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_sources/lecture-04/math-for-qc.md

@@ -1367,6 +1367,7 @@ The size of the basis, and hence the dimension of the Hilbert space of $n$ qubit
 The quantum state we referred to so far are what one calls pure states, and they are represented by a normalised vector in a complex Hilbert space. In the Dirac notation, they are described as a ket $|v\rangle$ or a bra $\langle v|$.
 
 There is another way to describe the state of quantum system which can be generalised to mixed states (defined below), and the is through **density operator**. For a system in pure quantum state $|v\rangle$, the density operator is defined as
+
 $$
 {\Huge
 \rho = |v\rangle \langle v|

lecture-04/math-for-qc.html

@@ -1708,12 +1708,13 @@ <h3>Pure vs Mixed States<a class="headerlink" href="#pure-vs-mixed-states" title
 <section id="pure-state">
 <h4>Pure state<a class="headerlink" href="#pure-state" title="Link to this heading">#</a></h4>
 <p>The quantum state we referred to so far are what one calls pure states, and they are represented by a normalised vector in a complex Hilbert space. In the Dirac notation, they are described as a ket <span class="math notranslate nohighlight">\(|v\rangle\)</span> or a bra <span class="math notranslate nohighlight">\(\langle v|\)</span>.</p>
-<p>There is another way to describe the state of quantum system which can be generalised to mixed states (defined below), and the is through <strong>density operator</strong>. For a system in pure quantum state <span class="math notranslate nohighlight">\(|v\rangle\)</span>, the density operator is defined as
-$<span class="math notranslate nohighlight">\(
+<p>There is another way to describe the state of quantum system which can be generalised to mixed states (defined below), and the is through <strong>density operator</strong>. For a system in pure quantum state <span class="math notranslate nohighlight">\(|v\rangle\)</span>, the density operator is defined as</p>
+<div class="math notranslate nohighlight">
+\[
 {\Huge
 \rho = |v\rangle \langle v|
 }
-\)</span>$</p>
+\]</div>
 <ul class="simple">
 <li><p>It is easy to see that trace of the density operator is 1, i.e., <span class="math notranslate nohighlight">\(\text{Tr}(\rho)=1\)</span>.</p></li>
 <li><p>For a pure state, <span class="math notranslate nohighlight">\(\rho^2 = \rho\)</span>, so <span class="math notranslate nohighlight">\(\text{Tr}(\rho^2)=1\)</span></p></li>

lecture-08/accessing-qc-systems.html

@@ -428,7 +428,7 @@ <h2>What are Classical Programs<a class="headerlink" href="#what-are-classical-p
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/49a31f021b1368ef172e5634d87908515d5c25262c7f1222956579054a78d1da.png" src="../_images/49a31f021b1368ef172e5634d87908515d5c25262c7f1222956579054a78d1da.png" />
+<img alt="../_images/405d6dcc8319dd5ef2848d725f075087a3eec47cacebfb16eba40e0272ecdbff.png" src="../_images/405d6dcc8319dd5ef2848d725f075087a3eec47cacebfb16eba40e0272ecdbff.png" />
 </div>
 </div>
 <p>The small piece of code above is a <code class="docutils literal notranslate"><span class="pre">program</span></code> written in Python programming language and uses the popular scientific library <code class="docutils literal notranslate"><span class="pre">Numpy</span></code>. The first line imports the module<a class="footnote-reference brackets" href="#lib" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>/library, and the second line imports the famous plotting library <code class="docutils literal notranslate"><span class="pre">matplotlib</span></code>.
@@ -492,7 +492,7 @@ <h2>What are Quantum Programs<a class="headerlink" href="#what-are-quantum-progr
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/06d86a2d1832054629c11019ce74eea737702ff1ac1e230d5e6e60f29b9cb64d.png" src="../_images/06d86a2d1832054629c11019ce74eea737702ff1ac1e230d5e6e60f29b9cb64d.png" />
+<img alt="../_images/2536a1a1ad345cc254deb4043a3a2a4998d469070c73587841e8910464b104eb.png" src="../_images/2536a1a1ad345cc254deb4043a3a2a4998d469070c73587841e8910464b104eb.png" />
 </div>
 </div>
 <p>The <code class="docutils literal notranslate"><span class="pre">Aer</span></code> module is part of qiskit software ecosystem, and provides us with a few backend software simulators for quantum computing. The line <code class="docutils literal notranslate"><span class="pre">sim</span> <span class="pre">=</span> <span class="pre">Aer.get_backend('aer_simulator')</span></code> defines a simulator object based on Aer’s ‘aer_simulator backend’.

