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+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
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+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
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+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
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 <meta name="author" content="Jannik Hellenkamp">
 <meta name="author" content="Dominique Unruh">
-<meta name="dcterms.date" content="2024-07-08">
+<meta name="dcterms.date" content="2024-07-09">
 
 <title>Introduction to Quantum Computing</title>
 <style>
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+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
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     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">July 8, 2024</p>
+      <p class="date">July 9, 2024</p>
     </div>
   </div>
   
diff --git a/_book/introduction.html b/_book/introduction.html
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+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
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   <ul>
   <li><a href="#electron-in-an-atom" id="toc-electron-in-an-atom" class="nav-link active" data-scroll-target="#electron-in-an-atom"><span class="header-section-number">13.1</span> Electron in an atom</a></li>
-  <li><a href="#setup-for-the-ion-traps" id="toc-setup-for-the-ion-traps" class="nav-link" data-scroll-target="#setup-for-the-ion-traps"><span class="header-section-number">13.2</span> Setup for the ion traps</a></li>
-  <li><a href="#cooling" id="toc-cooling" class="nav-link" data-scroll-target="#cooling"><span class="header-section-number">13.3</span> Cooling</a>
+  <li><a href="#setup-for-the-ion-traps" id="toc-setup-for-the-ion-traps" class="nav-link" data-scroll-target="#setup-for-the-ion-traps"><span class="header-section-number">13.2</span> Setup for the ion traps</a>
   <ul class="collapse">
-  <li><a href="#doppler-cooling" id="toc-doppler-cooling" class="nav-link" data-scroll-target="#doppler-cooling"><span class="header-section-number">13.3.1</span> Doppler cooling</a></li>
+  <li><a href="#cooling" id="toc-cooling" class="nav-link" data-scroll-target="#cooling"><span class="header-section-number">13.2.1</span> Cooling</a></li>
+  <li><a href="#unitaries" id="toc-unitaries" class="nav-link" data-scroll-target="#unitaries"><span class="header-section-number">13.2.2</span> Unitaries</a></li>
+  <li><a href="#measurements" id="toc-measurements" class="nav-link" data-scroll-target="#measurements"><span class="header-section-number">13.2.3</span> Measurements</a></li>
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-<p>So far we have looked at the principals of quantum mechanics and how to transfer these principals to out mathematical description of a quantum computer. While there are many different approaches on how to actually build a quantum computer, which are researched at the moment, we will only look at one approach. This approach is based on trapped ions.</p>
+<p>So far we have looked at the principles of quantum mechanics and how to transfer these principles to our mathematical description of quantum computing. While there are many different approaches on how to actually build a quantum computer, which are researched at the moment, we will only look at one approach. This approach is based on trapped ions.</p>
 <section id="electron-in-an-atom" class="level2" data-number="13.1">
 <h2 data-number="13.1" class="anchored" data-anchor-id="electron-in-an-atom"><span class="header-section-number">13.1</span> Electron in an atom</h2>
 <p>We look at a single atom with a nucleus with a positive charge and a single electron with negative charge “orbiting” the nucleus.</p>
-<p>The electromagnetic field generated by the nucleus is essentially a potential well for the electron, since the electron is drawn to the nucleus and the potential of the electron rises with bigger distance from the nucleus. We simplify a lot here and ignore e.g.&nbsp;the spin.</p>
-<p>We can solve the time-independent Schroedinger equation for this setup and by solving this, we will get the energy eigenstates of the Hamiltonian. These eigenstates are called <em>orbitals</em>.</p>
-<p>We can use a single ion as a qubit, where we define one of the energy eigenstates as <span class="math inline">\(\ket{0}\)</span> and a different eigenstate as <span class="math inline">\(\ket{1}\)</span>.</p>
+<p>The electromagnetic field generated by the nucleus is essentially a potential well for the electron, since the electron is drawn to the nucleus and the potential of the electron rises with bigger distance from the nucleus. We simplify a lot here and ignore ,e.g., the spin.</p>
+<p>We can solve the time-independent Schroedinger equation for this setup and by solving this, we will get the wave functions that are the energy eigenstates of the Hamiltonian. These wave functions are called <em>orbitals</em>.</p>
+<p>We can use a single atom as a qubit, where we define one of the energy eigenstates as <span class="math inline">\(\ket{0}\)</span> and a different eigenstate as <span class="math inline">\(\ket{1}\)</span>.</p>
+<p>In the following, we will specifically use electrically charged atoms, called ions, because they are easier to capture.</p>
 </section>
 <section id="setup-for-the-ion-traps" class="level2" data-number="13.2">
 <h2 data-number="13.2" class="anchored" data-anchor-id="setup-for-the-ion-traps"><span class="header-section-number">13.2</span> Setup for the ion traps</h2>
 <p>The setup for our quantum computer looks as follows:</p>
-<!-- figure -->
-<p>In the previous chapter, we have learned, that we need to be able to perform three different operations to build a quantum computer: we need to be able to perform three basic operations:</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="ionQC.svg" class="img-fluid figure-img" style="width:100.0%"></p>
+<figcaption>Setup for the ion based quantum computer</figcaption>
+</figure>
+</div>
+<p>In the previous chapter, we have learned that we need to be able to perform three different operations to build a quantum computer:</p>
 <ol type="1">
 <li>We need to initialize (cool) our qubit.</li>
 <li>We need to be able to apply a unitary on the qubit.</li>
 <li>We need to be able to measure the qubit.</li>
 </ol>
+<section id="cooling" class="level3" data-number="13.2.1">
+<h3 data-number="13.2.1" class="anchored" data-anchor-id="cooling"><span class="header-section-number">13.2.1</span> Cooling</h3>
+<p>We first look into cooling our system. For cooling, we use a useful fact: If <span class="math inline">\(E_i &lt; E_j\)</span> are different possible energy levels, an ion is in the energy level <span class="math inline">\(E_i\)</span> and then hit by a photon that has the energy <span class="math inline">\(E_j-E_i\)</span>, the ion will go to energy level <span class="math inline">\(E_j\)</span>.</p>
+<section id="doppler-cooling" class="level4" data-number="13.2.1.1">
+<h4 data-number="13.2.1.1" class="anchored" data-anchor-id="doppler-cooling"><span class="header-section-number">13.2.1.1</span> Doppler cooling</h4>
+<p>In our initial setup, we have an ion vibrating back and forth because it has too much energy. We shine a laser on it with slightly less energy than what is need for a transition. The energy of the photon of this laser is denoted by <span class="math inline">\(E=\hbar \cdot \omega\)</span>. When the ion moves towards the photon, the photon has a higher frequency from the point of view of the ion (Doppler effect). This means that the photon has a higher energy and therefore is more likely to be absorbed.</p>
+<p>So by shining a laser on the ion, the photons of the laser “push” the ion, when it “swings” <em>towards</em> the laser similar to a pendulum, where the pendulum gets a pushback with just enough energy so it stops. This reduces the vibrations energy down to a certain level.</p>
+</section>
+<section id="sideband-cooling" class="level4" data-number="13.2.1.2">
+<h4 data-number="13.2.1.2" class="anchored" data-anchor-id="sideband-cooling"><span class="header-section-number">13.2.1.2</span> Sideband cooling</h4>
+<p>Using the doppler cooling, we have reduced the vibration energy, but the electrons may still be excited. We now look at another technique called sideband cooling, which will set the energy of the electrons to a specific energy level <span class="math inline">\(E_0\)</span>.</p>
+<p>The electron can have any energy level <span class="math inline">\(E_i\)</span>. If this energy level is pretty low, the possibility of a spontaneous emission of a photon, which would reduce the energy to a lower level is also quite low. So an electron with energy level <span class="math inline">\(E_1\)</span> or <span class="math inline">\(E_2\)</span> will probably not change to level <span class="math inline">\(E_0\)</span>. If the energy level is big enough (we will call this energy level <span class="math inline">\(E_{\text{big}}\)</span>), the probability of a spontaneous emission of a photon, which would reduce the energy to a lower level is quite high. So when this spontaneous emission happens, the electron will reach an energy level of ,e.g., <span class="math inline">\(E_0\)</span>, <span class="math inline">\(E_1\)</span> or <span class="math inline">\(E_2\)</span>.</p>
+<p>Our goal is to get the energy level to <span class="math inline">\(E_0\)</span>. We know that the energy level of the electrons with <span class="math inline">\(E_{\text{big}}\)</span> will come down eventually, so we shine a laser with the energy per photon of <span class="math inline">\(E_{\text{big}} - E_1, E_{\text{big}} - E_2, \dots\)</span> but not with <span class="math inline">\(E_{\text{big}} - E_0\)</span>. This will “shoot” all the low energy electrons from <span class="math inline">\(E_1,E_2,\dots\)</span> to a higher lever where they will either fall down to <span class="math inline">\(E_1,E_2,\dots\)</span> where they will be energized again and the process is repeated, or they fall into <span class="math inline">\(E_0\)</span> which is our desired energy level.</p>
+<p>Using the sideband and the doppler cooling together, we can cool the vibration and electron excitation. This means that all the ions are in the state <span class="math inline">\(\ket{0}\)</span> and the vibrations energy is also in the state <span class="math inline">\(\ket{0}\)</span>.</p>
+</section>
+</section>
+<section id="unitaries" class="level3" data-number="13.2.2">
+<h3 data-number="13.2.2" class="anchored" data-anchor-id="unitaries"><span class="header-section-number">13.2.2</span> Unitaries</h3>
+<p>This section will be updated later.</p>
 </section>
-<section id="cooling" class="level2" data-number="13.3">
-<h2 data-number="13.3" class="anchored" data-anchor-id="cooling"><span class="header-section-number">13.3</span> Cooling</h2>
-<p>We first look into cooling our system. For cooling, we use a useful fact: If an atom is in the energy level <span class="math inline">\(E_i\)</span> and then hit by a photon of energy level <span class="math inline">\(E_j-E_i\)</span>, the atom will go to energy level <span class="math inline">\(E_j\)</span>.</p>
-<section id="doppler-cooling" class="level3" data-number="13.3.1">
-<h3 data-number="13.3.1" class="anchored" data-anchor-id="doppler-cooling"><span class="header-section-number">13.3.1</span> Doppler cooling</h3>
-<p>In oue initial setup, we have an ion vibrating back and forth. We shine a laser on it with slightly less energy than what is need for a transition.</p>
+<section id="measurements" class="level3" data-number="13.2.3">
+<h3 data-number="13.2.3" class="anchored" data-anchor-id="measurements"><span class="header-section-number">13.2.3</span> Measurements</h3>
+<p>We now look into performing a measurement on an ion-based quantum computer. So far we have defined that the quantum state <span class="math inline">\(\ket{0}\)</span> is represented by the energy level <span class="math inline">\(E_0\)</span> and the quantum state <span class="math inline">\(\ket{1}\)</span> is represented by the energy level <span class="math inline">\(E_1\)</span>.</p>
+<p>To perform a measurement, we also use an auxiliary energy level <span class="math inline">\(E_{\text{aux}} &gt; E_0,E_1\)</span>. Let <span class="math inline">\(\omega\)</span> be the frequency of a photon with energy <span class="math inline">\(E_\text{aux}-E_0 (E_\text{aux}-E_0=\Delta \omega)\)</span></p>
+<p>We now shine a laser with the frequency <span class="math inline">\(\omega\)</span> on the ion. If the state is <span class="math inline">\(\ket{0}\)</span>, the photon gets absorbed and the electron jumps to <span class="math inline">\(E_\text{aux}\)</span> and from there spontaneously back to <span class="math inline">\(E_0\)</span>.This process then repeats over and over emitting many photons. This will create fluorescence, which can be measured by light detectors. If the state is <span class="math inline">\(\ket{1}\)</span>, no ions get absorbed and no fluorescence can be seen.</p>
+<p>So all in all, we measure an ion by shining a photon of frequency <span class="math inline">\(\omega\)</span> onto it and then look whether it lights up.</p>
 
