From 18bfaad4a0b33ac85ce52a4dc75bee07cc220f81 Mon Sep 17 00:00:00 2001
From: Stefan Stump <stefan.stump@rwth-aachen.de>
Date: Wed, 4 Jun 2025 12:38:06 +0200
Subject: [PATCH] Added grovers algorithm

---
 _book/Introduction-to-Quantum-Computing.pdf   | Bin 902774 -> 1183007 bytes
 _book/content/bernsteinVazirani.html          |   6 +
 _book/content/compositeSystems.html           |   6 +
 _book/content/groversAlgorithm.html           | 820 ++++++++++++++++++
 _book/content/img/dft_3_qubits.svg            | 198 +++--
 _book/content/img/flip_star.svg               |  91 ++
 _book/content/img/flip_zero.svg               |  73 ++
 _book/content/img/grover.svg                  | 100 +++
 _book/content/img/grover_plot_1.svg           |  43 +
 _book/content/img/grover_plot_2.svg           |  56 ++
 _book/content/img/grover_plot_3.svg           |  76 ++
 _book/content/img/vf.svg                      |  65 ++
 _book/content/introduction.html               |   6 +
 _book/content/ketNotation.html                |   6 +
 _book/content/observingSystems.html           |   6 +
 _book/content/partialObserving.html           |   6 +
 _book/content/probabilisticSystems.html       |   6 +
 _book/content/quantumCircuits.html            |   6 +
 _book/content/quantumSystems.html             |   6 +
 _book/content/shorsAlgorithm.html             |  20 +-
 _book/index.html                              |  10 +-
 _book/search.json                             |  35 +-
 _book/sitemap.xml                             |   8 +-
 _quarto.yml                                   |   2 +-
 content/groversAlgorithm.qmd                  |  62 +-
 content/img/dft_3_qubits.pdf                  | Bin 37385 -> 37364 bytes
 content/img/dft_3_qubits.svg                  | 198 +++--
 content/img/flip.pdf                          | Bin 43625 -> 0 bytes
 content/img/flip.svg                          |   1 -
 content/img/flip_star.pdf                     | Bin 0 -> 37738 bytes
 content/img/flip_star.svg                     |  91 ++
 content/img/flip_zero.pdf                     | Bin 0 -> 28965 bytes
 content/img/flip_zero.svg                     |  73 ++
 content/img/grover.pdf                        | Bin 49289 -> 56703 bytes
 content/img/grover.svg                        | 101 ++-
 .../{grover_plt1.pdf => grover_plot_1.pdf}    | Bin 28846 -> 28546 bytes
 content/img/grover_plot_1.svg                 |  43 +
 .../{grover_plt2.pdf => grover_plot_2.pdf}    | Bin 37374 -> 37054 bytes
 content/img/grover_plot_2.svg                 |  56 ++
 content/img/grover_plot_3.pdf                 | Bin 0 -> 46179 bytes
 content/img/grover_plot_3.svg                 |  76 ++
 content/img/grover_plt1.svg                   |   1 -
 content/img/grover_plt2.svg                   |   1 -
 content/img/grover_plt3.pdf                   | Bin 55019 -> 0 bytes
 content/img/grover_plt3.svg                   |   1 -
 content/img/vf.pdf                            | Bin 37372 -> 36913 bytes
 content/img/vf.svg                            |  66 +-
 content/latex/dft_3_qubits.tex                |   2 +-
 content/latex/flip_star.tex                   |  13 +
 content/latex/flip_zero.tex                   |  13 +
 content/latex/grover.tex                      |  10 +
 content/latex/grover_plot_1.tex               |  20 +
 content/latex/grover_plot_2.tex               |  26 +
 content/latex/grover_plot_3.tex               |  32 +
 content/latex/vf.tex                          |  11 +
 content/shorsAlgorithm.qmd                    |  11 +-
 56 files changed, 2317 insertions(+), 242 deletions(-)
 create mode 100644 _book/content/groversAlgorithm.html
 create mode 100644 _book/content/img/flip_star.svg
 create mode 100644 _book/content/img/flip_zero.svg
 create mode 100644 _book/content/img/grover.svg
 create mode 100644 _book/content/img/grover_plot_1.svg
 create mode 100644 _book/content/img/grover_plot_2.svg
 create mode 100644 _book/content/img/grover_plot_3.svg
 create mode 100644 _book/content/img/vf.svg
 delete mode 100644 content/img/flip.pdf
 delete mode 100644 content/img/flip.svg
 create mode 100644 content/img/flip_star.pdf
 create mode 100644 content/img/flip_star.svg
 create mode 100644 content/img/flip_zero.pdf
 create mode 100644 content/img/flip_zero.svg
 rename content/img/{grover_plt1.pdf => grover_plot_1.pdf} (54%)
 create mode 100644 content/img/grover_plot_1.svg
 rename content/img/{grover_plt2.pdf => grover_plot_2.pdf} (63%)
 create mode 100644 content/img/grover_plot_2.svg
 create mode 100644 content/img/grover_plot_3.pdf
 create mode 100644 content/img/grover_plot_3.svg
 delete mode 100644 content/img/grover_plt1.svg
 delete mode 100644 content/img/grover_plt2.svg
 delete mode 100644 content/img/grover_plt3.pdf
 delete mode 100644 content/img/grover_plt3.svg
 create mode 100644 content/latex/flip_star.tex
 create mode 100644 content/latex/flip_zero.tex
 create mode 100644 content/latex/grover.tex
 create mode 100644 content/latex/grover_plot_1.tex
 create mode 100644 content/latex/grover_plot_2.tex
 create mode 100644 content/latex/grover_plot_3.tex
 create mode 100644 content/latex/vf.tex

