From f7f33c747cdc27b8a07006df16ad082a09daeed1 Mon Sep 17 00:00:00 2001
From: Jannik Hellenkamp <jannik.hellenkamp@rwth-aachen.de>
Date: Sat, 1 Jun 2024 11:34:58 +0200
Subject: [PATCH] fixed rendering

---
 _book/Introduction-to-Quantum-Computing.epub  | Bin 0 -> 114974 bytes
 _book/Introduction-to-Quantum-Computing.pdf   | Bin 0 -> 139430 bytes
 .../bernsteinVazirani.html                    |   0
 .../compositeSystems.html                     |   0
 index.html => _book/index.html                |   4 +-
 introduction.html => _book/introduction.html  |   0
 .../observingSystems.html                     |   0
 .../partialObserving.html                     |   0
 .../probabilisticSystems.html                 |   2 +-
 .../quantumAlgorithms.html                    |   0
 .../quantumBasics.html                        |   0
 .../quantumCircutsKetNotation.html            |   0
 .../quantumSystems.html                       |   0
 _book/robots.txt                              |   1 +
 _book/search.json                             | 154 ++++
 _book/shor.svg                                |   1 +
 .../shorsAlgorithm.html                       |   6 +-
 .../site_libs}/bootstrap/bootstrap-icons.css  |   0
 .../site_libs}/bootstrap/bootstrap-icons.woff | Bin
 .../site_libs}/bootstrap/bootstrap.min.css    |   0
 .../site_libs}/bootstrap/bootstrap.min.js     |   0
 .../site_libs}/clipboard/clipboard.min.js     |   0
 .../site_libs}/quarto-html/anchor.min.js      |   0
 .../site_libs}/quarto-html/popper.min.js      |   0
 .../quarto-syntax-highlighting.css            |   0
 .../site_libs}/quarto-html/quarto.js          |   0
 .../site_libs}/quarto-html/tippy.css          |   0
 .../site_libs}/quarto-html/tippy.umd.min.js   |   0
 .../site_libs}/quarto-nav/headroom.min.js     |   0
 .../site_libs}/quarto-nav/quarto-nav.js       |   0
 .../quarto-search/autocomplete.umd.js         |   0
 .../site_libs}/quarto-search/fuse.min.js      |   0
 .../site_libs}/quarto-search/quarto-search.js |   0
 _book/sitemap.xml                             |  59 ++
 _quarto.yml                                   |   3 -
 index.log                                     | 615 --------------
 index.tex                                     | 763 ------------------
 shorsAlgorithm.qmd                            |   2 +-
 38 files changed, 222 insertions(+), 1388 deletions(-)
 create mode 100644 _book/Introduction-to-Quantum-Computing.epub
 create mode 100644 _book/Introduction-to-Quantum-Computing.pdf
 rename bernsteinVazirani.html => _book/bernsteinVazirani.html (100%)
 rename compositeSystems.html => _book/compositeSystems.html (100%)
 rename index.html => _book/index.html (99%)
 rename introduction.html => _book/introduction.html (100%)
 rename observingSystems.html => _book/observingSystems.html (100%)
 rename partialObserving.html => _book/partialObserving.html (100%)
 rename probabilisticSystems.html => _book/probabilisticSystems.html (98%)
 rename quantumAlgorithms.html => _book/quantumAlgorithms.html (100%)
 rename quantumBasics.html => _book/quantumBasics.html (100%)
 rename quantumCircutsKetNotation.html => _book/quantumCircutsKetNotation.html (100%)
 rename quantumSystems.html => _book/quantumSystems.html (100%)
 create mode 100644 _book/robots.txt
 create mode 100644 _book/search.json
 create mode 100644 _book/shor.svg
 rename shorsAlgorithm.html => _book/shorsAlgorithm.html (98%)
 rename {site_libs => _book/site_libs}/bootstrap/bootstrap-icons.css (100%)
 rename {site_libs => _book/site_libs}/bootstrap/bootstrap-icons.woff (100%)
 rename {site_libs => _book/site_libs}/bootstrap/bootstrap.min.css (100%)
 rename {site_libs => _book/site_libs}/bootstrap/bootstrap.min.js (100%)
