Added shors algorithmus and minor fixes in bernstein-vazirani

933356d4 · Stefan Stump · b4a87c55 · 933356d4 · 933356d4 · 933356d4
Commit 933356d4 authored 1 month ago by Stefan Stump
--- a/_book/Introduction-to-Quantum-Computing.pdf
+++ b/_book/Introduction-to-Quantum-Computing.pdf
--- a/_book/content/bernsteinVazirani.html
+++ b/_book/content/bernsteinVazirani.html
@@ -30,6 +30,7 @@ ul.task-list li input[type="checkbox"] {
 <script src="../site_libs/quarto-search/fuse.min.js"></script>
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
+<link href="../content/shorsAlgorithm.html" rel="next">
 <link href="../content/ketNotation.html" rel="prev">
 <script src="../site_libs/quarto-html/quarto.js" type="module"></script>
 <script src="../site_libs/quarto-html/tabsets/tabsets.js" type="module"></script>
@@ -218,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link active">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>
@@ -263,13 +270,15 @@ window.Quarto = {
 </figure>
 </div>
 <p>Note that <span class="math inline">\(U_f\)</span> is defined with the explanation below.</p>
-<p>We start with <span class="math inline">\(n\)</span> qubits on the top wire. All of these qubits are in the state <span class="math inline">\(\ket{0}\)</span>, which we write <span class="math inline">\(\ket{0}^n = \ket{0} \otimes \dots \otimes \ket{0}\)</span>. The bottom wire is in the state <span class="math inline">\(\ket{1}\)</span>. Both wires composed together can be written as <span class="math inline">\(\psi_1 = \ket{0}^n \otimes \ket{1}\)</span>, which is the overall starting state of our algorithm. We now perform the following steps</p>
+<p>We start with <span class="math inline">\(n\)</span> qubits on the top wire. All of these qubits are in the state <span class="math inline">\(\ket{0}\)</span>, which we write <span class="math inline">\(\ket{0}^n = \ket{0} \otimes \dots \otimes \ket{0}\)</span>. The bottom wire is in the state <span class="math inline">\(\ket{1}\)</span>. Both wires composed together can be written as <span class="math inline">\(\ket{\psi_0} = \ket{0^n1} = \ket{0}^n \otimes \ket{1}\)</span>, which is the overall starting state of our algorithm. We now perform the following steps</p>
 <ol type="1">
 <li><p>First we apply a Hadamard gate on all qubits. This is denoted for the first <span class="math inline">\(n\)</span> qubits by the <span class="math inline">\(H^{\otimes n}\)</span> gate and for the last qubit by the <span class="math inline">\(H\)</span> gate on the bottom wire. The resulting quantum state is calculated as follows: <span class="math display">\[
 \begin{aligned}
-\psi_2 &amp;= \left(H^{\otimes n} \otimes H\right) \left(\ket{0}^n \otimes \ket{1}\right)\\
+\ket{\psi_1}
+&amp;= \left(H^{\otimes n} \otimes H\right) \left(\ket{\psi_0}\right)\\
+&amp;= \left(H^{\otimes n} \otimes H\right) \left(\ket{0}^n \otimes \ket{1}\right)\\
 &amp;=\left(H^{\otimes n}\ket{0}^n\right) \otimes H\ket{1}\\
-&amp;=\left(\ket{+}\right)^{\otimes n} \otimes \ket{-}\\
+&amp;=\ket{+}^{\otimes n} \otimes \ket{-}\\
 &amp;=\left(\frac{1}{\sqrt{2}} \ket{0} + \frac{1}{\sqrt{2}} \ket{1}  \right)^{\otimes n} \otimes \ket{-}\\
 &amp;=\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x}  \otimes \ket{-}
 \end{aligned}
@@ -278,15 +287,18 @@ window.Quarto = {
 U_f\,\ket{x,y} = \ket{x,y \oplus f(x)}
 \]</span> This unitary represents the function <span class="math inline">\(f\)</span> and combines the output of <span class="math inline">\(f(x)\)</span> with the bottom wire <span class="math inline">\(y\)</span>. For our quantum states, this means that the state after <span class="math inline">\(U_f\)</span> can be calculated as follows: <span class="math display">\[
 \begin{aligned}
-\psi_3&amp;=U_f\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x} \otimes \ket{-}\\
+\ket{\psi_2}
+&amp;= U_f\ket{\psi_1}\\
+&amp;= U_f\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x} \otimes \ket{-}\\
+&amp;= U_f\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x} \otimes \ket{-}\\
 &amp;=\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} U_f\Bigl (\ket{x} \otimes \ket{-}\Bigr) \\
 &amp;\stackrel{*}{=}\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} (-1)^{f(x)}\ket{x} \otimes \ket{-} \\
 &amp;=\left(\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} (-1)^{f(x)}\ket{x} \right) \otimes \ket{-}
 \end{aligned}