diff --git a/_book/Introduction-to-Quantum-Computing.pdf b/_book/Introduction-to-Quantum-Computing.pdf
index 2d42978b42c8bddd5959bf358073b04664d35804..0612f3ad435774e5a068cdd52c0e4e67478baf57 100644
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diff --git a/_book/content/bernsteinVazirani.html b/_book/content/bernsteinVazirani.html
index d4ce2b8..9607872 100644
--- a/_book/content/bernsteinVazirani.html
+++ b/_book/content/bernsteinVazirani.html
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diff --git a/_book/content/compositeSystems.html b/_book/content/compositeSystems.html
index fbb0ddd..80e10f0 100644
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+        <nav id="TOC" role="doc-toc" class="toc-active">
+    <h2 id="toc-title">Table of contents</h2>
+   
+  <ul>
+  <li><a href="#preparations" id="toc-preparations" class="nav-link active" data-scroll-target="#preparations"><span class="header-section-number">11.1</span> Preparations</a>
+  <ul class="collapse">
+  <li><a href="#constructing-the-oracle-v_f" id="toc-constructing-the-oracle-v_f" class="nav-link" data-scroll-target="#constructing-the-oracle-v_f"><span class="header-section-number">11.1.1</span> Constructing the oracle <span class="math inline">\(V_f\)</span></a></li>
+  <li><a href="#constructing-operatornameflip_" id="toc-constructing-operatornameflip_" class="nav-link" data-scroll-target="#constructing-operatornameflip_"><span class="header-section-number">11.1.2</span> Constructing <span class="math inline">\(\operatorname{FLIP}_*\)</span></a></li>
+  </ul></li>
+  <li><a href="#the-algorithm-for-searching" id="toc-the-algorithm-for-searching" class="nav-link" data-scroll-target="#the-algorithm-for-searching"><span class="header-section-number">11.2</span> The algorithm for searching</a>
+  <ul class="collapse">
+  <li><a href="#understanding-the-algorithm-for-searching" id="toc-understanding-the-algorithm-for-searching" class="nav-link" data-scroll-target="#understanding-the-algorithm-for-searching"><span class="header-section-number">11.2.1</span> Understanding the algorithm for searching</a></li>
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+<div class="quarto-title">
+<h1 class="title"><span class="chapter-number">11</span>&nbsp; <span class="chapter-title">Grover’s algorithm</span></h1>
+</div>
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+
+<p>Another well known quantum algorithm is Grover’s algorithm for searching. It was developed by Lov Grover in 1996.</p>
+<p>Grover’s algorithm takes a function <span class="math inline">\(f: \{0,1\}^n \rightarrow \{0,1\}\)</span>, where exactly one <span class="math inline">\(x_0\)</span> exists, such that <span class="math inline">\(f(x_0) = 1\)</span>. The goal is to find <span class="math inline">\(x_0\)</span>.</p>
+<p>There are a number of interesting problems, which can be reduced to this general definition. One of these problems is the breaking of a (symmetric) encryption. The function <span class="math inline">\(f\)</span> would take a key as an input and output a <span class="math inline">\(1\)</span>, if the decryption is successful. Otherwise it will output a <span class="math inline">\(0\)</span>.</p>
+<p>Classically, finding this <span class="math inline">\(x_0\)</span> takes approximately <span class="math inline">\(2^n\)</span> steps (when simply bruteforcing the function). Using Grover’s algorithm, we can reduce this runtime to approximately <span class="math inline">\(\sqrt{2^n}\)</span> steps. As an example, a <span class="math inline">\(128\)</span>-bit encryption would only take about <span class="math inline">\(2^{64}\)</span> steps to break it, instead of about <span class="math inline">\(2^{128}\)</span> steps for the classical bruteforce.</p>
+<section id="preparations" class="level2" data-number="11.1">
+<h2 data-number="11.1" class="anchored" data-anchor-id="preparations"><span class="header-section-number">11.1</span> Preparations</h2>
+<p>To construct Grover’s algorithm, we first need to introduce two new gates <span class="math inline">\(V_f\)</span> and <span class="math inline">\(\operatorname{FLIP}_*\)</span>.</p>
+<section id="constructing-the-oracle-v_f" class="level3" data-number="11.1.1">
+<h3 data-number="11.1.1" class="anchored" data-anchor-id="constructing-the-oracle-v_f"><span class="header-section-number">11.1.1</span> Constructing the oracle <span class="math inline">\(V_f\)</span></h3>
+<p>In the previous algorithms, we have learned that we can implement a function <span class="math inline">\(f\)</span> as a unitary <span class="math inline">\(U_f\)</span> with <span class="math inline">\(U_f\ket{x,y} = \ket{x, y \oplus f(x)}\)</span>. We construct a different unitary called <span class="math inline">\(V_f\)</span> from this, which has the following behavior:</p>
+<p><span class="math display">\[
+V_f \ket{x} =
+\begin{cases}
+    -\ket{x} &amp; \text{if } f(x) = 1 \\
+    \ket{x} &amp; \text{else}
+\end{cases}
+\]</span></p>
+<p>We can construct <span class="math inline">\(V_f\)</span> from <span class="math inline">\(U_f\)</span> using the following circuit:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="img/vf.svg" class="img-fluid figure-img" style="width:60.0%"></p>
+<figcaption>The circuit for <span class="math inline">\(V_f\)</span></figcaption>
+</figure>
+</div>
+<p>The bottom wire can be discarded, since it always contains a <span class="math inline">\(\ket{-}\)</span> and thus is not entangled with the upper wire.</p>
+</section>
+<section id="constructing-operatornameflip_" class="level3" data-number="11.1.2">
+<h3 data-number="11.1.2" class="anchored" data-anchor-id="constructing-operatornameflip_"><span class="header-section-number">11.1.2</span> Constructing <span class="math inline">\(\operatorname{FLIP}_*\)</span></h3>
+<p>As a second ingredient for Grover’s algorithm, we need the unitary <span class="math inline">\(\operatorname{FLIP}_*\)</span>. To realize this, we first need <span class="math inline">\(\operatorname{FLIP}_0\)</span>, which ist defined by the unitary <span class="math display">\[
+\operatorname{FLIP}_0 \ket{x} =
+\begin{cases}
+    \ket{0} &amp; \text{if } x = 0 \\
+    -\ket{x} &amp; \text{else}.
+\end{cases}
+\]</span></p>
+<p><span class="math inline">\(\operatorname{FLIP}_0\)</span> is implemented by the following quantum circuit:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="img/flip_zero.svg" class="img-fluid figure-img" style="width:50.0%"></p>
+<figcaption>The circuit for <span class="math inline">\(\operatorname{FLIP}_0\)</span></figcaption>
+</figure>
+</div>
+<p><span class="math inline">\(Z\)</span> is the Pauli matrix from definition <a href="quantumCircuits.html#def-gates-pauli" class="quarto-xref">Definition&nbsp;<span>7.2</span></a>. The empty circles indicates a negative control wire. So <span class="math inline">\(Z\)</span> is only applied if the other wires are <span class="math inline">\(\ket{0}\)</span>.</p>
+<p>Now we can define the unitary called <span class="math inline">\(\operatorname{FLIP}_*\)</span>. This unitary does nothing, if it is applied on the uniform superposition <span class="math inline">\(\ket{*}\)</span>. For any other quantum state <span class="math inline">\(\ket{\psi}\)</span> orthogonal to <span class="math inline">\(\ket{*}\)</span> it maps to <span class="math inline">\(-\ket{\psi}\)</span>. The uniform superposition <span class="math inline">\(\ket{*}\)</span> simply denotes the superposition over all classical possibilities <span class="math inline">\(\ket{*} = \frac{1}{\sqrt{2^n}} \sum_{x \in\{0,1\}^n} \ket{x}\)</span>. So <span class="math inline">\(\operatorname{FLIP}_*\)</span> is described by:</p>
+<p><span class="math display">\[
+\operatorname{FLIP}_* \ket{\psi} =
+\begin{cases}
+    \ket{*} &amp; \text{if } \ket{\psi} = \ket{*} \\
+    -\ket{x} &amp; \text{if } \ket{\psi} \perp \ket{*} \text{ (orthogonal)}.
+\end{cases}
+\]</span></p>
+<p>We can construct this <span class="math inline">\(\operatorname{FLIP}_*\)</span> by the following quantum circuit:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="img/flip_star.svg" class="img-fluid figure-img" style="width:60.0%"></p>
+<figcaption>The circuit for <span class="math inline">\(\operatorname{FLIP}_*\)</span></figcaption>
+</figure>
+</div>
+</section>
+</section>
+<section id="the-algorithm-for-searching" class="level2" data-number="11.2">
+<h2 data-number="11.2" class="anchored" data-anchor-id="the-algorithm-for-searching"><span class="header-section-number">11.2</span> The algorithm for searching</h2>
+<p>The actual algorithm takes a function <span class="math inline">\(f: \{0,1\}^n \rightarrow \{0,1\}\)</span> and outputs an <span class="math inline">\(x_0\)</span> with <span class="math inline">\(f(x_0)=1\)</span>. For simplicity, we assume that there is only one <span class="math inline">\(x_0\)</span> for which <span class="math inline">\(f(x_0) = 1\)</span> holds and for each other <span class="math inline">\(x \neq x_0\)</span> it holds that <span class="math inline">\(f(x)=0\)</span>.</p>
+<p>With the two new unitaries <span class="math inline">\(V_f\)</span> and <span class="math inline">\(\operatorname{FLIP}_*\)</span> defined, we can construct the circuit for Grover’s algorithm, which is shown in the following figure:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="img/grover.svg" class="img-fluid figure-img" style="width:70.0%"></p>
+<figcaption>The quantum circuit for Grover’s algorithm</figcaption>
+</figure>
+</div>
+<p>The algorithm works as follows:</p>
+<ol type="1">
+<li>We start with a <span class="math inline">\(\ket{0}\)</span> entry on every qubit.</li>
+<li>We bring the system into the superposition over all entries by applying <span class="math inline">\(H^{\otimes n}\)</span>. The quantum state is then <span class="math inline">\(\frac{1}{\sqrt{2^n}} \sum_{x \in \{0,1\}^n} \ket{x}\)</span> which we also call <span class="math inline">\(\ket{*}\)</span>.</li>
+<li>We apply the unitary <span class="math inline">\(V_f\)</span>.</li>
+<li>We apply the unitary <span class="math inline">\(\operatorname{FLIP}_*\)</span>.</li>
+<li>We repeat steps 3 and 4 <span class="math inline">\(t\)</span> times.</li>
+<li>We do a measurement.</li>
+</ol>
+<p>The measurement in step 6 will then give us <span class="math inline">\(x_0\)</span> with high probability.</p>
+<section id="understanding-the-algorithm-for-searching" class="level3" data-number="11.2.1">
+<h3 data-number="11.2.1" class="anchored" data-anchor-id="understanding-the-algorithm-for-searching"><span class="header-section-number">11.2.1</span> Understanding the algorithm for searching</h3>
+<p>When looking at the quantum circuit, it is not completely intuitive why the algorithm gives the correct result. We therefore now look into what is happening in each step.</p>
+<p>The desired quantum state after the algorithm finishes is <span class="math inline">\(\ket{x_0}\)</span>. At the beginning of the algorithm, we bring the system into the uniform superposition <span class="math inline">\(\ket{*} = \frac{1}{\sqrt{2^n}} \sum_{x \in \{0,1\}^n} \ket{x}\)</span>. We know that <span class="math inline">\(\ket{x_0}\)</span> is part of this superposition, therefore we can rewrite <span class="math inline">\(\ket{*}\)</span> as follows <span class="math display">\[
+\ket{*}
+= \frac{1}{\sqrt{2^n}} \sum_{x \in \{0,1\}^n} \ket{x}
+= \frac{1}{\sqrt{2^n}} \underbrace{\ket{x_0}}_{\textit{good}}  + \sqrt{\frac{2^n-1}{2^n}} \underbrace{\sum_{x\neq x_0} \frac{1}{\sqrt{2^n-1}}\ket{x}}_{\textit{bad}}
+\]</span></p>
+<p>So the current state can be seen as a superposition of a “good” state <em>good</em> and a “bad” state <em>bad</em>.</p>
+<p>The geometric interpretation of this superposition can be drawn as follows:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="img/grover_plot_1.svg" class="img-fluid figure-img" style="width:30.0%"></p>
+<figcaption>Geometric interpretation of <span class="math inline">\(\ket{*}\)</span></figcaption>
+</figure>
+</div>
+<p>The angel <span class="math inline">\(\theta\)</span> denotes, how “good” the resulting outcome will be. If <span class="math inline">\(\theta = 0\)</span>, the state is completely bad, if <span class="math inline">\(\theta = \frac{\pi}{2}\)</span>, the state is completely good.</p>
+<p>We can calculate <span class="math inline">\(\cos \theta = \lvert\braket{*|bad}\rvert = \frac{\sqrt{2^n-1}}{\sqrt{2^n}}\)</span>. From this we can derivate that the angle <span class="math inline">\(\theta\)</span> is <span class="math inline">\(\cos^{-1} \sqrt{\frac{2^n-1}{2^n}}\)</span> at the beginning, which is approximately <span class="math inline">\(\sqrt{\frac{1}{2^n}}\)</span>.</p>