 rename {site_libs => _book/site_libs}/clipboard/clipboard.min.js (100%)
 rename {site_libs => _book/site_libs}/quarto-html/anchor.min.js (100%)
 rename {site_libs => _book/site_libs}/quarto-html/popper.min.js (100%)
 rename {site_libs => _book/site_libs}/quarto-html/quarto-syntax-highlighting.css (100%)
 rename {site_libs => _book/site_libs}/quarto-html/quarto.js (100%)
 rename {site_libs => _book/site_libs}/quarto-html/tippy.css (100%)
 rename {site_libs => _book/site_libs}/quarto-html/tippy.umd.min.js (100%)
 rename {site_libs => _book/site_libs}/quarto-nav/headroom.min.js (100%)
 rename {site_libs => _book/site_libs}/quarto-nav/quarto-nav.js (100%)
 rename {site_libs => _book/site_libs}/quarto-search/autocomplete.umd.js (100%)
 rename {site_libs => _book/site_libs}/quarto-search/fuse.min.js (100%)
 rename {site_libs => _book/site_libs}/quarto-search/quarto-search.js (100%)
 create mode 100644 _book/sitemap.xml
 delete mode 100644 index.log
 delete mode 100644 index.tex

diff --git a/_book/Introduction-to-Quantum-Computing.epub b/_book/Introduction-to-Quantum-Computing.epub
new file mode 100644
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@@ -8,7 +8,7 @@
 
 <meta name="author" content="Jannik Hellenkamp">
 <meta name="author" content="Dominique Unruh">
-<meta name="dcterms.date" content="2024-05-31">
+<meta name="dcterms.date" content="2024-06-01">
 
 <title>Introduction to Quantum Computing</title>
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@@ -251,7 +251,7 @@ ul.task-list li input[type="checkbox"] {
     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">May 31, 2024</p>
+      <p class="date">June 1, 2024</p>
     </div>
   </div>
   
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@@ -371,7 +371,7 @@ d_{\text{2-bit}} = \begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3}
 </div>
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-<p>From <a href="#def-prob-distribution" class="quarto-xref">Definition&nbsp;<span class="quarto-unresolved-ref">def-prob-distribution</span></a> we know that a valid probability distribution <span class="math inline">\(a\)</span> has the properties <span class="math inline">\(\sum a_i = 1\)</span> and <span class="math inline">\(\forall i\)</span> <span class="math inline">\(a_i \geq 0\)</span>, therefore a matrix <span class="math inline">\(A\)</span> is a probabilistic process iff <span class="math inline">\(A \in \mathbb{R}^{n \times n}\)</span> with <span class="math inline">\(\sum a_i = 1\)</span> and <span class="math inline">\(\forall i\)</span> <span class="math inline">\(a_i \geq 0\)</span> . Such a matrix is also called a <em>stochastic matrix</em>.</p>
+<p>From <a href="#def-prob-distribution" class="quarto-xref">Definition&nbsp;<span>2.1</span></a> we know that a valid probability distribution <span class="math inline">\(a\)</span> has the properties <span class="math inline">\(\sum a_i = 1\)</span> and <span class="math inline">\(\forall i\)</span> <span class="math inline">\(a_i \geq 0\)</span>, therefore a matrix <span class="math inline">\(A\)</span> is a probabilistic process iff <span class="math inline">\(A \in \mathbb{R}^{n \times n}\)</span> with <span class="math inline">\(\sum a_i = 1\)</span> and <span class="math inline">\(\forall i\)</span> <span class="math inline">\(a_i \geq 0\)</span> . Such a matrix is also called a <em>stochastic matrix</em>.</p>
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+Sitemap: https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/sitemap.xml
diff --git a/_book/search.json b/_book/search.json
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+[
+  {
+    "objectID": "index.html",