-\]</span> Note that the <span class="math inline">\(*\)</span> holds since we can rewrite <span class="math inline">\(U_f(\ket{x}\otimes\ket{-})\)</span> as <span class="math display">\[
+\]</span> Note that the <span class="math inline">\(*\)</span> holds since we can rewrite <span class="math inline">\(U_f\Bigl(\ket{x}\otimes\ket{-}\Bigr)\)</span> as <span class="math display">\[
 \begin{aligned}
-&amp;U_f\Bigl(\ket{x}\otimes\ket{-}\Bigr)\\
-&amp;= \frac{1}{\sqrt{2}} U_f\ket{x,0} - \frac{1}{\sqrt{2}} \ket{x,1}\\
+U_f\Bigl(\ket{x}\otimes\ket{-}\Bigr)
+&amp;= \frac{1}{\sqrt{2}} U_f\ket{x,0} - \frac{1}{\sqrt{2}} U_f\ket{x,1}\\
 &amp;= \frac{1}{\sqrt{2}} \ket{x,f(x)} - \frac{1}{\sqrt{2}} \ket{x,\overline{f(x)}}\\
 &amp;= \begin{cases} \frac{1}{\sqrt{2}} \ket{x,0} - \frac{1}{\sqrt{2}} \ket{x,1} &amp; f(x)=0\\
 \frac{1}{\sqrt{2}} \ket{x,1} - \frac{1}{\sqrt{2}} \ket{x,0} &amp; f(x)=1 \end{cases}\\
@@ -295,9 +307,11 @@ U_f\,\ket{x,y} = \ket{x,y \oplus f(x)}
 &amp;= (-1)^{f(x)} \ket{x} \otimes \ket{-}
 \end{aligned}
 \]</span> The bottom wire has not changed and is still <span class="math inline">\(\ket{-}\)</span>. But on the top wire, we now have <span class="math inline">\(f(x)\)</span> somehow encoded into our quantum state. The phenomenon that the output of <span class="math inline">\(f\)</span> is encoded as a <span class="math inline">\(-1\)</span> in the input register is called phase kickback. Measuring this quantum state would not give us any advantage, since we would just get one random <span class="math inline">\(x\)</span>. We therefore perform one final step before measuring.</p></li>
-<li><p>As the final unitary, we perform another <span class="math inline">\(H^{\otimes n}\)</span> on the top wire. We hope that the result of this unitary transformation is the state <span class="math inline">\(\ket{s}\)</span>. To check, whether our hopes become reality, we can calculate <span class="math inline">\(\left(H^{\otimes n}\right)^\dagger \ket{s} \otimes \ket{-}\)</span> and check if it is equal to <span class="math inline">\(\psi_3\)</span>. We do it in this direction, since these calculations are a bit simpler: <span class="math display">\[
+<li><p>As the final unitary, we perform another <span class="math inline">\(H^{\otimes n}\)</span> on the top wire. We hope that the result of this unitary transformation is the state <span class="math inline">\(\ket{\psi_3} = \ket{s} \otimes \ket{-}\)</span>. To check, whether our hopes become reality, we can calculate <span class="math inline">\(\left(H^{\otimes n}\right)^\dagger \ket{s} \otimes \ket{-}\)</span> and check if it is equal to <span class="math inline">\(\ket{\psi_2}\)</span>. We do it in this direction, since these calculations are a bit simpler: <span class="math display">\[
 \begin{aligned}
-\left( H^{\otimes n}\right )^\dagger \ket{s}\otimes \ket{-} &amp;= H^{\otimes n} \ket{s}\otimes \ket{-} \\
+\left(\left( H^{\otimes n}\right )^\dagger \otimes I \right) \ket{\psi_3}
+&amp;= \left( H^{\otimes n}\right )^\dagger \ket{s}\otimes \ket{-}\\
+&amp;= H^{\otimes n} \ket{s}\otimes \ket{-} \\
 &amp;= H^{\otimes n} (\ket{s_1} \otimes \dots \otimes \ket{s_n})\otimes \ket{-}\\
 &amp;= H\ket{s_1} \otimes \dots \otimes H\ket{s_n}\otimes \ket{-}\\
 &amp;= \bigotimes_{i=1}^n \left(\frac{1}{\sqrt{2}} \ket{0} + (-1)^{s_i}\frac{1}{\sqrt{2}} \ket{1}\right) \otimes \ket{-}\\
@@ -306,7 +320,7 @@ U_f\,\ket{x,y} = \ket{x,y \oplus f(x)}
 &amp;=\frac{1}{\sqrt{2^{n}}}\sum_{x\in\{0,1\}^n} \left( (-1)^{\sum_i x_is_i \bmod 2}\ket{x}\right) \otimes \ket{-}\\
 &amp;=\frac{1}{\sqrt{2^{n}}}\sum_{x\in\{0,1\}^n}  (-1)^{s\cdot x}\ket{x} \otimes \ket{-}\\
 &amp;=\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} (-1)^{f(x)}\ket{x}  \otimes \ket{-}\\
-&amp;= \psi_3
+&amp;= \ket{\psi_2}
 \end{aligned}
 \]</span> This calculation shows that we have the quantum state <span class="math inline">\(\ket{s} \otimes \ket{-}\)</span> before the measurement.</p></li>
 <li><p>We now perform a measurement on the top wire and measure <span class="math inline">\(s\)</span> as a result.</p></li>
@@ -722,6 +736,9 @@ U_f\,\ket{x,y} = \ket{x,y \oplus f(x)}
      </a>          
  </div>
  <div class="nav-page nav-page-next">
+      <a href="../content/shorsAlgorithm.html" class="pagination-link" aria-label="Shor's Algorithm">
+        <span class="nav-page-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span> <i class="bi bi-arrow-right-short"></i>
+      </a>
  </div>
 </nav>
 </div> <!-- /content -->