 
 </section>
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diff --git a/_book/observingSystems.html b/_book/observingSystems.html
index d6b16452a7b0e1c09ff65cb3f2d7ae0a7df9bfd5..323ef84a57f740c3e759ac3703f88b1fbc60e5ae 100644
--- a/_book/observingSystems.html
+++ b/_book/observingSystems.html
@@ -234,6 +234,12 @@ ul.task-list li input[type="checkbox"] {
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  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/partialObserving.html b/_book/partialObserving.html
index 58731152f6eb1166b84bc315b3de712be33daf80..ef6bdcd351532aa0a0aeb1f935fe763f101f53f2 100644
--- a/_book/partialObserving.html
+++ b/_book/partialObserving.html
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  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/physicalBackground.html b/_book/physicalBackground.html
index ad813fa42eff500c989b604f8a007881606635cd..a146135c505719c4a781531e4858742748678641 100644
--- a/_book/physicalBackground.html
+++ b/_book/physicalBackground.html
@@ -234,6 +234,12 @@ ul.task-list li input[type="checkbox"] {
   <a href="./physicsToQC.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/physics.html b/_book/physics.html
index a510dbd1bd5c374e119ecd8613be353e7b1a768b..e96cac45bef6e13b523ebfd247fbff7fa917bb8a 100644
--- a/_book/physics.html
+++ b/_book/physics.html
@@ -326,6 +326,12 @@ window.Quarto = {
 \Pr[\text{Particle is in }[a,b]\text{ a time }t_0] = \int_a^b \lvert \psi_{t_0}(x) \rvert^2 dx
 \]</span></p>
 <p>From this we can see, that the integral <span class="math inline">\(\int \lvert \psi_{t_0} \rvert ^2 dx = 1\)</span>. The momentum is not needed for the wave function.</p>
+<p>The inner product of two wave functions is given by <span class="math display">\[
+\braket{\psi|\phi} := \int \overline{\psi(x)} \cdot \phi(x) dx
+\]</span></p>
+<p>The norm of a wave function is given by <span class="math display">\[
+\| \psi \|^2 :=\braket{\psi|\psi}  = \int \lvert \psi(x) \rvert ^2 dx
+\]</span></p>
 <p>In general, the wave function can have a different domain, e.g.&nbsp;<span class="math inline">\(\psi_t: \mathbb{R}^3 \rightarrow \mathbb{C}\)</span> for a particle in 3D-space. Everything below works analogously in that case.</p>
 </section>
 <section id="energy-hamiltonian" class="level2" data-number="11.2">
diff --git a/_book/physicsToQC.html b/_book/physicsToQC.html
index a56883408620b6f8583da848f01bb0367f6c57f3..ba7849f5721753056b70685b76bdf05f9ac5c0ca 100644
--- a/_book/physicsToQC.html
+++ b/_book/physicsToQC.html
@@ -312,7 +312,7 @@ window.Quarto = {
 \dots
 \end{aligned}
 \]</span></p>
-<p>For one qubit, we just look at <span class="math inline">\(\psi_0\)</span> and <span class="math inline">\(\psi_1\)</span> and ignore all other wave functions. Note that this can lead to errors, since those other wave function still exists and interacts with our system, even though they might have a very small probability.</p>
+<p>For one qubit, we just look at <span class="math inline">\(\psi_0\)</span> and <span class="math inline">\(\psi_1\)</span> and ignore all other wave functions. Note that this can lead to errors, since those other wave functions still exists and interact with our system, even though they might have a very small probability.</p>
 <p>To fully construct our one qubit quantum computer, we need to be able to perform three basic operations:</p>
 <ol type="1">
 <li>We need to initialize our qubit with <span class="math inline">\(\ket{0}\)</span>. This is also called cooling.</li>
@@ -340,7 +340,8 @@ E_1 &amp; 0 &amp; 0\\
 0 &amp; 0 &amp; \ddots
 \end{pmatrix}
 \]</span></p>
-<p>For this representation of <span class="math inline">\(H\)</span>, we immediately get We can now use a helpful theorem to get a solution for the differential equation.</p>
+<p>For this representation of <span class="math inline">\(H\)</span>, we immediately get <span class="math inline">\(H \ket{0} = \frac{\pi^2}{2}\)</span>, <span class="math inline">\(H \ket{1} =  2 \pi^2\)</span> and so on, so nothing has changed except that <span class="math inline">\(H\)</span> is represented more nicely.</p>
+<p>We can now use a helpful theorem to get a solution for the differential equation.</p>
 <div class="callout callout-style-simple callout-note no-icon">
 <div class="callout-body d-flex">
 <div class="callout-icon-container">
@@ -361,7 +362,9 @@ i \hbar \frac{\partial \psi_t}{\partial t} = H \psi_t
 <p>We use this theorem for our one qubit computer. The goal is to change the potential by some <span class="math inline">\(\delta V\)</span> and from this get a different <span class="math inline">\(U_t\)</span>.</p>
 <p>We try this by changing the potential to <span class="math inline">\(\delta V = \frac{9\pi^2}{16} (\frac{1}{2}-x)\cdot 1000\)</span> for <span class="math inline">\(x \in [0,1]\)</span> and <span class="math inline">\(\delta V = 0\)</span> for <span class="math inline">\(x \notin [0,1]\)</span>.</p>
 <!-- Figure? -->
-<p>We rewrite <span class="math inline">\(\delta V\)</span> as a matrix: <span class="math display">\[
+<p>We rewrite <span class="math inline">\(\delta V\)</span> as a matrix. To do so, we try to find a matrix in the base <span class="math inline">\(\ket{0}, \ket{1}\)</span>. If <span class="math inline">\(\delta V\ket{0}=a\ket{0}+b\ket{1}\)</span> and <span class="math inline">\(\delta V\ket{1}=c\ket{0}+d\ket{1}\)</span>, then <span class="math display">\[
+\delta V = \begin{pmatrix} a &amp; c \\ b &amp; d\end{pmatrix}
+\]</span> Since <span class="math inline">\(\ket{0}\)</span> and <span class="math inline">\(\ket{1}\)</span> are orthonormal, we know that <span class="math inline">\(\braket{0|\delta V|0} = \braket{0|a|0}  + \braket{0|b|1} = a + 0 = a\)</span>. We get <span class="math inline">\(b,c\)</span> and <span class="math inline">\(d\)</span> similar and from this the following matrix: <span class="math display">\[
 \delta V = \begin{pmatrix} \braket{0|\delta V|0} &amp; \braket{0|\delta V|1}\\ \braket{1|\delta V|0} &amp; \braket{1|\delta V|1}    \end{pmatrix}
 \]</span></p>
 <p>We calculate each of the entries of the matrix separately:</p>
diff --git a/_book/probabilisticSystems.html b/_book/probabilisticSystems.html
index 7e49d38b146e0fd3b8ad183a5385c8186c9a4e32..eb60a27e3e335c2b4a184688eec006a0ec110a13 100644
--- a/_book/probabilisticSystems.html
+++ b/_book/probabilisticSystems.html
@@ -263,6 +263,12 @@ window.Quarto = {
   <a href="./physicsToQC.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/quantumAlgorithms.html b/_book/quantumAlgorithms.html
index d701242c638e73f17756a572bb20fc14342cf39d..d03562837e5780ebcdcd5d89f0c7fd3520b17b9f 100644
--- a/_book/quantumAlgorithms.html
+++ b/_book/quantumAlgorithms.html
@@ -234,6 +234,12 @@ ul.task-list li input[type="checkbox"] {
   <a href="./physicsToQC.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/quantumBasics.html b/_book/quantumBasics.html
index b5209d3bab88acde12cea5f52c751f9eb781cc94..9e3af41e2d51bf0a9cc27013d259c490a130640b 100644
--- a/_book/quantumBasics.html
+++ b/_book/quantumBasics.html
@@ -234,6 +234,12 @@ ul.task-list li input[type="checkbox"] {
   <a href="./physicsToQC.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/quantumCircutsKetNotation.html b/_book/quantumCircutsKetNotation.html
index 768e577d3d5a9c44a65f3a6b8923a0afabc7da47..739a13b15e38ba08f6eb27a9fb43c24f44fd424f 100644
--- a/_book/quantumCircutsKetNotation.html
+++ b/_book/quantumCircutsKetNotation.html
@@ -234,6 +234,12 @@ ul.task-list li input[type="checkbox"] {
   <a href="./physicsToQC.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
@@ -247,7 +253,7 @@ ul.task-list li input[type="checkbox"] {
     <h2 id="toc-title">Table of contents</h2>
    