+    "href": "index.html",
+    "title": "Introduction to Quantum Computing",
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+    "text": "Welcome\nThese are the lecture notes for the “Introduction to Quantum Computing” lecture held by Dominique Unruh at RWTH Aachen in the summer term 2024. The lecture notes are updated throughout the semester and should be viewed as an addition to the handwritten notes and the lecture recordings.\nIf you prefer a .pdf or .epub file, there is a download available at the top left corner. Please note, that these files are autogenerated and some of the ebook readers have difficulties with the formulas. We are still working on a universal solution. You can also find the source code in the top left corner.\nIf you spot an error, please send Jannik Hellenkamp an e-mail. You can contact Jannik by sending an e-mail to firstname.lastname@rwth-aachen.de (please replace first and lastname with Jannik’s full name). If you have a question of understanding, please ask it in the Moodle forum.\nThese lecture notes are released under the CC BY-NC 4.0 license, which can be found here.",
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+      "Welcome"
+    ]
+  },
+  {
+    "objectID": "index.html#changelog",
+    "href": "index.html#changelog",
+    "title": "Introduction to Quantum Computing",
+    "section": "Changelog",
+    "text": "Changelog\n\nVersion 0.1.2 (31.05.2024)\n\nminor changes to chapter 2\nadded chapter 9\n\n\n\nVersion 0.1.1 (16.05.2024)\n\nStarted the lecture notes.",
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+      "Welcome"
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+    "href": "introduction.html",
+    "title": "1  Introduction",
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+    "text": "1.1 Double slit experiment\nThis section will be updated later on, since there is quite a lot of graphical stuff.",
+    "crumbs": [
+      "Quantum Basics",
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+    "href": "introduction.html#what-is-a-quantum-computer",
+    "title": "1  Introduction",
+    "section": "1.2 What is a quantum computer?",
+    "text": "1.2 What is a quantum computer?\nTo start into the topic of quantum computing and to understand the differences from classical computers, we first need to look at some of the basics of such classical computers.\nIn a classical computer the information is stored in bits which can either be in the state \\(0\\) or the state \\(1\\). These bits can be manipulated through different classical operations and we can look at these bits and read them, without interfering with the system or changing any states.\nIn a quantum computer the information is stored in a qubit which can be in a superposition between the state \\(0\\) and \\(1\\). Just as with classical computers, we can construct variables from these qubits to store bigger numbers. For example a 64-qubit integer would be described by 64 qubits which are in a superposition between \\(0\\) and \\(2^{64}-1\\). This can be imagined best as a variable where the universe has not yet decided on its value and therefore the variable has all possible values at the same time.\nWe can now use this superposition and manipulate it with different quantum operations. Contrary to a classical computer, in a quantum computer these operations are “applied” at all possible input values at the same time and the result is a superposition of all possible results of the operation. We call this effect quantum parallelism.\n\n\n\n\n\n\nExample: Quantum parallelism\n\n\n\nLet’s say you have a quantum variable \\(x\\) in a superposition of numbers between \\(0\\) and \\(2^{64}-1\\) (all possible 64-bit values) and some function \\(f(x)\\). You program a quantum computer to compute \\(f(x)\\).