--- a/_book/content/compositeSystems.html
+++ b/_book/content/compositeSystems.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/img/bernstein_vazirani.svg
+++ b/_book/content/img/bernstein_vazirani.svg
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+</g>
+</svg>
\ No newline at end of file
--- a/_book/content/img/dft_n_qubits.svg
+++ b/_book/content/img/dft_n_qubits.svg
--- a/_book/content/img/shor.svg
+++ b/_book/content/img/shor.svg
--- a/_book/content/introduction.html
+++ b/_book/content/introduction.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/ketNotation.html
+++ b/_book/content/ketNotation.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/observingSystems.html
+++ b/_book/content/observingSystems.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/partialObserving.html
+++ b/_book/content/partialObserving.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/probabilisticSystems.html
+++ b/_book/content/probabilisticSystems.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/quantumCircuits.html
+++ b/_book/content/quantumCircuits.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/quantumSystems.html
+++ b/_book/content/quantumSystems.html
@@ -219,6 +219,12 @@ window.Quarto = {
  <a href="../content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>

--- a/_book/content/shorsAlgorithm.html
+++ b/_book/content/shorsAlgorithm.html
--- a/_book/index.html
+++ b/_book/index.html
@@ -9,7 +9,7 @@
 <meta name="author" content="Jannik Hellenkamp">
 <meta name="author" content="Stefan Stump">
 <meta name="author" content="Dominique Unruh">
-<meta name="dcterms.date" content="2025-05-16">
+<meta name="dcterms.date" content="2025-06-03">

 <title>Introduction to Quantum Computing</title>
 <style>
@@ -193,6 +193,12 @@ ul.task-list li input[type="checkbox"] {
  <a href="./content/bernsteinVazirani.html" class="sidebar-item-text sidebar-link">
 <span class="menu-text"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Bernstein-Vazirani Algorithm</span></span></a>
  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
+  </div>
 </li>
      </ul>
  </li>
@@ -206,10 +212,7 @@ ul.task-list li input[type="checkbox"] {
    <h2 id="toc-title">Table of contents</h2>
   