   <ul>
-  <li><a href="#ket-notation" id="toc-ket-notation" class="nav-link active" data-scroll-target="#ket-notation"><span class="header-section-number">8</span> Ket Notation</a></li>
+  <li><a href="#ket-notation" id="toc-ket-notation" class="nav-link active" data-scroll-target="#ket-notation"><span class="header-section-number">7.1</span> Ket Notation</a></li>
   </ul>
 </nav>
     </div>
@@ -273,8 +279,8 @@ ul.task-list li input[type="checkbox"] {
 </header>
 
 
-<section id="ket-notation" class="level1" data-number="8">
-<h1 data-number="8"><span class="header-section-number">8</span> Ket Notation</h1>
+<section id="ket-notation" class="level2" data-number="7.1">
+<h2 data-number="7.1" class="anchored" data-anchor-id="ket-notation"><span class="header-section-number">7.1</span> Ket Notation</h2>
 
 
 </section>
diff --git a/_book/quantumSystems.html b/_book/quantumSystems.html
index 94fb23cd3661481437021c67ab9f054fe7852e2d..d7855fe4fd82c5912ae6166a3df63cc938c87335 100644
--- a/_book/quantumSystems.html
+++ b/_book/quantumSystems.html
@@ -263,6 +263,12 @@ window.Quarto = {
   <a href="./physicsToQC.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text"><span class="chapter-number">12</span>&nbsp; <span class="chapter-title">From to Quantum Physics to a Quantum Computer</span></span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./ionBasedQC.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text"><span class="chapter-number">13</span>&nbsp; <span class="chapter-title">Ion-based quantum computers</span></span></a>
+  </div>
 </li>
       </ul>
   </li>
diff --git a/_book/search.json b/_book/search.json
index b1ab98610d47c6e0a6076e51aa5d061193d5a736..57a1cf60001303bc6dc6c8ec5333e61a7fc09e72 100644
--- a/_book/search.json
+++ b/_book/search.json
@@ -123,7 +123,7 @@
     "href": "quantumCircutsKetNotation.html",
     "title": "7  Quantum Circuits",
     "section": "",
-    "text": "8 Ket Notation",
+    "text": "7.1 Ket Notation",
     "crumbs": [
       "Quantum Basics",
       "<span class='chapter-number'>7</span>  <span class='chapter-title'>Quantum Circuits</span>"
@@ -233,7 +233,7 @@
     "href": "physics.html",
     "title": "11  Quantum Physics",
     "section": "",
-    "text": "11.1 Wave function\nThe first concept we look into is the wave function of quantum mechanics. For this we will look at the experiment “particle in a well”. To keep the math simple, we assume the space to be 1-dimensional, so the particle is confined to a line.\nIn this experiment, we have one particle and a potential, which is denoted by a function \\(V: \\mathbb{R} \\rightarrow \\mathbb{R}\\). This function maps a position of a particle in \\(\\mathbb{R}\\) to the energy which is needed to hold the particle in that position.\nClassically a state of a system at time \\(t\\) is described by the position \\(x(t) \\in \\mathbb{R}\\) and the momentum \\(p(t) \\in \\mathbb{R}\\).\nIn the quantum world, we have a wave function \\(\\psi_t(x)\\) with \\(\\psi_t: \\mathbb{R} \\rightarrow \\mathbb{C}\\) under \\(t\\in\\mathbb{R}\\), which takes the position of a particle as an input and outputs the amplitude of that particle, with \\(t\\) as the time.\nTo calculate the probability of a particle being in the interval \\([a,b]\\) at time \\(t_0\\), we can use the integral over the wave function: \\[\n\\Pr[\\text{Particle is in }[a,b]\\text{ a time }t_0] = \\int_a^b \\lvert \\psi_{t_0}(x) \\rvert^2 dx\n\\]\nFrom this we can see, that the integral \\(\\int \\lvert \\psi_{t_0} \\rvert ^2 dx = 1\\). The momentum is not needed for the wave function.\nIn general, the wave function can have a different domain, e.g. \\(\\psi_t: \\mathbb{R}^3 \\rightarrow \\mathbb{C}\\) for a particle in 3D-space. Everything below works analogously in that case.",
+    "text": "11.1 Wave function\nThe first concept we look into is the wave function of quantum mechanics. For this we will look at the experiment “particle in a well”. To keep the math simple, we assume the space to be 1-dimensional, so the particle is confined to a line.\nIn this experiment, we have one particle and a potential, which is denoted by a function \\(V: \\mathbb{R} \\rightarrow \\mathbb{R}\\). This function maps a position of a particle in \\(\\mathbb{R}\\) to the energy which is needed to hold the particle in that position.\nClassically a state of a system at time \\(t\\) is described by the position \\(x(t) \\in \\mathbb{R}\\) and the momentum \\(p(t) \\in \\mathbb{R}\\).\nIn the quantum world, we have a wave function \\(\\psi_t(x)\\) with \\(\\psi_t: \\mathbb{R} \\rightarrow \\mathbb{C}\\) under \\(t\\in\\mathbb{R}\\), which takes the position of a particle as an input and outputs the amplitude of that particle, with \\(t\\) as the time.\nTo calculate the probability of a particle being in the interval \\([a,b]\\) at time \\(t_0\\), we can use the integral over the wave function: \\[\n\\Pr[\\text{Particle is in }[a,b]\\text{ a time }t_0] = \\int_a^b \\lvert \\psi_{t_0}(x) \\rvert^2 dx\n\\]\nFrom this we can see, that the integral \\(\\int \\lvert \\psi_{t_0} \\rvert ^2 dx = 1\\). The momentum is not needed for the wave function.\nThe inner product of two wave functions is given by \\[\n\\braket{\\psi|\\phi} := \\int \\overline{\\psi(x)} \\cdot \\phi(x) dx\n\\]\nThe norm of a wave function is given by \\[\n\\| \\psi \\|^2 :=\\braket{\\psi|\\psi}  = \\int \\lvert \\psi(x) \\rvert ^2 dx\n\\]\nIn general, the wave function can have a different domain, e.g. \\(\\psi_t: \\mathbb{R}^3 \\rightarrow \\mathbb{C}\\) for a particle in 3D-space. Everything below works analogously in that case.",
     "crumbs": [
       "Physical Background",
       "<span class='chapter-number'>11</span>  <span class='chapter-title'>Quantum Physics</span>"
@@ -288,7 +288,7 @@
     "href": "physicsToQC.html",
     "title": "12  From to Quantum Physics to a Quantum Computer",
     "section": "",