\nThe quantum computer would compute \\(f(x)\\) for \\(x=0,x=1,x=2,...\\) at the the same time and the result of this computation is a superposition of all possible values \\(f(x)\\).\n\n\nReading this, one might be tempted to utilize quantum parallelism to run any algorithm on a quantum computer in order to optimize runtime. Unfortunately there is a big catch with quantum computers: If we try to look at the state of a qubit (also called measuring), the universe decides randomly on an outcome and therefore when measuring we only get the result of one computation and all the rest of the information is lost.\n\n\n\n\n\n\nExample (continued): Quantum parallelism\n\n\n\nAfter your quantum computer has calculated a superposition of all possible values \\(f(x)\\), you want to get some information on the output and therefore you do a measurement on the resulting quantum state.\nYou will receive one random \\(f(x)\\) and all the other possible solutions are lost.\n\n\nDue to this restriction, naively running established algorithms on a quantum computer will not work. Fortunately there are some clever tricks to create some “interference” between different computations before measuring. This will give us useful information in some cases.",
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+    "text": "2.1 Deterministic possibilities\nAt first we need to define all the different possible outcomes of our system. For example, for a coin flip this could be heads or tails and for a dice this could be the labels of the different sides. We call these possibilities deterministic possibilities. Note that we will only be using a finite number of possibilities.",
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+    "text": "Example: Random 2-bit number\n\n\n\nImagine you have a random number generator, which outputs 2-bit numbers. The deterministic possibilities of this generator are \\(00\\), \\(01\\), \\(10\\) and \\(11\\).",
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+    "text": "2.2 Probability distribution\nNext, we need to assign each possibility a probability. We write this as \\(\\Pr[x]=p\\) where \\(p \\in [0,1]\\) is the probability of the deterministic possibility \\(x\\).\n\n\n\n\n\n\nExample: Coin flip\n\n\n\nFor a coin flip the probability of heads would be \\(\\Pr[\\text{heads}] = \\frac{1}{2}\\) and the probability for tails would be \\(\\Pr[\\text{tails}] = \\frac{1}{2}\\).\n\n\nIf we combine all probabilities for all the possible outcomes and write them as a vector, we get a probability distribution.\n\n\n\n\n\n\n\nDefinition 2.1 (Probability distribution) A vector \\(d \\in \\mathbb{R}^n\\) is a valid probability distribution iff \\(\\sum d_i = 1\\) and \\(\\forall i\\) \\(d_i \\geq 0\\).\n\n\n\n\nThis vector has \\(n\\) entries, where each entry corresponds to a deterministic possibility \\(X\\) and the probability of \\(X\\) is \\(\\Pr[X] = d_i\\). The sum over all probabilities has to be \\(1\\) and each entry needs to be nonnegative in order to be a valid probability.\n\n\n\n\n\n\nExample (continued): Coin flip\n\n\n\nFor a coin flip the probability distribution would be \\(d_{\\text{coin}} \\in \\mathbb{R}^2\\) with \\(d = \\begin{pmatrix}\\frac{1}{2}\\\\ \\frac{1}{2} \\end{pmatrix}\\)\n\n\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nRecall your random 2-bit number generator from above. Imagine your generator outputs each deterministic possibility with equal probability, except for the possibility \\(00\\), which is never generated. The corresponding probability distribution would be \\[\nd_{\\text{2-bit}} = \\begin{pmatrix} 0 \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{1}{3} \\end{pmatrix}\n\\]",