+<p>We now apply <span class="math inline">\(V_f\)</span> on this quantum state. This will negate the amplitude off our desired <span class="math inline">\(\ket{x_0}\)</span> and not change the amplitude to the rest of the state.</p>
+<p><span class="math display">\[
+V_f\ket{*}= -\frac{1}{\sqrt{2^n}} \underbrace{\ket{x_0}}_{\text{good}}  + \sqrt{\frac{2^n-1}{2^n}} \underbrace{\sum_{x\neq x_0} \ket{x}}_{\text{bad}}
+\]</span></p>
+<p>This looks like this in the geometric interpretation:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="img/grover_plot_2.svg" class="img-fluid figure-img" style="width:30.0%"></p>
+<figcaption>Geometric interpretation after <span class="math inline">\(V_f\)</span></figcaption>
+</figure>
+</div>
+<p>We can see that by applying <span class="math inline">\(V_f\)</span>, we mirror the vector across the <em>bad</em> axis.</p>
+<p>After <span class="math inline">\(V_f\)</span>, we apply the <span class="math inline">\(\operatorname{FLIP}_*\)</span> operation on the quantum state. Since <span class="math inline">\(\operatorname{FLIP}_*\)</span> does nothing on the <span class="math inline">\(\ket{*}\)</span> entries and negates the amplitude of any vector orthogonal to it, <span class="math inline">\(\operatorname{FLIP}_*\)</span> mirrors the vector across <span class="math inline">\(\ket{*}\)</span>. This can be seen in the following figure:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="img/grover_plot_3.svg" class="img-fluid figure-img" style="width:30.0%"></p>
+<figcaption>Geometric interpretation after <span class="math inline">\(\operatorname{FLIP}_*\)</span></figcaption>
+</figure>
+</div>
+<p>All in all, we have seen that by applying <span class="math inline">\(V_f\)</span> and <span class="math inline">\(\operatorname{FLIP}_*\)</span>, we can increase the angle of the quantum state in relation to the “good” and “bad” states by <span class="math inline">\(2\theta\)</span>. Therefore two reflection give rotation. By repeating this step often enough, we can get the amplitude of <span class="math inline">\(\ket{x_0}\)</span> close to 1.</p>
+<p>To be more precise: Since we know <span class="math inline">\(\theta\)</span> and we know that we will increase the <em>good</em>-ness of our quantum state by <span class="math inline">\(2\theta\)</span> each time, we can calculate that only <span class="math inline">\(t\)</span> iterations are necessary with <span class="math display">\[
+t \approx \frac{\frac{\pi / 2}{\theta} - 1}{2} \approx \frac{\pi}{4 \theta} \approx \frac{\pi}{4} \cdot \sqrt{2^n}.
+\]</span> Grover’s algorithm therefore takes <span class="math inline">\(O(\sqrt{2^n})\)</span> steps, where an evaluation of the circuit counts as one step.</p>
+
+
+</section>
+</section>
+
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+  <div class="sidebar-item-container"> 
+  <a href="../content/groversAlgorithm.html" class="sidebar-item-text sidebar-link">
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+  <div class="sidebar-item-container"> 
+  <a href="../content/groversAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">11</span>&nbsp; <span class="chapter-title">Grover’s algorithm</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/content/quantumCircuits.html b/_book/content/quantumCircuits.html
index 9635c39..9bf0c3e 100644
--- a/_book/content/quantumCircuits.html
+++ b/_book/content/quantumCircuits.html
@@ -225,6 +225,12 @@ window.Quarto = {
   <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>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/groversAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">11</span>&nbsp; <span class="chapter-title">Grover’s algorithm</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/content/quantumSystems.html b/_book/content/quantumSystems.html
index 3a03a29..07a3c4d 100644
--- a/_book/content/quantumSystems.html
+++ b/_book/content/quantumSystems.html
@@ -225,6 +225,12 @@ window.Quarto = {
   <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>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/groversAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">11</span>&nbsp; <span class="chapter-title">Grover’s algorithm</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/content/shorsAlgorithm.html b/_book/content/shorsAlgorithm.html
index 3ee35d6..47d07c6 100644
--- a/_book/content/shorsAlgorithm.html
+++ b/_book/content/shorsAlgorithm.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/groversAlgorithm.html" rel="next">
 <link href="../content/bernsteinVazirani.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>
@@ -224,6 +225,12 @@ window.Quarto = {
   <a href="../content/shorsAlgorithm.html" class="sidebar-item-text sidebar-link active">
  <span class="menu-text"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">Shor’s Algorithm</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../content/groversAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">11</span>&nbsp; <span class="chapter-title">Grover’s algorithm</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
@@ -528,10 +535,12 @@ Example: Construction of the DFT circuit for <span class="math inline">\(n=3\)</
 <figcaption>The DTF for three qubits</figcaption>
 </figure>
 </div>
-<p>First we construct the <span class="math inline">\(\psi_3\)</span> element. Contrary to the intuition, we use the top wire containing <span class="math inline">\(\ket{j_1}\)</span> for this. We use a Hadamad-gate to bring <span class="math inline">\(\ket{j_1}\)</span> into the superposition <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + (-1)^{j_1} \ket{1}) = \frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1} \ket{1})\)</span>. This looks close to <span class="math inline">\(\psi_3\)</span> already, we now need to add the last two decimal places <span class="math inline">\(j_2j_3\)</span> to the state. For this we use <span class="math inline">\(R_2\)</span> and <span class="math inline">\(R_3\)</span>. We apply <span class="math inline">\(R_2\)</span> controlled by the wire <span class="math inline">\(j_2\)</span> and <span class="math inline">\(R_3\)</span> controlled by the wire <span class="math inline">\(j_3\)</span>. This means, that we only apply the <span class="math inline">\(R\)</span>-gate, if the corresponding wire contains a <span class="math inline">\(1\)</span>. You can see this written as a quantum circuit at the figure below. After applying <span class="math inline">\(R_2\)</span> we have the state <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2} \ket{1})\)</span> and after applying <span class="math inline">\(R_3\)</span> we have the state <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2j_3} \ket{1})\)</span> on the top wire. This is the same as <span class="math inline">\(\psi_3\)</span>, so we are done on the first wire (We are at the first slice in the figure).</p>
-<p>The next step is to construct the <span class="math inline">\(\psi_2\)</span> state on the middle wire. We again use a Hadamad-gate to bring <span class="math inline">\(\ket{j_2}\)</span> into the superposition <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_2} \ket{1})\)</span>. We now need to include the last decimal point <span class="math inline">\(j_3\)</span>, for which we use <span class="math inline">\(R_2\)</span> again, this time controlled by <span class="math inline">\(j_3\)</span>. The resulting superposition is now <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} +  e^{2\pi i 0.j_2j_3} \ket{1})\)</span>, which is <span class="math inline">\(\psi_2\)</span>. (We are at the second slice in the figure).</p>
-<p>On the bottom wire, we can just do a Hadamad-gate to bring <span class="math inline">\(\ket{j_3}\)</span> into the superposition <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_3} \ket{1})\)</span>. We then have <span class="math inline">\(\psi_1\)</span> on the bottom wire. (We are at the third slice in the figure).</p>
-<p>When applying this circuit, we get the state <span class="math inline">\(\psi_3 \otimes \psi_2 \otimes \psi_1\)</span> as a result. This very close to our desired state <span class="math inline">\(\psi_1 \otimes \psi_2 \otimes \psi_3\)</span>, just the order of the wires is flipped. To solve this, we apply a <span class="math inline">\(\operatorname{SWAP}\)</span> onto all wires, which flips the order of the wires an delivers the correct output for <span class="math inline">\(\operatorname{DFT}_{2^3}\)</span>.</p>
+<ol type="1">
+<li>First we construct the <span class="math inline">\(\psi_3\)</span> element. Contrary to the intuition, we use the top wire containing <span class="math inline">\(\ket{j_1}\)</span> for this. We use a Hadamad-gate to bring <span class="math inline">\(\ket{j_1}\)</span> into the superposition <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + (-1)^{j_1} \ket{1}) = \frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1} \ket{1})\)</span>. This looks close to <span class="math inline">\(\psi_3\)</span> already, we now need to add the last two decimal places <span class="math inline">\(j_2j_3\)</span> to the state. For this we use <span class="math inline">\(R_2\)</span> and <span class="math inline">\(R_3\)</span>. We apply <span class="math inline">\(R_2\)</span> controlled by the wire <span class="math inline">\(j_2\)</span> and <span class="math inline">\(R_3\)</span> controlled by the wire <span class="math inline">\(j_3\)</span>. This means, that we only apply the <span class="math inline">\(R\)</span>-gate, if the corresponding wire contains a <span class="math inline">\(1\)</span>. You can see this written as a quantum circuit at the figure below. After applying <span class="math inline">\(R_2\)</span> we have the state <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2} \ket{1})\)</span> and after applying <span class="math inline">\(R_3\)</span> we have the state <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2j_3} \ket{1})\)</span> on the top wire. This is the same as <span class="math inline">\(\psi_3\)</span>, so we are done on the first wire (We are at the first slice in the figure).</li>
+<li>The next step is to construct the <span class="math inline">\(\psi_2\)</span> state on the middle wire. We again use a Hadamad-gate to bring <span class="math inline">\(\ket{j_2}\)</span> into the superposition <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_2} \ket{1})\)</span>. We now need to include the last decimal point <span class="math inline">\(j_3\)</span>, for which we use <span class="math inline">\(R_2\)</span> again, this time controlled by <span class="math inline">\(j_3\)</span>. The resulting superposition is now <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} +  e^{2\pi i 0.j_2j_3} \ket{1})\)</span>, which is <span class="math inline">\(\psi_2\)</span>. (We are at the second slice in the figure).</li>
+<li>On the bottom wire, we can just do a Hadamad-gate to bring <span class="math inline">\(\ket{j_3}\)</span> into the superposition <span class="math inline">\(\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_3} \ket{1})\)</span>. We then have <span class="math inline">\(\psi_1\)</span> on the bottom wire. (We are at the third slice in the figure).</li>
+<li>When applying this circuit, we get the state <span class="math inline">\(\psi_3 \otimes \psi_2 \otimes \psi_1\)</span> as a result. This very close to our desired state <span class="math inline">\(\psi_1 \otimes \psi_2 \otimes \psi_3\)</span>, just the order of the wires is flipped. To solve this, we apply a <span class="math inline">\(\operatorname{SWAP}\)</span> onto all wires, which flips the order of the wires an delivers the correct output for <span class="math inline">\(\operatorname{DFT}_{2^3}\)</span>.</li>
+</ol>
 </div>
 </div>
 <p>The more general approach to construct the <span class="math inline">\(\operatorname{DFT}_N\)</span> as a quantum circuit with <span class="math inline">\(n\)</span> qubits (<span class="math inline">\(N = 2^n\)</span>) works as follows:</p>
@@ -963,6 +972,9 @@ Example: Construction of the DFT circuit for <span class="math inline">\(n=3\)</
       </a>          
   </div>
   <div class="nav-page nav-page-next">
+      <a href="../content/groversAlgorithm.html" class="pagination-link" aria-label="Grover's algorithm">
+        <span class="nav-page-text"><span class="chapter-number">11</span>&nbsp; <span class="chapter-title">Grover’s algorithm</span></span> <i class="bi bi-arrow-right-short"></i>
+      </a>
   </div>
 </nav>
 </div> <!-- /content -->
diff --git a/_book/index.html b/_book/index.html
index ee5e147..f3d5ac2 100644
--- 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="2025-06-03">
+<meta name="dcterms.date" content="2025-06-04">
 