-    "text": "In the previous chapter, we have learned about the fundamentals of quantum physics. We now relate this to quantum computers.\nSo far we have seen solutions of the time-independent Schrödinger equation \\(\\psi_m(x)= \\sqrt{2} \\sin((k+1) \\pi x)\\) and \\(E_k=k^2\\frac{(k+1)\\pi}{2}\\). Note that we still assume \\(\\hbar = 1\\) and \\(m=1\\).\nWe can now combine physics and quantum computer science by setting\n\\[\n\\begin{aligned}\n\\ket{0} &:= \\psi_0\\\\\n\\ket{1} &:= \\psi_1\\\\\n\\dots\n\\end{aligned}\n\\]\nFor one qubit, we just look at \\(\\psi_0\\) and \\(\\psi_1\\) and ignore all other wave functions. Note that this can lead to errors, since those other wave function still exists and interacts with our system, even though they might have a very small probability.\nTo fully construct our one qubit quantum computer, we need to be able to perform three basic operations:\n\nWe need to initialize our qubit with \\(\\ket{0}\\). This is also called cooling.\nWe need to be able to apply a unitary on the qubit.\nWe need to be able to measure the qubit.\n\nWe look into how to construct a unitary on our particle. The cooling and measuring operations are out of scope od this chapter. We know that the time evolution of our quantum computer is \\(\\ket{0} \\mapsto  e^{-i \\frac{\\pi^2}{2}t} \\ket{0}\\) and \\(\\ket{1} \\mapsto  e^{-i 2\\pi^2 t} \\ket{1}\\).\nThis can be seen as a unitary \\(U_t\\), which is depending on \\(t\\) written as the matrix:\n\\[\nU_t=\\begin{pmatrix} e^{-i \\frac{\\pi^2}{2}t} & 0\\\\ 0 & e^{-i 2\\pi^2 t}\\end{pmatrix}\n\\]\nUsing \\(t= \\frac{6}{\\pi}\\), we get the unitary \\[\nU_{\\frac{6}{\\pi}}=\\begin{pmatrix} -1 & 0\\\\ 0 & 1 \\end{pmatrix}\n\\] which is equal to the unitary \\(-Z\\). This means, that the unitary \\(-Z\\) is applied every \\(\\frac{6}{\\pi}\\) steps “automatically”. The factor \\(-1\\) does not make a physical difference as it is a global phase factor, so the \\(-Z\\) is physically equal to the \\(Z\\) gate.\nBut how do we get new unitaries which are not \\(Z\\)? Since \\(U_t\\) is dependent on the evolution of \\(\\psi\\), which is dependent of \\(H\\), which is dependent of the potential \\(V(x)\\), we can change the potential \\(V(x)\\) to get a different \\(U_t\\).\nThe problem is that when changing the potential \\(V(x)\\), we need to solve the differential equation again. Luckily, there is a trick for that which avoids thinking about wave functions to much: If we have wave functions \\(\\ket{k}\\) with \\(k\\in\\{1,\\dots,N\\}\\) and \\(\\|\\ket{k}\\| = 1\\) and also \\(\\braket{k|l}=0\\) for \\(k\\neq l\\) so \\(\\ket{k}\\) and \\(\\ket{l}\\) are orthogonal to each other, we can rewrite \\(H\\) as a linear operator on the wave functions written as \\(\\operatorname{span}\\{\\ket{k}: k = 1\\dots N\\}\\), therefore we can write \\(H\\) as a matrix: \\[\nH = \\begin{pmatrix}\nE_1 & 0 & 0\\\\\n0 & E_2 & 0\\\\\n0 & 0 & \\ddots\n\\end{pmatrix}\n= \\begin{pmatrix}\n\\frac{\\pi^2}{2} & 0 & 0\\\\\n0 & 2 \\pi^2 & 0\\\\\n0 & 0 & \\ddots\n\\end{pmatrix}\n\\]\nFor this representation of \\(H\\), we immediately get We can now use a helpful theorem to get a solution for the differential equation.\n\n\n\n\n\n\n\nTheorem 12.1 The differential equation \\[\ni \\hbar \\frac{\\partial \\psi_t}{\\partial t} = H \\psi_t\n\\] with \\(H\\) as a \\(N \\times N\\) matrix and initial state \\(\\psi_0\\) as an \\(N\\)-dim vector has the solution \\[\n\\psi_t = e^{-i H t /\\hbar} \\psi_0\n\\]\n\n\n\n\n\nWe use this theorem for our one qubit computer. The goal is to change the potential by some \\(\\delta V\\) and from this get a different \\(U_t\\).\nWe try this by changing the potential to \\(\\delta V = \\frac{9\\pi^2}{16} (\\frac{1}{2}-x)\\cdot 1000\\) for \\(x \\in [0,1]\\) and \\(\\delta V = 0\\) for \\(x \\notin [0,1]\\).\n\nWe rewrite \\(\\delta V\\) as a matrix: \\[\n\\delta V = \\begin{pmatrix} \\braket{0|\\delta V|0} & \\braket{0|\\delta V|1}\\\\ \\braket{1|\\delta V|0} & \\braket{1|\\delta V|1}    \\end{pmatrix}\n\\]\nWe calculate each of the entries of the matrix separately:\n\\[\n\\begin{aligned}\n\\braket{0|\\delta V|0} =& \\int^1_0 \\sqrt{2} \\sin(\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(\\pi x) dx = 0\\\\\n\\braket{1|\\delta V|1} =& \\int^1_0 \\sqrt{2} \\sin(2\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(2\\pi x) dx = 0\\\\\n\\braket{1|\\delta V|0} =& \\int^1_0 \\sqrt{2} \\sin(2\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(\\pi x) dx = 1000 \\\\\n\\braket{0|\\delta V|1} =& \\int^1_0 \\sqrt{2} \\sin(\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(2\\pi x) dx = 1000 \\\\\n\\end{aligned}\n\\]\nFrom this we get \\(\\delta V\\) written as a matrix with readable numbers: \\[\n\\delta V = \\begin{pmatrix} 0  & 1000 \\\\ 1000 & 0 \\end{pmatrix}\n\\]\nWe can now add this matrix to the Hamiltonian \\(H\\) to get the Hamiltonian \\(H'\\) which is the Hamiltonian under the changed potential by calculating \\[\nH' = H + \\delta V =  \\begin{pmatrix} \\frac{\\pi^2}{2} & 1000 \\\\ 1000 & 2\\pi^2 \\end{pmatrix}\n\\]\nWe now need to solve Schrodinger equation with this new \\(H'\\).\nIf we would solve the Schrodinger equation with \\(\\delta V\\) as a Hamiltonian using Theorem 12.1, we would get the unitary \\(U_t' = e^{-i \\delta V t}\\). After \\(t=\\frac{\\pi}{2000}\\) this would be the unitary \\(\\begin{pmatrix} 0 & -i \\\\ -i & 0\\end{pmatrix} = -i X\\), essentially an \\(X\\) gate.\nIf we now apply \\(H' = H + \\delta V\\) as a Hamiltonian, we would get \\(U_t' = e^{-i H' t}\\) and this is \\[\nU_t' = \\begin{pmatrix} e^{-i \\frac{\\pi^2}{2} t} & e^{-i 1000 t}\\\\\ne^{-i 1000 t} & e^{-i 2\\pi^2 t}\\end{pmatrix}\n\\] so approximately the unitary \\(\\begin{pmatrix} 0 & -i \\\\ -i & 0\\end{pmatrix} = -i X\\) up to about \\(2\\%\\) error.",
+    "text": "In the previous chapter, we have learned about the fundamentals of quantum physics. We now relate this to quantum computers.\nSo far we have seen solutions of the time-independent Schrödinger equation \\(\\psi_m(x)= \\sqrt{2} \\sin((k+1) \\pi x)\\) and \\(E_k=k^2\\frac{(k+1)\\pi}{2}\\). Note that we still assume \\(\\hbar = 1\\) and \\(m=1\\).\nWe can now combine physics and quantum computer science by setting\n\\[\n\\begin{aligned}\n\\ket{0} &:= \\psi_0\\\\\n\\ket{1} &:= \\psi_1\\\\\n\\dots\n\\end{aligned}\n\\]\nFor one qubit, we just look at \\(\\psi_0\\) and \\(\\psi_1\\) and ignore all other wave functions. Note that this can lead to errors, since those other wave functions still exists and interact with our system, even though they might have a very small probability.\nTo fully construct our one qubit quantum computer, we need to be able to perform three basic operations:\n\nWe need to initialize our qubit with \\(\\ket{0}\\). This is also called cooling.\nWe need to be able to apply a unitary on the qubit.\nWe need to be able to measure the qubit.