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+    "text": "2.3 Probabilistic processes\nWith a probability distribution, we can only describe the probabilities of possibilities without any knowledge of a prior state. We therefore add another element to our toolbox of probabilistic systems called a probabilistic process.\nA probabilistic process is a collection of \\(n\\) probability distributions, where for each deterministic possibility \\(i\\) there is a probability distribution \\(a_i\\). This means, that if the system is in deterministic possibility \\(i\\) before the process is applied, the system will afterwards be distributed according to \\(a_i\\). We can write this as a matrix, where each column is a probability distribution \\(a_i\\).\n\n\n\n\n\n\n\nDefinition 2.2 (Probabilistic process) A matrix \\(A \\in \\mathbb{R}^{n\\times n}\\) is a valid probabilistic process iff for every column \\(a\\) of \\(A\\), \\(a\\) is a valid probability distribution.\n\n\n\n\nFrom Definition 2.1 we know that a valid probability distribution \\(a\\) has the properties \\(\\sum a_i = 1\\) and \\(\\forall i\\) \\(a_i \\geq 0\\), therefore a matrix \\(A\\) is a probabilistic process iff \\(A \\in \\mathbb{R}^{n \\times n}\\) with \\(\\sum a_i = 1\\) and \\(\\forall i\\) \\(a_i \\geq 0\\) . Such a matrix is also called a stochastic matrix.\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nImagine a second device, which receives a 2-bit number as an input and flips both bits at the same time with a probability of \\(\\frac{1}{3}\\). The probability distributions for each of the deterministic possibility would then be \\[\na_{00} = \\begin{pmatrix} \\frac{2}{3} \\\\ 0 \\\\ 0 \\\\ \\frac{1}{3} \\end{pmatrix}, a_{01} =\\begin{pmatrix} 0 \\\\ \\frac{2}{3} \\\\ \\frac{1}{3} \\\\ 0  \\end{pmatrix}, a_{10} =\\begin{pmatrix} 0 \\\\ \\frac{1}{3} \\\\ \\frac{2}{3} \\\\ 0  \\end{pmatrix}, a_{11} = \\begin{pmatrix} \\frac{1}{3} \\\\ 0 \\\\ 0 \\\\ \\frac{2}{3} \\end{pmatrix}\n\\] From this we can construct the process as a matrix from these processes as follows: \\[\nA_{\\text{flip}} = \\begin{pmatrix} a_{00} & a_{01} & a_{10} & a_{11} \\end{pmatrix} = \\begin{pmatrix} \\frac{2}{3} & 0 & 0 &  \\frac{1}{3} \\\\ 0 &  \\frac{2}{3} &  \\frac{1}{3} & 0 \\\\ 0 &  \\frac{1}{3} &  \\frac{2}{3} & 0 \\\\  \\frac{1}{3} & 0 & 0 &  \\frac{2}{3} \\end{pmatrix}\n\\]\n\n\n\nApplying a probabilistic process\nHaving defined probability distributions and probabilistic processes, we can now combine these two elements and apply a probabilistic process on a probability distribution.\n\n\n\n\n\n\n\nDefinition 2.3 (Applying a probabilistic process) Given an initial probability distribution \\(x \\in \\mathbb{R}^n\\) and a probabilistic process \\(A \\in \\mathbb{R}^{n\\times n}\\), the result \\(y \\in \\mathbb{R}^n\\) of applying the process \\(A\\) is defined as \\[\ny = Ax\n\\]\n\n\n\n\n\n\n\n\n\n\nExample (continued): Random 2-bit number\n\n\n\nRecall the 2-bit number generator and the bit flip from above. Imagine you would first draw a random 2-bit number from the generator and then run the bit flip device. We already know that the probability distribution of the generator is \\(d_\\text{2-bit}\\). Using \\(A_\\text{flip}\\) we can calculate the final probability distribution: \\[\nA_\\text{flip} \\cdot d_\\text{2-bit} = \\begin{pmatrix} \\frac{2}{3} & 0 & 0 &  \\frac{1}{3} \\\\ 0 &  \\frac{2}{3} &  \\frac{1}{3} & 0 \\\\ 0 &  \\frac{1}{3} &  \\frac{2}{3} & 0 \\\\  \\frac{1}{3} & 0 & 0 &  \\frac{2}{3} \\end{pmatrix}\\begin{pmatrix} 0 \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{1}{3} \\end{pmatrix} = \\begin{pmatrix} \\frac{1}{9} \\\\ \\frac{1}{3}\\\\ \\frac{1}{3} \\\\ \\frac{2}{9} \\end{pmatrix}\n\\]",