 <title>Introduction to Quantum Computing</title>
 <style>
@@ -199,6 +199,12 @@ ul.task-list li input[type="checkbox"] {
   <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>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./content/groversAlgorithm.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">11</span>&nbsp; <span class="chapter-title">Grover’s algorithm</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
@@ -241,7 +247,7 @@ 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">June 3, 2025</p>
+      <p class="date">June 4, 2025</p>
     </div>
   </div>
   
diff --git a/_book/search.json b/_book/search.json
index ef5a4c0..c426502 100644
--- a/_book/search.json
+++ b/_book/search.json
@@ -377,10 +377,43 @@
     "href": "content/shorsAlgorithm.html#constructing-the-dft",
     "title": "10  Shor’s Algorithm",
     "section": "10.5 Constructing the DFT",
-    "text": "10.5 Constructing the DFT\nSo far we have described everything necessary for Shor’s algorithm, but only described the matrix representation of the \\(\\operatorname{DFT}_N\\). We will now take a closer look into implementing the \\(\\operatorname{DFT}_M\\) as a quantum circuit. Since we only use the \\(\\operatorname{DFT}_N\\) for Shor’s algorithm so far, we will only look at \\(N=2^n\\), which is the \\(\\operatorname{DFT}\\) applied on \\(n\\) qubits.\nTo start the circuit, we recall the definition of the \\(\\operatorname{DFT}_N\\) from Definition 10.1: \\(\\operatorname{DFT}_N := \\frac{1}{\\sqrt{{2^n}}} (\\omega^{kl})_{kl}\\) with \\(\\omega:= e^{2\\pi i / 2^n}\\). To apply the \\(\\operatorname{DFT}_N\\) to a quantum state \\(\\ket{j}\\) we calculate \\[\n\\begin{aligned}\n\\operatorname{DFT}_{N}\\ket{j}\n=& \\frac{1}{\\sqrt{N}} \\sum_{k=0}^{N-1} \\omega^{jk} \\ket{k}\\\\\n=& \\frac{1}{\\sqrt{N}} \\sum_{k=0}^{N-1} e^{2\\pi i j k / N} \\ket{k}\\\\\n=& \\frac{1}{\\sqrt{N}} \\sum_{k_1 \\in \\{0,1\\}} \\dots \\sum_{k_n \\in \\{0,1\\}} e^{2\\pi i j (\\sum_{l=1}^n k_l 2^{-l})} \\ket{k_1 \\dots k_n}\\\\\n=& \\frac{1}{\\sqrt{N}} \\sum_{k_1 \\in \\{0,1\\}} \\dots \\sum_{k_n \\in \\{0,1\\}} \\bigotimes_{l=1}^n e^{2\\pi i j k_l 2^{-l}} \\ket{k_l}\\\\\n=& \\frac{1}{\\sqrt{N}} \\bigotimes_{l=1}^n \\sum_{k_l \\in \\{0,1\\}} e^{2\\pi i j k_l 2^{-l}} \\ket{k_l}\\\\\n=& \\frac{1}{\\sqrt{N}} \\bigotimes_{l=1}^n (\\ket{0} +  e^{2\\pi i j 2^{-l}} \\ket{1})\\\\\n=& \\frac{1}{\\sqrt{N}} \\bigotimes_{l=1}^n (\\ket{0} +  e^{2\\pi i 0.j_{n-l+1} \\dots j_{n}} \\ket{1}).\n\\end{aligned}\n\\] The expression \\(0.j\\) expresses a binary fraction (e.g. \\(0.101 = \\frac{1}{2} + \\frac{1}{8} = \\frac{5}{8}\\)).\nWith this we have shown, that we can write \\(\\operatorname{DFT}_N \\ket{j}\\) as the following tensor product of quantum states\n\\[\n\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_n} \\ket{1}) \\otimes \\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_{n-1}j_n} \\ket{1}) \\otimes \\dots \\otimes \\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_1\\dots j_n} \\ket{1}).\n\\]\nFrom this rewritten tensor product, we can get an idea on how to construct the quantum circuit for the \\(\\operatorname{DFT}_N\\). Namely, we can construct a quantum circuit for each element of the tensor product and from this build the general circuit.\nFor this, we segment the tensor product into different elements \\(\\psi\\) as follows:\n\\[\n\\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_n} \\ket{1})}_{\\psi_1} \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_{n-1}j_n} \\ket{1})}_{\\psi_2} \\otimes \\dots \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_1\\dots j_n} \\ket{1})}_{\\psi_n}.\n\\]\nWe also introduce a new gate \\(R_k\\) which is defined by the following matrix:\n\\[\nR_k:=\\begin{pmatrix} 1 & 0 \\\\ 0 & e^{2 \\pi i / 2^k}  \\end{pmatrix}.\n\\]\nTo understand the construction of the circuit from these elements, we will look at an example for \\(n=3\\) first:\n\n\n\n\n\n\nExample: Construction of the DFT circuit for \\(n=3\\)\n\n\n\nWe start by building the tensor product for \\(n=3\\). The input for the DFT circuit is \\(\\ket{j} = \\ket{j_1j_2j_3}\\). Using the formula from above, we can write the result of \\(\\operatorname{DFT}_{2^3} \\ket{j}\\) as the following tensor product:\n\\[\n\\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_3} \\ket{1})}_{\\psi_1} \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_2j_3} \\ket{1})}_{\\psi_2} \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_1j_2j_3} \\ket{1})}_{\\psi_3}.\n\\]\n\n\n\nThe DTF for three qubits\n\n\nFirst we construct the \\(\\psi_3\\) element. Contrary to the intuition, we use the top wire containing \\(\\ket{j_1}\\) for this. We use a Hadamad-gate to bring \\(\\ket{j_1}\\) into the superposition \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + (-1)^{j_1} \\ket{1}) = \\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_1} \\ket{1})\\). This looks close to \\(\\psi_3\\) already, we now need to add the last two decimal places \\(j_2j_3\\) to the state. For this we use \\(R_2\\) and \\(R_3\\). We apply \\(R_2\\) controlled by the wire \\(j_2\\) and \\(R_3\\) controlled by the wire \\(j_3\\). This means, that we only apply the \\(R\\)-gate, if the corresponding wire contains a \\(1\\). You can see this written as a quantum circuit at the figure below. After applying \\(R_2\\) we have the state \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_1j_2} \\ket{1})\\) and after applying \\(R_3\\) we have the state \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_1j_2j_3} \\ket{1})\\) on the top wire. This is the same as \\(\\psi_3\\), so we are done on the first wire (We are at the first slice in the figure).\nThe next step is to construct the \\(\\psi_2\\) state on the middle wire. We again use a Hadamad-gate to bring \\(\\ket{j_2}\\) into the superposition \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_2} \\ket{1})\\). We now need to include the last decimal point \\(j_3\\), for which we use \\(R_2\\) again, this time controlled by \\(j_3\\). The resulting superposition is now \\(\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_2j_3} \\ket{1})\\), which is \\(\\psi_2\\). (We are at the second slice in the figure).\nOn the bottom wire, we can just do a Hadamad-gate to bring \\(\\ket{j_3}\\) into the superposition \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_3} \\ket{1})\\). We then have \\(\\psi_1\\) on the bottom wire. (We are at the third slice in the figure).\nWhen applying this circuit, we get the state \\(\\psi_3 \\otimes \\psi_2 \\otimes \\psi_1\\) as a result. This very close to our desired state \\(\\psi_1 \\otimes \\psi_2 \\otimes \\psi_3\\), just the order of the wires is flipped. To solve this, we apply a \\(\\operatorname{SWAP}\\) onto all wires, which flips the order of the wires an delivers the correct output for \\(\\operatorname{DFT}_{2^3}\\).\n\n\nThe more general approach to construct the \\(\\operatorname{DFT}_N\\) as a quantum circuit with \\(n\\) qubits (\\(N = 2^n\\)) works as follows:\n\nInitialize wires with input \\(\\ket{j}\\), so that \\(\\ket{j_1}\\) is on the top wire and \\(\\ket{j_n}\\) is on the bottom wire. Note, that this is not part of the circuit yet.\nStart with the top wire. For each wire \\(j_i\\) do the following:\n\nApply a Hadamad-gate on the wire \\(j_i\\).\nFor each wire \\(j_k\\) below the current wire \\(j_i\\) (with \\(i &lt; k \\leq n\\)), add a \\(R_{k-i+1}\\)-gate controlled by \\(j_k\\). Start with \\(k = i+1\\) (if \\(i&lt;n\\), else stop).\n\nPerform a \\(\\operatorname{SWAP}\\) to flip all the wires. This means, that the first wire is swapped with the last wire, the second wire is swapped with the second to last wire and so on.\n\nNote: If the the output of the DFT circuit is measured right after applying it (as in Shor’s algorithm) or if the rest of the algorithm allows for it, it is more efficient to perform the \\(\\operatorname{SWAP}\\) classically, since this is considered to be the cheaper operation.\nThe more general layout of the quantum circuit for the \\(\\operatorname{DFT_N}\\) with the \\(\\operatorname{SWAP}\\) is shown in this figure.\n\n\n\nThe DTF for \\(n\\) qubits",
+    "text": "10.5 Constructing the DFT\nSo far we have described everything necessary for Shor’s algorithm, but only described the matrix representation of the \\(\\operatorname{DFT}_N\\). We will now take a closer look into implementing the \\(\\operatorname{DFT}_M\\) as a quantum circuit. Since we only use the \\(\\operatorname{DFT}_N\\) for Shor’s algorithm so far, we will only look at \\(N=2^n\\), which is the \\(\\operatorname{DFT}\\) applied on \\(n\\) qubits.\nTo start the circuit, we recall the definition of the \\(\\operatorname{DFT}_N\\) from Definition 10.1: \\(\\operatorname{DFT}_N := \\frac{1}{\\sqrt{{2^n}}} (\\omega^{kl})_{kl}\\) with \\(\\omega:= e^{2\\pi i / 2^n}\\). To apply the \\(\\operatorname{DFT}_N\\) to a quantum state \\(\\ket{j}\\) we calculate \\[\n\\begin{aligned}\n\\operatorname{DFT}_{N}\\ket{j}\n=& \\frac{1}{\\sqrt{N}} \\sum_{k=0}^{N-1} \\omega^{jk} \\ket{k}\\\\\n=& \\frac{1}{\\sqrt{N}} \\sum_{k=0}^{N-1} e^{2\\pi i j k / N} \\ket{k}\\\\\n=& \\frac{1}{\\sqrt{N}} \\sum_{k_1 \\in \\{0,1\\}} \\dots \\sum_{k_n \\in \\{0,1\\}} e^{2\\pi i j (\\sum_{l=1}^n k_l 2^{-l})} \\ket{k_1 \\dots k_n}\\\\\n=& \\frac{1}{\\sqrt{N}} \\sum_{k_1 \\in \\{0,1\\}} \\dots \\sum_{k_n \\in \\{0,1\\}} \\bigotimes_{l=1}^n e^{2\\pi i j k_l 2^{-l}} \\ket{k_l}\\\\\n=& \\frac{1}{\\sqrt{N}} \\bigotimes_{l=1}^n \\sum_{k_l \\in \\{0,1\\}} e^{2\\pi i j k_l 2^{-l}} \\ket{k_l}\\\\\n=& \\frac{1}{\\sqrt{N}} \\bigotimes_{l=1}^n (\\ket{0} +  e^{2\\pi i j 2^{-l}} \\ket{1})\\\\\n=& \\frac{1}{\\sqrt{N}} \\bigotimes_{l=1}^n (\\ket{0} +  e^{2\\pi i 0.j_{n-l+1} \\dots j_{n}} \\ket{1}).\n\\end{aligned}\n\\] The expression \\(0.j\\) expresses a binary fraction (e.g. \\(0.101 = \\frac{1}{2} + \\frac{1}{8} = \\frac{5}{8}\\)).\nWith this we have shown, that we can write \\(\\operatorname{DFT}_N \\ket{j}\\) as the following tensor product of quantum states\n\\[\n\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_n} \\ket{1}) \\otimes \\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_{n-1}j_n} \\ket{1}) \\otimes \\dots \\otimes \\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_1\\dots j_n} \\ket{1}).\n\\]\nFrom this rewritten tensor product, we can get an idea on how to construct the quantum circuit for the \\(\\operatorname{DFT}_N\\). Namely, we can construct a quantum circuit for each element of the tensor product and from this build the general circuit.\nFor this, we segment the tensor product into different elements \\(\\psi\\) as follows:\n\\[\n\\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_n} \\ket{1})}_{\\psi_1} \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_{n-1}j_n} \\ket{1})}_{\\psi_2} \\otimes \\dots \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_1\\dots j_n} \\ket{1})}_{\\psi_n}.\n\\]\nWe also introduce a new gate \\(R_k\\) which is defined by the following matrix:\n\\[\nR_k:=\\begin{pmatrix} 1 & 0 \\\\ 0 & e^{2 \\pi i / 2^k}  \\end{pmatrix}.\n\\]\nTo understand the construction of the circuit from these elements, we will look at an example for \\(n=3\\) first:\n\n\n\n\n\n\nExample: Construction of the DFT circuit for \\(n=3\\)\n\n\n\nWe start by building the tensor product for \\(n=3\\). The input for the DFT circuit is \\(\\ket{j} = \\ket{j_1j_2j_3}\\). Using the formula from above, we can write the result of \\(\\operatorname{DFT}_{2^3} \\ket{j}\\) as the following tensor product:\n\\[\n\\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_3} \\ket{1})}_{\\psi_1} \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_2j_3} \\ket{1})}_{\\psi_2} \\otimes \\underbrace{\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_1j_2j_3} \\ket{1})}_{\\psi_3}.\n\\]\n\n\n\nThe DTF for three qubits\n\n\n\nFirst we construct the \\(\\psi_3\\) element. Contrary to the intuition, we use the top wire containing \\(\\ket{j_1}\\) for this. We use a Hadamad-gate to bring \\(\\ket{j_1}\\) into the superposition \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + (-1)^{j_1} \\ket{1}) = \\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_1} \\ket{1})\\). This looks close to \\(\\psi_3\\) already, we now need to add the last two decimal places \\(j_2j_3\\) to the state. For this we use \\(R_2\\) and \\(R_3\\). We apply \\(R_2\\) controlled by the wire \\(j_2\\) and \\(R_3\\) controlled by the wire \\(j_3\\). This means, that we only apply the \\(R\\)-gate, if the corresponding wire contains a \\(1\\). You can see this written as a quantum circuit at the figure below. After applying \\(R_2\\) we have the state \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_1j_2} \\ket{1})\\) and after applying \\(R_3\\) we have the state \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_1j_2j_3} \\ket{1})\\) on the top wire. This is the same as \\(\\psi_3\\), so we are done on the first wire (We are at the first slice in the figure).