\n\nWe look into how to construct a unitary on our particle. The cooling and measuring operations are out of scope od this chapter. We know that the time evolution of our quantum computer is \\(\\ket{0} \\mapsto  e^{-i \\frac{\\pi^2}{2}t} \\ket{0}\\) and \\(\\ket{1} \\mapsto  e^{-i 2\\pi^2 t} \\ket{1}\\).\nThis can be seen as a unitary \\(U_t\\), which is depending on \\(t\\) written as the matrix:\n\\[\nU_t=\\begin{pmatrix} e^{-i \\frac{\\pi^2}{2}t} & 0\\\\ 0 & e^{-i 2\\pi^2 t}\\end{pmatrix}\n\\]\nUsing \\(t= \\frac{6}{\\pi}\\), we get the unitary \\[\nU_{\\frac{6}{\\pi}}=\\begin{pmatrix} -1 & 0\\\\ 0 & 1 \\end{pmatrix}\n\\] which is equal to the unitary \\(-Z\\). This means, that the unitary \\(-Z\\) is applied every \\(\\frac{6}{\\pi}\\) steps “automatically”. The factor \\(-1\\) does not make a physical difference as it is a global phase factor, so the \\(-Z\\) is physically equal to the \\(Z\\) gate.\nBut how do we get new unitaries which are not \\(Z\\)? Since \\(U_t\\) is dependent on the evolution of \\(\\psi\\), which is dependent of \\(H\\), which is dependent of the potential \\(V(x)\\), we can change the potential \\(V(x)\\) to get a different \\(U_t\\).\nThe problem is that when changing the potential \\(V(x)\\), we need to solve the differential equation again. Luckily, there is a trick for that which avoids thinking about wave functions to much: If we have wave functions \\(\\ket{k}\\) with \\(k\\in\\{1,\\dots,N\\}\\) and \\(\\|\\ket{k}\\| = 1\\) and also \\(\\braket{k|l}=0\\) for \\(k\\neq l\\) so \\(\\ket{k}\\) and \\(\\ket{l}\\) are orthogonal to each other, we can rewrite \\(H\\) as a linear operator on the wave functions written as \\(\\operatorname{span}\\{\\ket{k}: k = 1\\dots N\\}\\), therefore we can write \\(H\\) as a matrix: \\[\nH = \\begin{pmatrix}\nE_1 & 0 & 0\\\\\n0 & E_2 & 0\\\\\n0 & 0 & \\ddots\n\\end{pmatrix}\n= \\begin{pmatrix}\n\\frac{\\pi^2}{2} & 0 & 0\\\\\n0 & 2 \\pi^2 & 0\\\\\n0 & 0 & \\ddots\n\\end{pmatrix}\n\\]\nFor this representation of \\(H\\), we immediately get \\(H \\ket{0} = \\frac{\\pi^2}{2}\\), \\(H \\ket{1} =  2 \\pi^2\\) and so on, so nothing has changed except that \\(H\\) is represented more nicely.\nWe can now use a helpful theorem to get a solution for the differential equation.\n\n\n\n\n\n\n\nTheorem 12.1 The differential equation \\[\ni \\hbar \\frac{\\partial \\psi_t}{\\partial t} = H \\psi_t\n\\] with \\(H\\) as a \\(N \\times N\\) matrix and initial state \\(\\psi_0\\) as an \\(N\\)-dim vector has the solution \\[\n\\psi_t = e^{-i H t /\\hbar} \\psi_0\n\\]\n\n\n\n\n\nWe use this theorem for our one qubit computer. The goal is to change the potential by some \\(\\delta V\\) and from this get a different \\(U_t\\).\nWe try this by changing the potential to \\(\\delta V = \\frac{9\\pi^2}{16} (\\frac{1}{2}-x)\\cdot 1000\\) for \\(x \\in [0,1]\\) and \\(\\delta V = 0\\) for \\(x \\notin [0,1]\\).\n\nWe rewrite \\(\\delta V\\) as a matrix. To do so, we try to find a matrix in the base \\(\\ket{0}, \\ket{1}\\). If \\(\\delta V\\ket{0}=a\\ket{0}+b\\ket{1}\\) and \\(\\delta V\\ket{1}=c\\ket{0}+d\\ket{1}\\), then \\[\n\\delta V = \\begin{pmatrix} a & c \\\\ b & d\\end{pmatrix}\n\\] Since \\(\\ket{0}\\) and \\(\\ket{1}\\) are orthonormal, we know that \\(\\braket{0|\\delta V|0} = \\braket{0|a|0}  + \\braket{0|b|1} = a + 0 = a\\). We get \\(b,c\\) and \\(d\\) similar and from this the following matrix: \\[\n\\delta V = \\begin{pmatrix} \\braket{0|\\delta V|0} & \\braket{0|\\delta V|1}\\\\ \\braket{1|\\delta V|0} & \\braket{1|\\delta V|1}    \\end{pmatrix}\n\\]\nWe calculate each of the entries of the matrix separately:\n\\[\n\\begin{aligned}\n\\braket{0|\\delta V|0} =& \\int^1_0 \\sqrt{2} \\sin(\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(\\pi x) dx = 0\\\\\n\\braket{1|\\delta V|1} =& \\int^1_0 \\sqrt{2} \\sin(2\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(2\\pi x) dx = 0\\\\\n\\braket{1|\\delta V|0} =& \\int^1_0 \\sqrt{2} \\sin(2\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(\\pi x) dx = 1000 \\\\\n\\braket{0|\\delta V|1} =& \\int^1_0 \\sqrt{2} \\sin(\\pi x) \\cdot \\delta V(x) \\cdot \\sqrt{2} \\sin(2\\pi x) dx = 1000 \\\\\n\\end{aligned}\n\\]\nFrom this we get \\(\\delta V\\) written as a matrix with readable numbers: \\[\n\\delta V = \\begin{pmatrix} 0  & 1000 \\\\ 1000 & 0 \\end{pmatrix}\n\\]\nWe can now add this matrix to the Hamiltonian \\(H\\) to get the Hamiltonian \\(H'\\) which is the Hamiltonian under the changed potential by calculating \\[\nH' = H + \\delta V =  \\begin{pmatrix} \\frac{\\pi^2}{2} & 1000 \\\\ 1000 & 2\\pi^2 \\end{pmatrix}\n\\]\nWe now need to solve Schrodinger equation with this new \\(H'\\).\nIf we would solve the Schrodinger equation with \\(\\delta V\\) as a Hamiltonian using Theorem 12.1, we would get the unitary \\(U_t' = e^{-i \\delta V t}\\). After \\(t=\\frac{\\pi}{2000}\\) this would be the unitary \\(\\begin{pmatrix} 0 & -i \\\\ -i & 0\\end{pmatrix} = -i X\\), essentially an \\(X\\) gate.\nIf we now apply \\(H' = H + \\delta V\\) as a Hamiltonian, we would get \\(U_t' = e^{-i H' t}\\) and this is \\[\nU_t' = \\begin{pmatrix} e^{-i \\frac{\\pi^2}{2} t} & e^{-i 1000 t}\\\\\ne^{-i 1000 t} & e^{-i 2\\pi^2 t}\\end{pmatrix}\n\\] so approximately the unitary \\(\\begin{pmatrix} 0 & -i \\\\ -i & 0\\end{pmatrix} = -i X\\) up to about \\(2\\%\\) error.",
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-    "text": "13.1 Electron in an atom\nWe look at a single atom with a nucleus with a positive charge and a single electron with negative charge “orbiting” the nucleus.\nThe electromagnetic field generated by the nucleus is essentially a potential well for the electron, since the electron is drawn to the nucleus and the potential of the electron rises with bigger distance from the nucleus. We simplify a lot here and ignore e.g. the spin.\nWe can solve the time-independent Schroedinger equation for this setup and by solving this, we will get the energy eigenstates of the Hamiltonian. These eigenstates are called orbitals.\nWe can use a single ion as a qubit, where we define one of the energy eigenstates as \\(\\ket{0}\\) and a different eigenstate as \\(\\ket{1}\\).",