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+    "text": "9.1 Discrete Fourier Transformation\nOne of the tools required for Shor’s algorithm is the Discrete Fourier Transformation (DFT). Generally, a Fourier transformation is a mathematical technique that decomposes a function into its constituent frequencies. We use the DFT to find the period of a vector.\nThe DFT is defined as follows:\nThis transformation is best imagined as a process, which takes a periodic vector as an input and outputs the period of that vector. The DFT has some important properties, which help us later on.",
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+    "text": "Definition 9.1 (Discrete Fourier Transformation (DFT)) The discrete Fourier transform (DFT) is a linear transformation on \\(\\mathbb{C}^N\\) represented by the matrix \\[\n\\operatorname{DFT}_N = \\frac{1}{\\sqrt{N}} (\\omega^{kl})_{kl} \\in \\mathbb{C}^{N\\times N}\n\\] with \\(\\omega = e^{2i\\pi/N}\\), which is the \\(N\\)-th root of unity.\n\n\n\n\n\n\n\n\n\n\n\n\nTheorem 9.1 (Properties of the DFT) Here are some properties of the DFT which can be used without further proof.\n\nThe DFT is unitary.\n\\(\\omega^t = \\omega^{t\\mod N}\\) for all \\(t \\in \\mathbb{Z}\\).\nGiven a quantum state \\(\\psi \\in \\mathbb{C}^N\\) which is \\(r\\)-periodic and where \\(r\\mid N\\), \\(\\operatorname{DFT}_N \\psi\\) will compute a quantum state \\(\\phi \\in \\mathbb{C}^N\\), which has non-zero values on the multiples of \\(\\frac{N}{r}\\). Note that \\(\\frac{N}{r}\\) intuitively represents the frequency of \\(\\psi\\). This means, that \\[\n|\\phi_i| = \\begin{cases} \\frac{1}{\\sqrt{t}}, & \\text{if}\\ \\frac{N}{t}\\mid i \\\\ 0, & \\text{otherwise} \\end{cases}\n\\]",
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+    "section": "9.2 Reducing factoring to period finding",
+    "text": "9.2 Reducing factoring to period finding\nWith the DFT, we have seen, that we can use a unitary to find the period of a quantum state. We now look into using period finding to factor integers. We first look at the definition of the two problems:\n\n\n\n\n\n\n\nDefinition 9.2 (Factoring problem) Given integer \\(N\\) with two prime factors \\(p,q\\) such that \\(pq=N\\) and \\(p \\neq q\\), find \\(p\\) and \\(q\\).\n\n\n\n\nNote that this definition of the factoring problem is a simplified version of the factoring problem, where \\(N\\) has only 2 prime factors.\n\n\n\n\n\n\n\nDefinition 9.3 (Period finding problem) Given \\(f: \\mathbb{Z} \\to X\\) with \\(f(x) = f(y)\\) iff \\(x \\equiv y \\bmod r\\) for some fixed secret \\(r\\). \\(r\\) is called the period of \\(f\\). Find \\(r\\).\n\n\n\n\nTo start the reduction, we need a special case of the period finding problem called order finding:\n\n\n\n\n\n\n\nDefinition 9.4 (Order finding problem) For known \\(a\\) and \\(N\\) which are relatively prime, find the period \\(r\\) of \\(f(i) = a^i \\bmod n\\). We call \\(r\\) the order of \\(a\\) written \\(r = \\text{ ord } a\\). (This is similar to finding the smallest \\(i &gt; 0\\) with \\(f(i) = 1\\)).\n\n\n\n\nSince the order finding problem is just the period finding problem for a specific \\(f(x)\\), we know that if we can solve the period finding problem within reasonable runtime, we can also solve the order finding problem within reasonable runtime. We now reduce the factoring problem to the order finding problem:\nWe have a integer \\(N\\) as an input for the factoring problem.