\nThe next step is to construct the \\(\\psi_2\\) state on the middle wire. We again use a Hadamad-gate to bring \\(\\ket{j_2}\\) into the superposition \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_2} \\ket{1})\\). We now need to include the last decimal point \\(j_3\\), for which we use \\(R_2\\) again, this time controlled by \\(j_3\\). The resulting superposition is now \\(\\frac{1}{\\sqrt{2}}(\\ket{0} +  e^{2\\pi i 0.j_2j_3} \\ket{1})\\), which is \\(\\psi_2\\). (We are at the second slice in the figure).\nOn the bottom wire, we can just do a Hadamad-gate to bring \\(\\ket{j_3}\\) into the superposition \\(\\frac{1}{\\sqrt{2}}(\\ket{0} + e^{2 \\pi i 0.j_3} \\ket{1})\\). We then have \\(\\psi_1\\) on the bottom wire. (We are at the third slice in the figure).\nWhen applying this circuit, we get the state \\(\\psi_3 \\otimes \\psi_2 \\otimes \\psi_1\\) as a result. This very close to our desired state \\(\\psi_1 \\otimes \\psi_2 \\otimes \\psi_3\\), just the order of the wires is flipped. To solve this, we apply a \\(\\operatorname{SWAP}\\) onto all wires, which flips the order of the wires an delivers the correct output for \\(\\operatorname{DFT}_{2^3}\\).\n\n\n\nThe more general approach to construct the \\(\\operatorname{DFT}_N\\) as a quantum circuit with \\(n\\) qubits (\\(N = 2^n\\)) works as follows:\n\nInitialize wires with input \\(\\ket{j}\\), so that \\(\\ket{j_1}\\) is on the top wire and \\(\\ket{j_n}\\) is on the bottom wire. Note, that this is not part of the circuit yet.\nStart with the top wire. For each wire \\(j_i\\) do the following:\n\nApply a Hadamad-gate on the wire \\(j_i\\).\nFor each wire \\(j_k\\) below the current wire \\(j_i\\) (with \\(i &lt; k \\leq n\\)), add a \\(R_{k-i+1}\\)-gate controlled by \\(j_k\\). Start with \\(k = i+1\\) (if \\(i&lt;n\\), else stop).\n\nPerform a \\(\\operatorname{SWAP}\\) to flip all the wires. This means, that the first wire is swapped with the last wire, the second wire is swapped with the second to last wire and so on.\n\nNote: If the the output of the DFT circuit is measured right after applying it (as in Shor’s algorithm) or if the rest of the algorithm allows for it, it is more efficient to perform the \\(\\operatorname{SWAP}\\) classically, since this is considered to be the cheaper operation.\nThe more general layout of the quantum circuit for the \\(\\operatorname{DFT_N}\\) with the \\(\\operatorname{SWAP}\\) is shown in this figure.\n\n\n\nThe DTF for \\(n\\) qubits",
     "crumbs": [
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+    "text": "11.1 Preparations\nAnother well known quantum algorithm is Grover’s algorithm for searching. It was developed by Lov Grover in 1996.\nGrover’s algorithm takes a function \\(f: \\{0,1\\}^n \\rightarrow \\{0,1\\}\\), where exactly one \\(x_0\\) exists, such that \\(f(x_0) = 1\\). The goal is to find \\(x_0\\).\nThere are a number of interesting problems, which can be reduced to this general definition. One of these problems is the breaking of a (symmetric) encryption. The function \\(f\\) would take a key as an input and output a \\(1\\), if the decryption is successful. Otherwise it will output a \\(0\\).\nClassically, finding this \\(x_0\\) takes approximately \\(2^n\\) steps (when simply bruteforcing the function). Using Grover’s algorithm, we can reduce this runtime to approximately \\(\\sqrt{2^n}\\) steps. As an example, a \\(128\\)-bit encryption would only take about \\(2^{64}\\) steps to break it, instead of about \\(2^{128}\\) steps for the classical bruteforce.\nTo construct Grover’s algorithm, we first need to introduce two new gates \\(V_f\\) and \\(\\operatorname{FLIP}_*\\).",
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+    "title": "11  Grover’s algorithm",
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+    "text": "11.1.1 Constructing the oracle \\(V_f\\)\nIn the previous algorithms, we have learned that we can implement a function \\(f\\) as a unitary \\(U_f\\) with \\(U_f\\ket{x,y} = \\ket{x, y \\oplus f(x)}\\). We construct a different unitary called \\(V_f\\) from this, which has the following behavior:\n\\[\nV_f \\ket{x} =\n\\begin{cases}\n    -\\ket{x} & \\text{if } f(x) = 1 \\\\\n    \\ket{x} & \\text{else}\n\\end{cases}\n\\]\nWe can construct \\(V_f\\) from \\(U_f\\) using the following circuit:\n\n\n\nThe circuit for \\(V_f\\)\n\n\nThe bottom wire can be discarded, since it always contains a \\(\\ket{-}\\) and thus is not entangled with the upper wire.\n\n\n11.1.2 Constructing \\(\\operatorname{FLIP}_*\\)\nAs a second ingredient for Grover’s algorithm, we need the unitary \\(\\operatorname{FLIP}_*\\). To realize this, we first need \\(\\operatorname{FLIP}_0\\), which ist defined by the unitary \\[\n\\operatorname{FLIP}_0 \\ket{x} =\n\\begin{cases}\n    \\ket{0} & \\text{if } x = 0 \\\\\n    -\\ket{x} & \\text{else}.\n\\end{cases}\n\\]\n\\(\\operatorname{FLIP}_0\\) is implemented by the following quantum circuit:\n\n\n\nThe circuit for \\(\\operatorname{FLIP}_0\\)\n\n\n\\(Z\\) is the Pauli matrix from definition Definition 7.2. The empty circles indicates a negative control wire. So \\(Z\\) is only applied if the other wires are \\(\\ket{0}\\).\nNow we can define the unitary called \\(\\operatorname{FLIP}_*\\). This unitary does nothing, if it is applied on the uniform superposition \\(\\ket{*}\\). For any other quantum state \\(\\ket{\\psi}\\) orthogonal to \\(\\ket{*}\\) it maps to \\(-\\ket{\\psi}\\). The uniform superposition \\(\\ket{*}\\) simply denotes the superposition over all classical possibilities \\(\\ket{*} = \\frac{1}{\\sqrt{2^n}} \\sum_{x \\in\\{0,1\\}^n} \\ket{x}\\). So \\(\\operatorname{FLIP}_*\\) is described by:\n\\[\n\\operatorname{FLIP}_* \\ket{\\psi} =\n\\begin{cases}\n    \\ket{*} & \\text{if } \\ket{\\psi} = \\ket{*} \\\\\n    -\\ket{x} & \\text{if } \\ket{\\psi} \\perp \\ket{*} \\text{ (orthogonal)}.\n\\end{cases}\n\\]\nWe can construct this \\(\\operatorname{FLIP}_*\\) by the following quantum circuit:\n\n\n\nThe circuit for \\(\\operatorname{FLIP}_*\\)",
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+    "title": "11  Grover’s algorithm",
+    "section": "11.2 The algorithm for searching",
+    "text": "11.2 The algorithm for searching\nThe actual algorithm takes a function \\(f: \\{0,1\\}^n \\rightarrow \\{0,1\\}\\) and outputs an \\(x_0\\) with \\(f(x_0)=1\\). For simplicity, we assume that there is only one \\(x_0\\) for which \\(f(x_0) = 1\\) holds and for each other \\(x \\neq x_0\\) it holds that \\(f(x)=0\\).\nWith the two new unitaries \\(V_f\\) and \\(\\operatorname{FLIP}_*\\) defined, we can construct the circuit for Grover’s algorithm, which is shown in the following figure:\n\n\n\nThe quantum circuit for Grover’s algorithm\n\n\nThe algorithm works as follows:\n\nWe start with a \\(\\ket{0}\\) entry on every qubit.\nWe bring the system into the superposition over all entries by applying \\(H^{\\otimes n}\\). The quantum state is then \\(\\frac{1}{\\sqrt{2^n}} \\sum_{x \\in \\{0,1\\}^n} \\ket{x}\\) which we also call \\(\\ket{*}\\).\nWe apply the unitary \\(V_f\\).\nWe apply the unitary \\(\\operatorname{FLIP}_*\\).\nWe repeat steps 3 and 4 \\(t\\) times.\nWe do a measurement.\n\nThe measurement in step 6 will then give us \\(x_0\\) with high probability.\n\n11.2.1 Understanding the algorithm for searching\nWhen looking at the quantum circuit, it is not completely intuitive why the algorithm gives the correct result. We therefore now look into what is happening in each step.\nThe desired quantum state after the algorithm finishes is \\(\\ket{x_0}\\). At the beginning of the algorithm, we bring the system into the uniform superposition \\(\\ket{*} = \\frac{1}{\\sqrt{2^n}} \\sum_{x \\in \\{0,1\\}^n} \\ket{x}\\). We know that \\(\\ket{x_0}\\) is part of this superposition, therefore we can rewrite \\(\\ket{*}\\) as follows \\[\n\\ket{*}\n= \\frac{1}{\\sqrt{2^n}} \\sum_{x \\in \\{0,1\\}^n} \\ket{x}\n= \\frac{1}{\\sqrt{2^n}} \\underbrace{\\ket{x_0}}_{\\textit{good}}  + \\sqrt{\\frac{2^n-1}{2^n}} \\underbrace{\\sum_{x\\neq x_0} \\frac{1}{\\sqrt{2^n-1}}\\ket{x}}_{\\textit{bad}}\n\\]\nSo the current state can be seen as a superposition of a “good” state good and a “bad” state bad.\nThe geometric interpretation of this superposition can be drawn as follows:\n\n\n\nGeometric interpretation of \\(\\ket{*}\\)\n\n\nThe angel \\(\\theta\\) denotes, how “good” the resulting outcome will be. If \\(\\theta = 0\\), the state is completely bad, if \\(\\theta = \\frac{\\pi}{2}\\), the state is completely good.\nWe can calculate \\(\\cos \\theta = \\lvert\\braket{*|bad}\\rvert = \\frac{\\sqrt{2^n-1}}{\\sqrt{2^n}}\\). From this we can derivate that the angle \\(\\theta\\) is \\(\\cos^{-1} \\sqrt{\\frac{2^n-1}{2^n}}\\) at the beginning, which is approximately \\(\\sqrt{\\frac{1}{2^n}}\\).\nWe now apply \\(V_f\\) on this quantum state. This will negate the amplitude off our desired \\(\\ket{x_0}\\) and not change the amplitude to the rest of the state.\n\\[\nV_f\\ket{*}= -\\frac{1}{\\sqrt{2^n}} \\underbrace{\\ket{x_0}}_{\\text{good}}  + \\sqrt{\\frac{2^n-1}{2^n}} \\underbrace{\\sum_{x\\neq x_0} \\ket{x}}_{\\text{bad}}\n\\]\nThis looks like this in the geometric interpretation:\n\n\n\nGeometric interpretation after \\(V_f\\)\n\n\nWe can see that by applying \\(V_f\\), we mirror the vector across the bad axis.\nAfter \\(V_f\\), we apply the \\(\\operatorname{FLIP}_*\\) operation on the quantum state. Since \\(\\operatorname{FLIP}_*\\) does nothing on the \\(\\ket{*}\\) entries and negates the amplitude of any vector orthogonal to it, \\(\\operatorname{FLIP}_*\\) mirrors the vector across \\(\\ket{*}\\). This can be seen in the following figure:\n\n\n\nGeometric interpretation after \\(\\operatorname{FLIP}_*\\)\n\n\nAll in all, we have seen that by applying \\(V_f\\) and \\(\\operatorname{FLIP}_*\\), we can increase the angle of the quantum state in relation to the “good” and “bad” states by \\(2\\theta\\). Therefore two reflection give rotation. By repeating this step often enough, we can get the amplitude of \\(\\ket{x_0}\\) close to 1.\nTo be more precise: Since we know \\(\\theta\\) and we know that we will increase the good-ness of our quantum state by \\(2\\theta\\) each time, we can calculate that only \\(t\\) iterations are necessary with \\[\nt \\approx \\frac{\\frac{\\pi / 2}{\\theta} - 1}{2} \\approx \\frac{\\pi}{4 \\theta} \\approx \\frac{\\pi}{4} \\cdot \\sqrt{2^n}.\n\\] Grover’s algorithm therefore takes \\(O(\\sqrt{2^n})\\) steps, where an evaluation of the circuit counts as one step.",
+    "crumbs": [
+      "Quantum algorithms",
+      "<span class='chapter-number'>11</span>  <span class='chapter-title'>Grover's algorithm</span>"
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\ No newline at end of file
diff --git a/_book/sitemap.xml b/_book/sitemap.xml
index a37c9ac..f094cfe 100644
--- a/_book/sitemap.xml
+++ b/_book/sitemap.xml
@@ -42,10 +42,14 @@
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/shorsAlgorithm.html</loc>
-    <lastmod>2025-06-03T17:03:57.368Z</lastmod>
+    <lastmod>2025-06-03T19:53:54.622Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/content/groversAlgorithm.html</loc>
+    <lastmod>2025-06-04T09:27:01.821Z</lastmod>
   </url>
   <url>
     <loc>https://qis.rwth-aachen.de/teaching/25ss/intro-quantum-computing/script/Introduction-to-Quantum-Computing.pdf</loc>
-    <lastmod>2025-06-03T17:05:56.764Z</lastmod>
+    <lastmod>2025-06-04T10:37:30.177Z</lastmod>
   </url>
 </urlset>
diff --git a/_quarto.yml b/_quarto.yml
index 2378d89..f03ad30 100644
--- a/_quarto.yml
+++ b/_quarto.yml
@@ -32,7 +32,7 @@ book:
       chapters:
         - content/bernsteinVazirani.qmd
         - content/shorsAlgorithm.qmd
-    #    - content/groversAlgorithm.qmd
+        - content/groversAlgorithm.qmd
     #- part: "Physical background"
     #  chapters:
     #    - content/physics.qmd
diff --git a/content/groversAlgorithm.qmd b/content/groversAlgorithm.qmd
index dfa0c94..6f8c5f7 100644
--- a/content/groversAlgorithm.qmd
+++ b/content/groversAlgorithm.qmd
@@ -4,9 +4,10 @@ Another well known quantum algorithm is Grover's algorithm for searching. It was
 