+    "text": "13.1 Electron in an atom\nWe look at a single atom with a nucleus with a positive charge and a single electron with negative charge “orbiting” the nucleus.\nThe electromagnetic field generated by the nucleus is essentially a potential well for the electron, since the electron is drawn to the nucleus and the potential of the electron rises with bigger distance from the nucleus. We simplify a lot here and ignore ,e.g., the spin.\nWe can solve the time-independent Schroedinger equation for this setup and by solving this, we will get the wave functions that are the energy eigenstates of the Hamiltonian. These wave functions are called orbitals.\nWe can use a single atom as a qubit, where we define one of the energy eigenstates as \\(\\ket{0}\\) and a different eigenstate as \\(\\ket{1}\\).\nIn the following, we will specifically use electrically charged atoms, called ions, because they are easier to capture.",
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     "section": "13.2 Setup for the ion traps",
-    "text": "13.2 Setup for the ion traps\nThe setup for our quantum computer looks as follows:\n\nIn the previous chapter, we have learned, that we need to be able to perform three different operations to build a quantum computer: we need to be able to perform three basic operations:\n\nWe need to initialize (cool) our qubit.\nWe need to be able to apply a unitary on the qubit.\nWe need to be able to measure the qubit.",
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-    "text": "13.3 Cooling\nWe first look into cooling our system. For cooling, we use a useful fact: If an atom is in the energy level \\(E_i\\) and then hit by a photon of energy level \\(E_j-E_i\\), the atom will go to energy level \\(E_j\\).\n\n13.3.1 Doppler cooling\nIn oue initial setup, we have an ion vibrating back and forth. We shine a laser on it with slightly less energy than what is need for a transition.",
+    "text": "13.2 Setup for the ion traps\nThe setup for our quantum computer looks as follows:\n\n\n\nSetup for the ion based quantum computer\n\n\nIn the previous chapter, we have learned that we need to be able to perform three different operations to build a quantum computer:\n\nWe need to initialize (cool) our qubit.\nWe need to be able to apply a unitary on the qubit.\nWe need to be able to measure the qubit.\n\n\n13.2.1 Cooling\nWe first look into cooling our system. For cooling, we use a useful fact: If \\(E_i &lt; E_j\\) are different possible energy levels, an ion is in the energy level \\(E_i\\) and then hit by a photon that has the energy \\(E_j-E_i\\), the ion will go to energy level \\(E_j\\).\n\n13.2.1.1 Doppler cooling\nIn our initial setup, we have an ion vibrating back and forth because it has too much energy. We shine a laser on it with slightly less energy than what is need for a transition. The energy of the photon of this laser is denoted by \\(E=\\hbar \\cdot \\omega\\). When the ion moves towards the photon, the photon has a higher frequency from the point of view of the ion (Doppler effect). This means that the photon has a higher energy and therefore is more likely to be absorbed.\nSo by shining a laser on the ion, the photons of the laser “push” the ion, when it “swings” towards the laser similar to a pendulum, where the pendulum gets a pushback with just enough energy so it stops. This reduces the vibrations energy down to a certain level.\n\n\n13.2.1.2 Sideband cooling\nUsing the doppler cooling, we have reduced the vibration energy, but the electrons may still be excited. We now look at another technique called sideband cooling, which will set the energy of the electrons to a specific energy level \\(E_0\\).\nThe electron can have any energy level \\(E_i\\). If this energy level is pretty low, the possibility of a spontaneous emission of a photon, which would reduce the energy to a lower level is also quite low. So an electron with energy level \\(E_1\\) or \\(E_2\\) will probably not change to level \\(E_0\\). If the energy level is big enough (we will call this energy level \\(E_{\\text{big}}\\)), the probability of a spontaneous emission of a photon, which would reduce the energy to a lower level is quite high. So when this spontaneous emission happens, the electron will reach an energy level of ,e.g., \\(E_0\\), \\(E_1\\) or \\(E_2\\).\nOur goal is to get the energy level to \\(E_0\\). We know that the energy level of the electrons with \\(E_{\\text{big}}\\) will come down eventually, so we shine a laser with the energy per photon of \\(E_{\\text{big}} - E_1, E_{\\text{big}} - E_2, \\dots\\) but not with \\(E_{\\text{big}} - E_0\\). This will “shoot” all the low energy electrons from \\(E_1,E_2,\\dots\\) to a higher lever where they will either fall down to \\(E_1,E_2,\\dots\\) where they will be energized again and the process is repeated, or they fall into \\(E_0\\) which is our desired energy level.\nUsing the sideband and the doppler cooling together, we can cool the vibration and electron excitation. This means that all the ions are in the state \\(\\ket{0}\\) and the vibrations energy is also in the state \\(\\ket{0}\\).\n\n\n\n13.2.2 Unitaries\nThis section will be updated later.\n\n\n13.2.3 Measurements\nWe now look into performing a measurement on an ion-based quantum computer. So far we have defined that the quantum state \\(\\ket{0}\\) is represented by the energy level \\(E_0\\) and the quantum state \\(\\ket{1}\\) is represented by the energy level \\(E_1\\).\nTo perform a measurement, we also use an auxiliary energy level \\(E_{\\text{aux}} &gt; E_0,E_1\\). Let \\(\\omega\\) be the frequency of a photon with energy \\(E_\\text{aux}-E_0 (E_\\text{aux}-E_0=\\Delta \\omega)\\)\nWe now shine a laser with the frequency \\(\\omega\\) on the ion. If the state is \\(\\ket{0}\\), the photon gets absorbed and the electron jumps to \\(E_\\text{aux}\\) and from there spontaneously back to \\(E_0\\).This process then repeats over and over emitting many photons. This will create fluorescence, which can be measured by light detectors. If the state is \\(\\ket{1}\\), no ions get absorbed and no fluorescence can be seen.\nSo all in all, we measure an ion by shining a photon of frequency \\(\\omega\\) onto it and then look whether it lights up.",
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diff --git a/index.qmd b/index.qmd
index 6703d46addb11b3279abcfba46400c2c70bdf88e..233e9e30424b4182e59ac2f701fe743330a9dce1 100644
--- a/index.qmd
+++ b/index.qmd
@@ -12,6 +12,10 @@ These lecture notes are released under the CC BY-NC 4.0 license, which can be fo
 