\n\nPick an \\(a \\in \\{1,...,N-1\\}\\) with \\(a\\) relatively prime to \\(n\\).\nCompute the order of \\(a\\), so that \\(r = \\text{ ord } a\\) (using the solver for the order finding problem).\nIf the order \\(r\\) is odd, we abort.\nCalculate \\(x:= a^{\\frac{r}{2}}+1 \\bmod N\\) and \\(y:= a^{\\frac{r}{2}}-1 \\bmod N\\).\nIf \\(\\gcd(x,N) \\in \\{1,N\\}\\), we abort.\nWe compute \\(p = \\gcd(x,N)\\) and \\(q = \\gcd(y,N)\\).\n\nThe output of the reduction are \\(p,q\\), such that \\(pq = N\\). This holds, since \\[\nxy = (a^{\\frac{r}{2}}+1) (a^{\\frac{r}{2}}-1) = a^r - 1 \\equiv 1-1 = 0 \\pmod N\n\\]\n\n\n\n\n\n\n\nTheorem 9.2 (Probability of an abort) If \\(N\\) has at least two different prime factors and \\(N\\) is odd, then the probability to abort is \\(\\leq \\frac{1}{2}\\).\n\n\n\n\nAll in all this reduction shows, that if we have an oracle which can solve the period finding problem within reasonable runtime, we can also solve the factoring problem within reasonable runtime (since all other operations are classically fast to compute).",
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+    "text": "9.3 The quantum algorithm for period finding\nWe now look into an quantum algorithm that solves the period finding problem within reasonable runtime. The quantum circuit for Shor’s algorithm requires a \\(f:\\{0,1\\}^n\\rightarrow\\{0,1\\}^m\\) which is \\(r\\)-periodic and is show in this figure:\n\n\n\nShor’s algorithm (quantum part)\n\n\nThe algorithm works as follows:\n\nWe start with a \\(\\ket{0}\\) entry on every wire.\nWe bring the top wire into the superposition over all entries. The quantum state is then \\(2^\\frac{-n}{2}\\sum_x \\ket{x} \\otimes \\ket{0^m}\\).\nWe apply \\(U_f\\), which is the unitary of \\(f:\\{0,1\\}^n\\rightarrow\\{0,1\\}^m\\). This calculates the superposition over all possible values \\(f(x)\\) on the bottom wire. The resulting quantum state is \\(\\frac{-n}{2}\\sum_x \\ket{x,f(x)}\\).\nTo understand the algorithm better, we measure the bottom wire at this point. This will give us one random value \\(f(x_0)\\) for some \\(x_0\\). The top wire will then contain a superposition over all values \\(x\\) where \\(f(x) = f(x_0)\\). Since \\(f\\) is know to be \\(r\\)-periodic, we know, that \\(f(x) = f(x_0)\\) iff \\(x \\equiv x_0 \\bmod r\\). This means, that on the resulting quantum state on the top wire is periodic and can be written as \\(\\frac{1}{\\sqrt{2^\\frac{n}{r}}} \\sum_{x\\equiv x_0 \\bmod r} \\ket{x} \\otimes \\ket{f(x_0)}\\).\nWe apply the Discrete Fourier Transform on the top wire. This will “analyze” the top wire for the period and output a vector with entries at multiples of \\(\\frac{2^n}{r}\\) as seen in Theorem 9.1. For simplicity we assume, that \\(r \\mid 2^n\\) holds.\nWe measure the top wire and get one random multiple of \\(\\frac{2^n}{r}\\), which we can denote as \\(a\\cdot\\frac{2^n}{r}\\)\n\nSince we get a multiple of \\(\\frac{2^n}{r}\\) on each run, we can simply run the algorithm multiple times to get different multiples and then compute \\(\\frac{2^n}{r}\\) by taking the gcd of those multiples. From that we compute \\(r\\).\nUnfortunately this only works because we assumed \\(r \\mid 2^n\\). Since this does usually not hold, we only get approximate multiples of \\(\\frac{2^n}{r}\\) (which is not even an integer) and thus post processing is a bit more complex.",