 Grover's algorithm takes a function $f: \{0,1\}^n \rightarrow \{0,1\}$, where exactly one $x_0$ exists, such that $f(x_0) = 1$. The goal is to find $x_0$. 
 
-There are a number of interesting problems, which can be reduced to this general definition. One of these problems is the breaking of a (symmetric) encryption. The function $f$ would take a key as an input and output a $1$, if the decryption is successful.
+There are a number of interesting problems, which can be reduced to this general definition. One of these problems is the breaking of a (symmetric) encryption. The function $f$ would take a key as an input and output a $1$, if the decryption is successful. Otherwise it will output a $0$.
+
+Classically, finding this $x_0$ takes approximately $2^n$ steps (when simply bruteforcing the function). Using Grover's algorithm, we can reduce this runtime to approximately $\sqrt{2^n}$ steps. As an example, a $128$-bit encryption would only take about $2^{64}$ steps to break it, instead of about $2^{128}$ steps for the classical bruteforce.
 
-Classically, finding this $x_0$ takes approximately $2^n$ steps (when simply bruteforcing the function). Using Grover's algorithm, we can reduce this runtime to approximately $2^{\frac{n}{2}}$ steps. As an example, a $128$-bit encryption would only take about $2^{64}$ steps to break it, instead of about $2^{128}$ steps for the classical bruteforce.
 
