0\rangle[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{-1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{e^{i\pi /2}+1}{2} |0\rangle + \frac{e^{i\pi /2}-1}{2i}|1\rangle[/mathjaxinline]</text>
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Bloch sphere states
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Drag the correct labels into the three empty boxes, denoting states on the Bloch sphere at the tips of the three arrows (and note the orientation of these axes, compared with those drawn by Prof. Shor in lecture): </p>
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<h2 class="hd hd-2 unit-title">CQ: Valid unitary matrices</h2>
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Valid unitary matrices
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Which of the following are valid unitary matrices? </p>
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<text>[mathjaxinline]\displaystyle \left( \begin{array}{cc}0 &amp; 1 \\ i &amp; 0\end{array}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left( \begin{array}{cc}1 &amp; -1 \\ 0 &amp; 1\end{array}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left( \begin{array}{cc}1 &amp; 1 \\ 1 &amp; -1 \end{array}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{e^{i\pi /4}}{\sqrt{2}} \left( \begin{array}{cc} 1 &amp; -1 \\ -1 &amp; -1 \end{array}\right)[/mathjaxinline]</text>
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Rotation axes of a Hadamard gate
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
On the Bloch sphere, the axis of rotation of a single-qubit gate [mathjaxinline]U[/mathjaxinline] is the line going from the center of the Bloch sphere, and a point [mathjaxinline]\vec{p}=(p_ x, p_ y, p_ z)[/mathjaxinline] on the Bloch sphere, corresponding to an eigenstate of [mathjaxinline]U[/mathjaxinline]. </p>
<p>
Which of the following points gives the point [mathjaxinline]\vec{p}[/mathjaxinline] defining the axis of rotation for the <a href="https://en.wikipedia.org/wiki/Quantum_gate#Hadamard_gate" target="_blank">Hadamard gate</a>? </p>
<p>
<div class="hideshowbox">
<h4 onclick="hideshow(this);" style="margin: 0px">Hint<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">
<p>
An arbitrary single qubit (&#8220;pure") state [mathjaxinline]|\psi \rangle[/mathjaxinline] may be written as </p>
<table id="a0000000002" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\psi \rangle = \cos \frac{\theta }{2} |0\rangle + e^{i\phi } \sin \frac{\theta }{2} |1\rangle \, ,[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.1)</td>
</tr>
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<p>
corresponding to a point on the Bloch sphere located at [mathjaxinline](\theta , \phi )[/mathjaxinline] in <a href="https://en.wikipedia.org/wiki/Spherical_coordinate_system" target="_blank">spherical coordinates</a>, with [mathjaxinline]r=1[/mathjaxinline]. </p>
</div>
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