  <ul>
-  <li><a href="#welcome" id="toc-welcome" class="nav-link active" data-scroll-target="#welcome">Welcome</a>
-  <ul class="collapse">
-  <li><a href="#changelog" id="toc-changelog" class="nav-link" data-scroll-target="#changelog">Changelog</a></li>
-  </ul></li>
+  <li><a href="#welcome" id="toc-welcome" class="nav-link active" data-scroll-target="#welcome">Welcome</a></li>
  </ul>
 </nav>
    </div>
@@ -238,7 +241,7 @@ ul.task-list li input[type="checkbox"] {
    <div>
    <div class="quarto-title-meta-heading">Published</div>
    <div class="quarto-title-meta-contents">
-      <p class="date">May 16, 2025</p>
+      <p class="date">June 3, 2025</p>
    </div>
  </div>
  
@@ -259,18 +262,8 @@ ul.task-list li input[type="checkbox"] {
 <p>If you spot an error, please report it to <a href="https://git.rwth-aachen.de/unruh/script-intro-qc/">Gitlab</a>. Alternatively, you can send Stefan Stump an e-mail (stefan.stump@rwth-aachen.de).If you have a question of understanding, please ask it in the Moodle forum.</p>
 <p>These lecture notes are released under the CC BY-NC 4.0 license, which can be found <a href="https://creativecommons.org/licenses/by-nc/4.0/">here</a>.</p>
 <p>The Jupyter notebooks created during the lectures can be found in the <a href="https://jupyter.rwth-aachen.de/">JupyterHub</a> in the course “[IQC] Introduction to Quantum Computing” or in Moodle. These will be added shortly after the lectures. If changes are made to the files, they can be easily reset with the usual git commands using “Git” and then “Open Git Repository in Terminal.</p>
-<section id="changelog" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="changelog">Changelog</h2>
-<section id="section" class="level4">
-<h4 class="anchored" data-anchor-id="section">16.05.2025</h4>
-<ul>
-<li>started the lecture notes for the summer term 2025</li>
-<li>added chapters 1 to 9 (except 4.3)</li>
-</ul>


-</section>
-</section>
 </section>

 </main> <!-- /main -->

--- a/_book/search.json
+++ b/_book/search.json
--- a/_book/sitemap.xml
+++ b/_book/sitemap.xml
@@ -2,46 +2,50 @@
 <urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/index.html</loc>
-    <lastmod>2025-05-16T15:50:32.669Z</lastmod>
+    <lastmod>2025-06-03T15:52:33.090Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/introduction.html</loc>
-    <lastmod>2025-05-10T11:43:43.511Z</lastmod>
+    <lastmod>2025-05-19T12:07:45.152Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/probabilisticSystems.html</loc>
-    <lastmod>2025-05-16T14:08:33.952Z</lastmod>
+    <lastmod>2025-05-19T12:07:45.236Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/quantumSystems.html</loc>
-    <lastmod>2025-05-13T11:25:06.952Z</lastmod>
+    <lastmod>2025-05-19T12:07:45.249Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/observingSystems.html</loc>
-    <lastmod>2025-05-16T14:03:14.603Z</lastmod>
+    <lastmod>2025-05-19T12:07:45.214Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/partialObserving.html</loc>
-    <lastmod>2025-05-16T14:09:38.516Z</lastmod>
+    <lastmod>2025-05-19T12:07:45.216Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/compositeSystems.html</loc>
-    <lastmod>2025-05-16T15:34:57.178Z</lastmod>
+    <lastmod>2025-05-19T12:07:44.906Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/quantumCircuits.html</loc>
-    <lastmod>2025-05-16T15:35:58.896Z</lastmod>
+    <lastmod>2025-05-19T12:07:45.236Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/ketNotation.html</loc>
-    <lastmod>2025-05-16T15:43:17.881Z</lastmod>
+    <lastmod>2025-05-19T12:07:45.172Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/bernsteinVazirani.html</loc>
-    <lastmod>2025-05-16T15:41:32.344Z</lastmod>
+    <lastmod>2025-05-30T18:54:32.195Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/shorsAlgorithm.html</loc>
+    <lastmod>2025-06-03T15:46:14.791Z</lastmod>
  </url>
  <url>
    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/Introduction-to-Quantum-Computing.pdf</loc>
-    <lastmod>2025-05-16T15:51:08.539Z</lastmod>
+    <lastmod>2025-06-03T15:53:37.480Z</lastmod>
  </url>
 </urlset>
--- a/_quarto.yml
+++ b/_quarto.yml
@@ -31,7 +31,7 @@ book:
    - part: "Quantum algorithms"
      chapters:
        - content/bernsteinVazirani.qmd
-    #    - content/shorsAlgorithm.qmd
+        - content/shorsAlgorithm.qmd
    #    - content/groversAlgorithm.qmd
    #- part: "Physical background"
    #  chapters:

--- a/content/bernsteinVazirani.qmd
+++ b/content/bernsteinVazirani.qmd
@@ -14,15 +14,17 @@ We will now look at a quantum algorithm which will find $s$ with only one evalua

 Note that $U_f$ is defined with the explanation below.

-We start with $n$ qubits on the top wire. All of these qubits are in the state $\ket{0}$, which we write $\ket{0}^n = \ket{0} \otimes \dots \otimes \ket{0}$. The bottom wire is in the state $\ket{1}$. Both wires composed together can be written as $\psi_1 = \ket{0}^n \otimes \ket{1}$, which is the overall starting state of our algorithm.
+We start with $n$ qubits on the top wire. All of these qubits are in the state $\ket{0}$, which we write $\ket{0}^n = \ket{0} \otimes \dots \otimes \ket{0}$. The bottom wire is in the state $\ket{1}$. Both wires composed together can be written as $\ket{\psi_0} = \ket{0^n1} = \ket{0}^n \otimes \ket{1}$, which is the overall starting state of our algorithm.
 We now perform the following steps

 1. First we apply a Hadamard gate on all qubits. This is denoted for the first $n$ qubits by the $H^{\otimes n}$ gate and for the last qubit by the $H$ gate on the bottom wire. The resulting quantum state is calculated as follows:
 $$
 \begin{aligned}
-\psi_2 &= \left(H^{\otimes n} \otimes H\right) \left(\ket{0}^n \otimes \ket{1}\right)\\
+\ket{\psi_1}
+&= \left(H^{\otimes n} \otimes H\right) \left(\ket{\psi_0}\right)\\
+&= \left(H^{\otimes n} \otimes H\right) \left(\ket{0}^n \otimes \ket{1}\right)\\
 &=\left(H^{\otimes n}\ket{0}^n\right) \otimes H\ket{1}\\
-&=\left(\ket{+}\right)^{\otimes n} \otimes \ket{-}\\
+&=\ket{+}^{\otimes n} \otimes \ket{-}\\
 &=\left(\frac{1}{\sqrt{2}} \ket{0} + \frac{1}{\sqrt{2}} \ket{1}  \right)^{\otimes n} \otimes \ket{-}\\
 &=\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x}  \otimes \ket{-}
 \end{aligned}
@@ -36,17 +38,20 @@ $$
 This unitary represents the function $f$ and combines the output of $f(x)$ with the bottom wire $y$. For our quantum states, this means that the state after $U_f$ can be calculated as follows:
 $$
 \begin{aligned}
-\psi_3&=U_f\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x} \otimes \ket{-}\\
+\ket{\psi_2}
+&= U_f\ket{\psi_1}\\
+&= U_f\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x} \otimes \ket{-}\\
+&= U_f\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} \ket{x} \otimes \ket{-}\\
 &=\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} U_f\Bigl (\ket{x} \otimes \ket{-}\Bigr) \\
 &\stackrel{*}{=}\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} (-1)^{f(x)}\ket{x} \otimes \ket{-} \\
 &=\left(\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} (-1)^{f(x)}\ket{x} \right) \otimes \ket{-}
 \end{aligned}
 $$
-Note that the $*$ holds since we can rewrite $U_f(\ket{x}\otimes\ket{-})$ as
+Note that the $*$ holds since we can rewrite $U_f\Bigl(\ket{x}\otimes\ket{-}\Bigr)$ as
 $$
 \begin{aligned}
-&U_f\Bigl(\ket{x}\otimes\ket{-}\Bigr)\\
-&= \frac{1}{\sqrt{2}} U_f\ket{x,0} - \frac{1}{\sqrt{2}} \ket{x,1}\\ 
+U_f\Bigl(\ket{x}\otimes\ket{-}\Bigr)
+&= \frac{1}{\sqrt{2}} U_f\ket{x,0} - \frac{1}{\sqrt{2}} U_f\ket{x,1}\\ 
 &= \frac{1}{\sqrt{2}} \ket{x,f(x)} - \frac{1}{\sqrt{2}} \ket{x,\overline{f(x)}}\\
 &= \begin{cases} \frac{1}{\sqrt{2}} \ket{x,0} - \frac{1}{\sqrt{2}} \ket{x,1} & f(x)=0\\
 \frac{1}{\sqrt{2}} \ket{x,1} - \frac{1}{\sqrt{2}} \ket{x,0} & f(x)=1 \end{cases}\\
@@ -57,10 +62,12 @@ $$
 $$
 The bottom wire has not changed and is still $\ket{-}$. But on the top wire, we now have $f(x)$ somehow encoded into our quantum state. The phenomenon that the output of $f$ is encoded as a $-1$ in the input register is called phase kickback. Measuring this quantum state would not give us any advantage, since we would just get one random $x$. We therefore perform one final step before measuring.