 ## Preparations
 
@@ -17,7 +18,11 @@ To construct Grover's algorithm, we first need to introduce two new gates $V_f$
 In the previous algorithms, we have learned that we can implement a function $f$ as a unitary $U_f$ with $U_f\ket{x,y} = \ket{x, y \oplus f(x)}$. We construct a different unitary called $V_f$ from this, which has the following behavior:
 
 $$
-V_f \ket{x} = \begin{cases} -\ket{x} & \text{if } f(x) = 1\\ \ket{x} & \text{else} \end{cases}
+V_f \ket{x} =
+\begin{cases}
+    -\ket{x} & \text{if } f(x) = 1 \\
+    \ket{x} & \text{else}
+\end{cases}
 $$
 
 We can construct $V_f$ from $U_f$ using the following circuit:
@@ -28,16 +33,36 @@ The bottom wire can be discarded, since it always contains a $\ket{-}$ and thus
 
 ### Constructing $\operatorname{FLIP}_*$
 
-As a second ingredient for Grover's algorithm, we define a unitary called $\operatorname{FLIP}_*$. This unitary does nothing, if it is applied on the uniform superposition $\ket{*}$. For any other quantum state $\psi$ with $\psi$ orthogonal to $\ket{*}$ it maps $\psi$ to $-\psi$. The uniform superposition $\ket{*}$ simply denotes the superposition over all classical possibilities $2^{-\frac{n}{2}}\sum_x \ket{x}$. We can construct this $\operatorname{FLIP}_*$ by the following quantum circuit:
+As a second ingredient for Grover's algorithm, we need the unitary $\operatorname{FLIP}_*$. To realize this, we first need $\operatorname{FLIP}_0$, which ist defined by the unitary
+$$
+\operatorname{FLIP}_0 \ket{x} =
+\begin{cases}
+    \ket{0} & \text{if } x = 0 \\
+    -\ket{x} & \text{else}.
+\end{cases}
+$$
+
+$\operatorname{FLIP}_0$ is implemented by the following quantum circuit:
+
+![The circuit for $\operatorname{FLIP}_0$](img/flip_zero){width=50%}
 
-![The circuit for $\operatorname{FLIP}_*$](img/flip){width=60%}
+$Z$ is the Pauli matrix from definition @def-gates-pauli. The empty circles indicates a negative control wire. So $Z$ is only applied if the other wires are $\ket{0}$.
 
+Now we can define the unitary called $\operatorname{FLIP}_*$. This unitary does nothing, if it is applied on the uniform superposition $\ket{*}$. For any other quantum state $\ket{\psi}$ orthogonal to $\ket{*}$ it maps to $-\ket{\psi}$. The uniform superposition $\ket{*}$ simply denotes the superposition over all classical possibilities $\ket{*} = \frac{1}{\sqrt{2^n}} \sum_{x \in\{0,1\}^n} \ket{x}$. So $\operatorname{FLIP}_*$ is described by:
 
-$\operatorname{FLIP}_0$ is here definied by the unitary
 $$
-\operatorname{FLIP}_0 \ket{x} = \begin{cases} \ket{0} & \text{if } x = 0\\ -\ket{x} & \text{else} \end{cases}
+\operatorname{FLIP}_* \ket{\psi} =
+\begin{cases}
+    \ket{*} & \text{if } \ket{\psi} = \ket{*} \\
+    -\ket{x} & \text{if } \ket{\psi} \perp \ket{*} \text{ (orthogonal)}.
+\end{cases}
 $$
 
+We can construct this $\operatorname{FLIP}_*$ by the following quantum circuit:
+
+![The circuit for $\operatorname{FLIP}_*$](img/flip_star){width=60%}
+
+
 ## The algorithm for searching
 
 The actual algorithm takes a function  $f: \{0,1\}^n \rightarrow \{0,1\}$ and outputs an $x_0$ with $f(x_0)=1$. For simplicity, we assume that there is only one $x_0$ for which $f(x_0) = 1$ holds and for each other $x \neq x_0$ it holds that $f(x)=0$.  
@@ -49,29 +74,30 @@ With the two new unitaries $V_f$ and $\operatorname{FLIP}_*$ defined, we can con
 The algorithm works as follows:
 
 1. We start with a $\ket{0}$ entry on every qubit.
-2. We bring the system into the superposition over all entries by applying $H^{\otimes n}$. The quantum state is then $2^\frac{-n}{2}\sum_x \ket{x}$ which we also call $\ket{*}$.
+2. We bring the system into the superposition over all entries by applying $H^{\otimes n}$. The quantum state is then $\frac{1}{\sqrt{2^n}} \sum_{x \in \{0,1\}^n} \ket{x}$ which we also call $\ket{*}$.
 3. We apply the unitary $V_f$. 
 4. We apply the unitary $\operatorname{FLIP}_*$. 
-5. We repeat steps 3 and 4 $t$ times, and then do a measurement.
+5. We repeat steps 3 and 4 $t$ times.
+6. We do a measurement.
 
-The measurement in step 5 will then give us $x_0$ with high probability. 
+The measurement in step 6 will then give us $x_0$ with high probability. 
 
 ### Understanding the algorithm for searching
 
 When looking at the quantum circuit, it is not completely intuitive why the algorithm gives the correct result. We therefore now look into what is happening in each step. 
 
-The desired quantum state after the algorithm finishes is $\ket{x_0}$. At the beginning of the algorithm, we bring the system into the uniform superposition $\ket{*} = 2^\frac{-n}{2}\sum_x \ket{x}$. We know that $\ket{x_0}$ is part of this superposition, therefore we can rewrite $\ket{*}$ as follows
+The desired quantum state after the algorithm finishes is $\ket{x_0}$. At the beginning of the algorithm, we bring the system into the uniform superposition $\ket{*} = \frac{1}{\sqrt{2^n}} \sum_{x \in \{0,1\}^n} \ket{x}$. We know that $\ket{x_0}$ is part of this superposition, therefore we can rewrite $\ket{*}$ as follows
 $$
-\ket{*} = 2^{-\frac{n}{2}}\sum_x \ket{x}
+\ket{*}
+= \frac{1}{\sqrt{2^n}} \sum_{x \in \{0,1\}^n} \ket{x}
 = \frac{1}{\sqrt{2^n}} \underbrace{\ket{x_0}}_{\textit{good}}  + \sqrt{\frac{2^n-1}{2^n}} \underbrace{\sum_{x\neq x_0} \frac{1}{\sqrt{2^n-1}}\ket{x}}_{\textit{bad}}
 $$
 
-
 So the current state can be seen as a superposition of a "good" state *good* and a "bad" state *bad*.
 
 The geometric interpretation of this superposition can be drawn as follows:
 
-![Geometric interpretation of $\ket{*}$](img/grover_plt1){width=30%}
+![Geometric interpretation of $\ket{*}$](img/grover_plot_1){width=30%}
 
 
 The angel $\theta$ denotes, how "good" the resulting outcome will be. If $\theta = 0$, the state is completely bad, if $\theta = \frac{\pi}{2}$, the state is completely good.
@@ -87,19 +113,19 @@ $$
 
 This looks like this in the geometric interpretation:
 
-![Geometric interpretation after $V_f$](img/grover_plt2){width=30%}
+![Geometric interpretation after $V_f$](img/grover_plot_2){width=30%}
 
 We can see that by applying $V_f$, we mirror the vector across the *bad* axis. 
 
 After $V_f$, we apply the $\operatorname{FLIP}_*$ operation on the quantum state. Since $\operatorname{FLIP}_*$ does nothing on the $\ket{*}$ entries and negates the amplitude of any vector orthogonal to it, $\operatorname{FLIP}_*$  mirrors the vector across $\ket{*}$. This can be seen in the following figure:
 
-![Geometric interpretation after $\operatorname{FLIP}_*$](img/grover_plt3){width=30%}
+![Geometric interpretation after $\operatorname{FLIP}_*$](img/grover_plot_3){width=30%}
 
 
 All in all, we have seen that by applying $V_f$ and $\operatorname{FLIP}_*$, we can increase the angle of the quantum state in relation to the "good" and "bad" states by $2\theta$. Therefore two reflection give rotation. By repeating this step often enough, we can get the amplitude of $\ket{x_0}$ close to 1. 
 
 To be more precise: Since we know $\theta$ and we know that we will increase the *good*-ness of our quantum state by $2\theta$ each time, we can calculate that only $t$ iterations are necessary with 
 $$
-t \approx \frac{\frac{\pi / 2}{\theta} - 1}{2} \approx \frac{\pi}{4 \theta} \approx \frac{\pi}{4} \cdot \sqrt{2^n}
+t \approx \frac{\frac{\pi / 2}{\theta} - 1}{2} \approx \frac{\pi}{4 \theta} \approx \frac{\pi}{4} \cdot \sqrt{2^n}.
 $$
- Grover's algorithm therefore takes $O(\sqrt{2^n})$ steps, where an evaluation of the circuit counts as one step. 
+Grover's algorithm therefore takes $O(\sqrt{2^n})$ steps, where an evaluation of the circuit counts as one step. 
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diff --git a/content/latex/dft_3_qubits.tex b/content/latex/dft_3_qubits.tex
index e2dd682..61abe15 100644
--- a/content/latex/dft_3_qubits.tex
+++ b/content/latex/dft_3_qubits.tex
@@ -4,7 +4,7 @@
 \begin{document}
 