lecture-11-note.html

@@ -377,8 +377,8 @@ <h2>Example 1<a class="headerlink" href="#example-1" title="Link to this heading
 S_A = \sum_{i\in U} n_i x_i \\S_B = \sum_{i\in U}n_i (1-x_i)
 \)</span>$</p>
 <p>The cost function can now be written as</p>
-<div class="amsmath math notranslate nohighlight" id="equation-e5ba0c6f-a7cf-4f87-81b7-906fe9110aba">
-<span class="eqno">()<a class="headerlink" href="#equation-e5ba0c6f-a7cf-4f87-81b7-906fe9110aba" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-64581398-2a47-467d-8f06-81caa79a242f">
+<span class="eqno">()<a class="headerlink" href="#equation-64581398-2a47-467d-8f06-81caa79a242f" title="Permalink to this equation">#</a></span>\[\begin{align}
 f(\{n_i\}) &amp;= (S_A - S_B)^2 = S_A^2 + S_B^2 - 2S_A S_B\\
 &amp;= (\sum_{i\in U} n_i x_i)^2 + (\sum_{i\in U}n_i (1-x_i))^2 - 2 (\sum_{i\in U} n_i x_i)\sum_{j\in U}n_j (1-x_j)\\
 &amp;= \sum_{ij\in U}\left[

testpad.html

@@ -387,7 +387,7 @@ <h2> Contents </h2>
 </div>
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-<img alt="_images/5bef002fda27344fbb48df57a64f0688adf85fea48a34f6dd3cf09decb9b0718.png" src="_images/5bef002fda27344fbb48df57a64f0688adf85fea48a34f6dd3cf09decb9b0718.png" />
+<img alt="_images/86909bb56d6053f3266dd80db5cfa6c153f4f5a8eca2553f3727fe32a431353a.png" src="_images/86909bb56d6053f3266dd80db5cfa6c153f4f5a8eca2553f3727fe32a431353a.png" />
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@@ -397,22 +397,22 @@ <h2> Contents </h2>
 </div>
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-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{&#39;1111&#39;: 596,
- &#39;1110&#39;: 580,
- &#39;1000&#39;: 628,
- &#39;0110&#39;: 627,
- &#39;1101&#39;: 587,
- &#39;0000&#39;: 626,
- &#39;0010&#39;: 676,
- &#39;1010&#39;: 630,
- &#39;0001&#39;: 629,
- &#39;1011&#39;: 600,
- &#39;0011&#39;: 661,
- &#39;0111&#39;: 614,
- &#39;1100&#39;: 623,
- &#39;0101&#39;: 664,
- &#39;1001&#39;: 627,
- &#39;0100&#39;: 632}
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{&#39;0001&#39;: 631,
+ &#39;0011&#39;: 643,
+ &#39;0111&#39;: 585,
+ &#39;1011&#39;: 652,
+ &#39;0100&#39;: 641,
+ &#39;1101&#39;: 655,
+ &#39;1000&#39;: 619,
+ &#39;0110&#39;: 613,
+ &#39;0000&#39;: 631,
+ &#39;0010&#39;: 597,
+ &#39;1001&#39;: 641,
+ &#39;0101&#39;: 620,
+ &#39;1110&#39;: 640,
+ &#39;1010&#39;: 615,
+ &#39;1100&#39;: 635,
+ &#39;1111&#39;: 582}
 </pre></div>
 </div>
 </div>