 ## Changelog {.unnumbered}
 
+#### Version 0.1.7 (09.07.2024)
+- finished chapter 12
+- added chapter 12
+
 #### Version 0.1.6 (03.07.2024)
 - finished chapter 11
 - started chapter 12
diff --git a/ionBasedQC.qmd b/ionBasedQC.qmd
index 8746a77396c59da4d8c757f6243f97d3c7193af2..0d1beabd23795645b86686280fd3dc6af00ae03c 100644
--- a/ionBasedQC.qmd
+++ b/ionBasedQC.qmd
@@ -1,34 +1,63 @@
 # Ion-based quantum computers
 
-So far we have looked at the principals of quantum mechanics and how to transfer these principals to out mathematical description of a quantum computer. While there are many different approaches on how to actually build a quantum computer, which are researched at the moment, we will only look at one approach. This approach is based on trapped ions. 
+So far we have looked at the principles of quantum mechanics and how to transfer these principles to our mathematical description of quantum computing. While there are many different approaches on how to actually build a quantum computer, which are researched at the moment, we will only look at one approach. This approach is based on trapped ions. 
 
 ## Electron in an atom
 
 We look at a single atom with a nucleus with a positive charge and a single electron with negative charge "orbiting" the nucleus. 
 
-The electromagnetic field generated by the nucleus is essentially a potential well for the electron, since the electron is drawn to the nucleus and the potential of the electron rises with bigger distance from the nucleus. We simplify a lot here and ignore e.g. the spin. 
+The electromagnetic field generated by the nucleus is essentially a potential well for the electron, since the electron is drawn to the nucleus and the potential of the electron rises with bigger distance from the nucleus. We simplify a lot here and ignore ,e.g., the spin. 
 
-We can solve the time-independent Schroedinger equation for this setup and by solving this, we will get the energy eigenstates of the Hamiltonian. These eigenstates are called *orbitals*.
+We can solve the time-independent Schroedinger equation for this setup and by solving this, we will get the wave functions that are the energy eigenstates of the Hamiltonian. These wave functions are called *orbitals*.
 
-We can use a single ion as a qubit, where we define one of the energy eigenstates as $\ket{0}$ and a different eigenstate as $\ket{1}$.
+We can use a single atom as a qubit, where we define one of the energy eigenstates as $\ket{0}$ and a different eigenstate as $\ket{1}$.
+
+In the following, we will specifically use electrically charged atoms, called ions, because they are easier to capture. 
 
 ## Setup for the ion traps
 
 The setup for our quantum computer looks as follows: 
 
-<!-- figure -->
+![Setup for the ion based quantum computer](ionQC){width=100%}
+
+In the previous chapter, we have learned that we need to be able to perform three different operations to build a quantum computer:
 
-In the previous chapter, we have learned, that we need to be able to perform three different operations to build a quantum computer:
-we need to be able to perform three basic operations:
 
 1. We need to initialize (cool) our qubit.
 2. We need to be able to apply a unitary on the qubit.
 3. We need to be able to measure the qubit.
 