 \begin{quantikz}
-    \lstick{$\ket{j_1}$} &\gate{H} &\gate{R_2} &\gate{R_3}\slice{1} &\qw &\qw\slice{\ket{2}} &\qw\slice{3} &\swap{2} & \rstick{$\psi_1$} \\
+    \lstick{$\ket{j_1}$} &\gate{H} &\gate{R_2} &\gate{R_3}\slice{1} &\qw &\qw\slice{2} &\qw\slice{3} &\swap{2} & \rstick{$\psi_1$} \\
     \lstick{$\ket{j_2}$} &\qw &\ctrl{-1} &\qw &\gate{H} &\gate{R_2} &\qw &\qw & \rstick{$\psi_2$} \\
     \lstick{$\ket{j_3}$} &\qw &\qw &\ctrl{-2} &\qw &\ctrl{-1} &\gate{H} &\targX{} & \rstick{$\psi_3$}
 \end{quantikz}
diff --git a/content/latex/flip_star.tex b/content/latex/flip_star.tex
new file mode 100644
index 0000000..9136bfc
--- /dev/null
+++ b/content/latex/flip_star.tex
@@ -0,0 +1,13 @@
+\documentclass{standalone}
+\usepackage{quantikz}
+
+\begin{document}
+
+\begin{quantikz}
+    \lstick[4]{$\ket{\psi}$} &\gate{H} &\qw &\gate[4]{FLIP_0} &\qw &\gate{H} & \\
+    &\gate{H} &\qw &\qw &\qw &\gate{H} & \\
+    \setwiretype{n} &\push{\vdots} & & & &\push{\vdots} \\
+    &\gate{H} &\qw &\qw &\qw &\gate{H} &
+\end{quantikz}
+
+\end{document}
\ No newline at end of file
diff --git a/content/latex/flip_zero.tex b/content/latex/flip_zero.tex
new file mode 100644
index 0000000..5796d71
--- /dev/null
+++ b/content/latex/flip_zero.tex
@@ -0,0 +1,13 @@
+\documentclass{standalone}
+\usepackage{quantikz}
+
+\begin{document}
+
+\begin{quantikz}
+    \lstick[4]{$\ket{x}$} &\qw &\gate{-Z} &\qw &\gate[4]{-I} & \\
+    &\qw &\ctrl[open]{-1} &\qw &\qw & \\
+    \setwiretype{n} &\push{\vdots} & &\push{\vdots} \\
+    &\qw &\ctrl[open]{-3} &\qw &\qw &
+\end{quantikz}
+
+\end{document}
\ No newline at end of file
diff --git a/content/latex/grover.tex b/content/latex/grover.tex
new file mode 100644
index 0000000..8be355f
--- /dev/null
+++ b/content/latex/grover.tex
@@ -0,0 +1,10 @@
+\documentclass{standalone}
+\usepackage{quantikz}
+
+\begin{document}
+
+\begin{quantikz}
+     \lstick{$\ket{0^n}$} &\qwbundle{n} &\gate{H^{\otimes n}} \gategroup[steps=2]{repeat $t$ times} &\gate{V_f} &\gate{\operatorname{FLIP}_*} &\metercw{x_0}
+\end{quantikz}
+
+\end{document}
\ No newline at end of file
diff --git a/content/latex/grover_plot_1.tex b/content/latex/grover_plot_1.tex
new file mode 100644
index 0000000..517ca9a
--- /dev/null
+++ b/content/latex/grover_plot_1.tex
@@ -0,0 +1,20 @@
+\documentclass{standalone}
+\usepackage{tikz}
+\usepackage{amsmath}
+\usepackage{braket}
+
+\begin{document}
+
+\begin{tikzpicture}
+  % Axis
+  \draw[->] (0,0) -- (5,0) node[anchor=west] {bad};
+  \draw[->] (0,0) -- (0,5) node[anchor=south] {good};
+  
+  % Blue line
+  \draw[->, thick, blue] (0,0) -- (5,1) node[anchor=west] {\(\ket{*}\)};
+  % Angle blue line
+  \draw (2,0) arc[start angle=0, end angle=11, radius=2cm];
+  \node at (2.2,0.2) {\(\theta\)};
+\end{tikzpicture}
+
+\end{document}
\ No newline at end of file
diff --git a/content/latex/grover_plot_2.tex b/content/latex/grover_plot_2.tex
new file mode 100644
index 0000000..ec41908
--- /dev/null
+++ b/content/latex/grover_plot_2.tex
@@ -0,0 +1,26 @@
+\documentclass{standalone}
+\usepackage{tikz}
+\usepackage{amsmath}
+\usepackage{braket}
+
+\begin{document}
+
+\begin{tikzpicture}
+  % Axis
+  \draw[->] (0,0) -- (5,0) node[anchor=west] {bad};
+  \draw[->] (0,0) -- (0,5) node[anchor=south] {good};
+  
+  % Blue line
+  \draw[->, thick, blue] (0,0) -- (5,1) node[anchor=west] {\(\ket{*}\)};
+  % Angle blue line
+  \draw (2,0) arc[start angle=0, end angle=11, radius=2cm];
+  \node at (2.2,0.2) {\(\theta\)};
+  
+  % Red line
+  \draw[->, thick, red] (0,0) -- (5,-1) node[anchor=west] {\(V_f \ket{*}\)};
+  % Angle red line
+  \draw (1.5,0) arc[start angle=0, end angle=-11, radius=1.5cm];
+  \node at (1.7,-0.15) {\(\theta\)};
+\end{tikzpicture}
+
+\end{document}
\ No newline at end of file
diff --git a/content/latex/grover_plot_3.tex b/content/latex/grover_plot_3.tex
new file mode 100644
index 0000000..0125d5b
--- /dev/null
+++ b/content/latex/grover_plot_3.tex
@@ -0,0 +1,32 @@
+\documentclass{standalone}
+\usepackage{tikz}
+\usepackage{amsmath}
+\usepackage{braket}
+
+\begin{document}
+
+\begin{tikzpicture}
+  % Axis
+  \draw[->] (0,0) -- (5,0) node[anchor=west] {bad};
+  \draw[->] (0,0) -- (0,5) node[anchor=south] {good};
+  
+  % Blue line
+  \draw[->, thick, blue] (0,0) -- (5,1) node[anchor=west] {\(\ket{*}\)};
+  % Angle blue line
+  \draw (2,0) arc[start angle=0, end angle=11, radius=2cm];
+  \node at (2.2,0.2) {\(\theta\)};
+  
+  % Red line
+  \draw[->, thick, red] (0,0) -- (5,-1) node[anchor=west] {\(V_f \ket{*}\)};
+  % Angle red line
+  \draw (1.5,0) arc[start angle=0, end angle=-11, radius=1.5cm];
+  \node at (1.7,-0.15) {\(\theta\)};
+
+  % Green line
+  \draw[->, thick, green!50!black] (0,0) -- (5,3) node[anchor=south west] {\(\operatorname{FLIP_* } V_f \ket{*}\)};
+  % Angle green line
+  \draw (2.5,0.5) arc[start angle=11.31, end angle=31, radius=2.5cm];
+  \node at (2.7,1) {\(2\theta\)};
+\end{tikzpicture}
+
+\end{document}
\ No newline at end of file
diff --git a/content/latex/vf.tex b/content/latex/vf.tex
new file mode 100644
index 0000000..4e6816e
--- /dev/null
+++ b/content/latex/vf.tex
@@ -0,0 +1,11 @@
+\documentclass{standalone}
+\usepackage{quantikz}
+
+\begin{document}
+
+\begin{quantikz}
+    \lstick{\ket{x}} &\qw &\gate[2]{U_f} &\qw & \rstick{\ket{x} or -\ket{x}} \\
+    \lstick{\ket{-}} &\qw &\qw &\qw & \rstick{\ket{-}}
+\end{quantikz}
+
+\end{document}
\ No newline at end of file
diff --git a/content/shorsAlgorithm.qmd b/content/shorsAlgorithm.qmd
index c4f504c..356eb84 100644
--- a/content/shorsAlgorithm.qmd
+++ b/content/shorsAlgorithm.qmd
@@ -277,13 +277,10 @@ $$
 
 ![The DTF for three qubits](img/dft_3_qubits){width=100%}
 
-First we construct the $\psi_3$ element. Contrary to the intuition, we use the top wire containing $\ket{j_1}$ for this. We use a Hadamad-gate to bring $\ket{j_1}$ into the superposition $\frac{1}{\sqrt{2}}(\ket{0} + (-1)^{j_1} \ket{1}) = \frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1} \ket{1})$. This looks close to $\psi_3$ already, we now need to add the last two decimal places $j_2j_3$ to the state. For this we use $R_2$ and $R_3$. We apply $R_2$ controlled by the wire $j_2$ and $R_3$ controlled by the wire $j_3$. This means, that we only apply the $R$-gate, if the corresponding wire contains a $1$. You can see this written as a quantum circuit at the figure below. After applying $R_2$ we have the state $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2} \ket{1})$ and after applying $R_3$ we have the state $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2j_3} \ket{1})$ on the top wire. This is the same as $\psi_3$, so we are done on the first wire (We are at the first slice in the figure). 
-
-The next step is to construct the $\psi_2$ state on the middle wire. We again use a Hadamad-gate to bring $\ket{j_2}$ into the superposition $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_2} \ket{1})$. We now need to include the last decimal point $j_3$, for which we use $R_2$ again, this time controlled by $j_3$. The resulting superposition is now $\frac{1}{\sqrt{2}}(\ket{0} +  e^{2\pi i 0.j_2j_3} \ket{1})$, which is $\psi_2$. (We are at the second slice in the figure).
-
-On the bottom wire, we can just do a Hadamad-gate to bring $\ket{j_3}$ into the superposition $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_3} \ket{1})$. We then have $\psi_1$ on the bottom wire. (We are at the third slice in the figure).
-
-When applying this circuit, we get the state $\psi_3 \otimes \psi_2 \otimes \psi_1$ as a result. This very close to our desired state $\psi_1 \otimes \psi_2 \otimes \psi_3$, just the order of the wires is flipped. To solve this, we apply a $\operatorname{SWAP}$ onto all wires, which flips the order of the wires an delivers the correct output for $\operatorname{DFT}_{2^3}$. 
+1. First we construct the $\psi_3$ element. Contrary to the intuition, we use the top wire containing $\ket{j_1}$ for this. We use a Hadamad-gate to bring $\ket{j_1}$ into the superposition $\frac{1}{\sqrt{2}}(\ket{0} + (-1)^{j_1} \ket{1}) = \frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1} \ket{1})$. This looks close to $\psi_3$ already, we now need to add the last two decimal places $j_2j_3$ to the state. For this we use $R_2$ and $R_3$. We apply $R_2$ controlled by the wire $j_2$ and $R_3$ controlled by the wire $j_3$. This means, that we only apply the $R$-gate, if the corresponding wire contains a $1$. You can see this written as a quantum circuit at the figure below. After applying $R_2$ we have the state $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2} \ket{1})$ and after applying $R_3$ we have the state $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_1j_2j_3} \ket{1})$ on the top wire. This is the same as $\psi_3$, so we are done on the first wire (We are at the first slice in the figure). 
+2. The next step is to construct the $\psi_2$ state on the middle wire. We again use a Hadamad-gate to bring $\ket{j_2}$ into the superposition $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_2} \ket{1})$. We now need to include the last decimal point $j_3$, for which we use $R_2$ again, this time controlled by $j_3$. The resulting superposition is now $\frac{1}{\sqrt{2}}(\ket{0} +  e^{2\pi i 0.j_2j_3} \ket{1})$, which is $\psi_2$. (We are at the second slice in the figure).
+3. On the bottom wire, we can just do a Hadamad-gate to bring $\ket{j_3}$ into the superposition $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_3} \ket{1})$. We then have $\psi_1$ on the bottom wire. (We are at the third slice in the figure).
+4. When applying this circuit, we get the state $\psi_3 \otimes \psi_2 \otimes \psi_1$ as a result. This very close to our desired state $\psi_1 \otimes \psi_2 \otimes \psi_3$, just the order of the wires is flipped. To solve this, we apply a $\operatorname{SWAP}$ onto all wires, which flips the order of the wires an delivers the correct output for $\operatorname{DFT}_{2^3}$. 
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 The more general approach to construct the $\operatorname{DFT}_N$ as a quantum circuit with $n$ qubits ($N = 2^n$) works as follows:
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