-3. As the final unitary, we perform another $H^{\otimes n}$ on the top wire. We hope that the result of this unitary transformation is the state $\ket{s}$. To check, whether our hopes become reality, we can calculate  $\left(H^{\otimes n}\right)^\dagger \ket{s} \otimes \ket{-}$ and check if it is equal to $\psi_3$. We do it in this direction, since these calculations are a bit simpler:
+3. As the final unitary, we perform another $H^{\otimes n}$ on the top wire. We hope that the result of this unitary transformation is the state $\ket{\psi_3} = \ket{s} \otimes \ket{-}$. To check, whether our hopes become reality, we can calculate  $\left(H^{\otimes n}\right)^\dagger \ket{s} \otimes \ket{-}$ and check if it is equal to $\ket{\psi_2}$. We do it in this direction, since these calculations are a bit simpler:
 $$
 \begin{aligned}
-\left( H^{\otimes n}\right )^\dagger \ket{s}\otimes \ket{-} &= H^{\otimes n} \ket{s}\otimes \ket{-} \\
+\left(\left( H^{\otimes n}\right )^\dagger \otimes I \right) \ket{\psi_3}
+&= \left( H^{\otimes n}\right )^\dagger \ket{s}\otimes \ket{-}\\
+&= H^{\otimes n} \ket{s}\otimes \ket{-} \\
 &= H^{\otimes n} (\ket{s_1} \otimes \dots \otimes \ket{s_n})\otimes \ket{-}\\
 &= H\ket{s_1} \otimes \dots \otimes H\ket{s_n}\otimes \ket{-}\\
 &= \bigotimes_{i=1}^n \left(\frac{1}{\sqrt{2}} \ket{0} + (-1)^{s_i}\frac{1}{\sqrt{2}} \ket{1}\right) \otimes \ket{-}\\
@@ -69,7 +76,7 @@ $$
 &=\frac{1}{\sqrt{2^{n}}}\sum_{x\in\{0,1\}^n} \left( (-1)^{\sum_i x_is_i \bmod 2}\ket{x}\right) \otimes \ket{-}\\
 &=\frac{1}{\sqrt{2^{n}}}\sum_{x\in\{0,1\}^n}  (-1)^{s\cdot x}\ket{x} \otimes \ket{-}\\
 &=\frac{1}{\sqrt{2^{n}}}\sum_{x\in \{0,1\}^n} (-1)^{f(x)}\ket{x}  \otimes \ket{-}\\
-&= \psi_3
+&= \ket{\psi_2}
 \end{aligned}
 $$
 This calculation shows that we have the quantum state $\ket{s} \otimes \ket{-}$ before the measurement.