-## Cooling
+### Cooling
+
+We first look into cooling our system. For cooling, we use a useful fact: If $E_i < E_j$ are different possible energy levels, an ion is in the energy level $E_i$ and then hit by a photon that has the energy $E_j-E_i$, the ion will go to energy level $E_j$.
+
+#### Doppler cooling
+
+In our initial setup, we have an ion vibrating back and forth because it has too much energy. We shine a laser on it with slightly less energy than what is need for a transition. The energy of the photon of this laser is denoted by $E=\hbar \cdot \omega$. When the ion moves towards the photon, the photon has a higher frequency from the point of view of the ion (Doppler effect). This means that the photon has a higher energy and therefore is more likely to be absorbed. 
+
+So by shining a laser on the ion, the photons of the laser "push" the ion, when it "swings" *towards* the laser similar to a pendulum, where the pendulum gets a pushback with just enough energy so it stops. This reduces the vibrations energy down to a certain level.
+
+#### Sideband cooling
+
+ Using the doppler cooling, we have reduced the vibration energy, but the electrons may still be excited. We now look at another technique called sideband cooling, which will set the energy of the electrons to a specific energy level $E_0$.
+
+ The electron can have any energy level $E_i$. If this energy level is pretty low, the possibility of a spontaneous emission of a photon, which would reduce the energy to a lower level is also quite low. So an electron with energy level $E_1$ or $E_2$ will probably not change to level $E_0$. If the energy level is big enough (we will call this energy level $E_{\text{big}}$), the probability of a spontaneous emission of a photon, which would reduce the energy to a lower level is quite high. So when this spontaneous emission happens, the electron will reach an energy level of ,e.g., $E_0$, $E_1$ or $E_2$. 
+
+ Our goal is to get the energy level to $E_0$. We know that the energy level of the electrons with $E_{\text{big}}$ will come down eventually, so we shine a laser with the energy per photon of $E_{\text{big}} - E_1, E_{\text{big}} - E_2, \dots$ but not with $E_{\text{big}} - E_0$. This will "shoot" all the low energy electrons from $E_1,E_2,\dots$ to a higher lever where they will either fall down to $E_1,E_2,\dots$ where they  will be energized again and the process is repeated, or they fall into $E_0$ which is our desired energy level.
+
+ Using the sideband and the doppler cooling together, we can cool the vibration and electron excitation. This means that all the ions are in the state $\ket{0}$ and the vibrations energy is also in the state $\ket{0}$.
+
+
+### Unitaries
+
+This section will be updated later.
+
+### Measurements
+
+We now look into performing a measurement on an ion-based quantum computer. So far we have defined that the quantum state $\ket{0}$ is represented by the energy level $E_0$ and the quantum state $\ket{1}$ is represented by the energy level $E_1$.
 
-We first look into cooling our system. For cooling, we use a useful fact: If an atom is in the energy level $E_i$ and then hit by a photon of energy level $E_j-E_i$, the atom will go to energy level $E_j$.
+To perform a measurement, we also use an auxiliary energy level $E_{\text{aux}} > E_0,E_1$. Let $\omega$ be the frequency of a photon with energy $E_\text{aux}-E_0 (E_\text{aux}-E_0=\Delta \omega)$
 
-### Doppler cooling
+We now shine a laser with the frequency $\omega$ on the ion. If the state is $\ket{0}$, the photon gets absorbed and the electron jumps to $E_\text{aux}$ and from there spontaneously back to $E_0$.This process then repeats over and over emitting many photons. This will create fluorescence, which can be measured by light detectors. If the state is $\ket{1}$, no ions get absorbed and no fluorescence can be seen. 
 
-In oue initial setup, we have an ion vibrating back and forth. We shine a laser on it with slightly less energy than what is need for a transition. 
\ No newline at end of file
+So all in all, we measure an ion by shining a photon of frequency $\omega$ onto it and then look whether it lights up.
diff --git a/ionQC.pdf b/ionQC.pdf
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diff --git a/ionQC.svg b/ionQC.svg
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diff --git a/physics.qmd b/physics.qmd
index 8577045ec0fea47d28937338871f4d463679c42b..b2799c414f73a232684412d1f4d6472ebbd150bc 100644
--- a/physics.qmd
+++ b/physics.qmd
@@ -19,8 +19,19 @@ $$
 
 From this we can see, that the integral $\int \lvert \psi_{t_0} \rvert ^2 dx = 1$. The momentum is not needed for the wave function.
 
+The inner product of two wave functions is given by 
+$$
+\braket{\psi|\phi} := \int \overline{\psi(x)} \cdot \phi(x) dx 
+$$
+
+The norm of a wave function is given by 
+$$
+\| \psi \|^2 :=\braket{\psi|\psi}  = \int \lvert \psi(x) \rvert ^2 dx
+$$
+
 In general, the wave function can have a different domain, e.g. $\psi_t: \mathbb{R}^3 \rightarrow \mathbb{C}$ for a particle in 3D-space. Everything below works analogously in that case. 
 
+
 ## Energy / Hamiltonian
 
 Given a particle in a state, this particle has some energy. Classically this energy is calculated from the potential and the momentum, with 
diff --git a/physicsToQC.qmd b/physicsToQC.qmd
index b1e9cda8a9be52a4a12a6ed0a44709035d461819..7c5fb3ba453da87ae3847643b4c15d748b9f65ae 100644
--- a/physicsToQC.qmd
+++ b/physicsToQC.qmd
@@ -14,7 +14,7 @@ $$
 \end{aligned}
 $$
 
-For one qubit, we just look at $\psi_0$ and $\psi_1$ and ignore all other wave functions. Note that this can lead to errors, since those other wave function still exists and interacts with our system, even though they might have a very small probability.
+For one qubit, we just look at $\psi_0$ and $\psi_1$ and ignore all other wave functions. Note that this can lead to errors, since those other wave functions still exists and interact with our system, even though they might have a very small probability.
 
 To fully construct our one qubit quantum computer, we need to be able to perform three basic operations:
 
@@ -53,7 +53,8 @@ E_1 & 0 & 0\\
 \end{pmatrix}
 $$
 
-For this representation of $H$, we immediately get
+For this representation of $H$, we immediately get $H \ket{0} = \frac{\pi^2}{2}$, $H \ket{1} =  2 \pi^2$ and so on, so nothing has changed except that $H$ is represented more nicely.
+
 We can now use a helpful theorem to get a solution for the differential equation.
 
 ::: {.callout-note appearance="minimal" icon=false}
@@ -80,7 +81,11 @@ We try this by changing the potential to $\delta V = \frac{9\pi^2}{16} (\frac{1}
 
 <!-- Figure? -->
 
-We rewrite $\delta V$ as a matrix:
+We rewrite $\delta V$ as a matrix. To do so, we try to find a matrix in the base $\ket{0}, \ket{1}$. If $\delta V\ket{0}=a\ket{0}+b\ket{1}$ and $\delta V\ket{1}=c\ket{0}+d\ket{1}$, then
+$$
+\delta V = \begin{pmatrix} a & c \\ b & d\end{pmatrix}
+$$
+Since $\ket{0}$ and $\ket{1}$ are orthonormal, we know that $\braket{0|\delta V|0} = \braket{0|a|0}  + \braket{0|b|1} = a + 0 = a$. We get $b,c$ and $d$ similar and from this the following matrix: 
 $$
 \delta V = \begin{pmatrix} \braket{0|\delta V|0} & \braket{0|\delta V|1}\\ \braket{1|\delta V|0} & \braket{1|\delta V|1}    \end{pmatrix}
 $$