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+    "text": "9.4 Post processing\nSo far we have seen the DFT to analyze the period of a quantum state, we have seen a way to reduce the factoring problem to the period finding and we have seen a quantum algorithm for finding an approximate multiple of such a period of a function. We just need one final step to find \\(r\\). For this we start with a theorem:\n\n\n\n\n\n\n\nTheorem 9.3 If \\(\\{0,1\\}^n \\rightarrow \\{0,1\\}^n\\) is \\(r\\)-periodic with probability \\(\\Omega(1/\\log\\log r)\\) the following holds: \\[\n\\frac{-r}{2} \\leq rc\\bmod 2^n \\leq \\frac{r}{2}\n\\] where \\(c\\) is the output of the second measurement of the quantum circuit described in Section 9.3.\n\n\n\n\nWe assume that the theorem holds for our outcome of the second measurement (If that is not the case, our result will be wrong and we can just run the quantum algorithm again to get a different outcome):\nThen exists a \\(d\\) such that: \\[\n\\begin{aligned}\n&\\lvert rc - d2^n\\rvert \\leq \\frac{r}{2} \\\\\n\\Leftrightarrow&\\lvert \\frac{c}{2^n} - \\frac{d}{r}\\rvert \\leq \\frac{1}{2^{n+1}}\n\\end{aligned}\n\\] The fraction \\(\\frac{c}{2^n}\\) is known, so the goal is to find a fraction \\(\\frac{d}{r}\\) that is \\(\\frac{1}{2^{n+1}}\\) close to \\(\\frac{c}{2^n}\\).\nThe rest of postprocessing will be updated after the next lecture.",
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diff --git a/shorsAlgorithm.html b/_book/shorsAlgorithm.html
similarity index 98%
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index 0a4af32..3e16d3c 100644
--- a/shorsAlgorithm.html
+++ b/_book/shorsAlgorithm.html
@@ -398,7 +398,7 @@ xy = (a^{\frac{r}{2}}+1) (a^{\frac{r}{2}}-1) = a^r - 1 \equiv 1-1 = 0 \pmod N
 <li>We bring the top wire into the superposition over all entries. The quantum state is then <span class="math inline">\(2^\frac{-n}{2}\sum_x \ket{x} \otimes \ket{0^m}\)</span>.</li>
 <li>We apply <span class="math inline">\(U_f\)</span>, which is the unitary of <span class="math inline">\(f:\{0,1\}^n\rightarrow\{0,1\}^m\)</span>. This calculates the superposition over all possible values <span class="math inline">\(f(x)\)</span> on the bottom wire. The resulting quantum state is <span class="math inline">\(\frac{-n}{2}\sum_x \ket{x,f(x)}\)</span>.</li>
 <li>To understand the algorithm better, we measure the bottom wire at this point. This will give us one random value <span class="math inline">\(f(x_0)\)</span> for some <span class="math inline">\(x_0\)</span>. The top wire will then contain a superposition over all values <span class="math inline">\(x\)</span> where <span class="math inline">\(f(x) = f(x_0)\)</span>. Since <span class="math inline">\(f\)</span> is know to be <span class="math inline">\(r\)</span>-periodic, we know, that <span class="math inline">\(f(x) = f(x_0)\)</span> iff <span class="math inline">\(x \equiv x_0 \bmod r\)</span>. This means, that on the resulting quantum state on the top wire is periodic and can be written as <span class="math inline">\(\frac{1}{\sqrt{2^\frac{n}{r}}} \sum_{x\equiv x_0 \bmod r} \ket{x} \otimes \ket{f(x_0)}\)</span>.</li>
-<li>We apply the Discrete Fourier Transform on the top wire. This will “analyze” the top wire for the period and output a vector with entries at multiples of <span class="math inline">\(\frac{2^n}{r}\)</span> as seen in <a href="#thm-dft-properties" class="quarto-xref">Theorem&nbsp;<span class="quarto-unresolved-ref">thm-dft-properties</span></a>. For simplicity we assume, that <span class="math inline">\(r \mid 2^n\)</span> holds.</li>
+<li>We apply the Discrete Fourier Transform on the top wire. This will “analyze” the top wire for the period and output a vector with entries at multiples of <span class="math inline">\(\frac{2^n}{r}\)</span> as seen in <a href="#thm-dft-properties" class="quarto-xref">Theorem&nbsp;<span>9.1</span></a>. For simplicity we assume, that <span class="math inline">\(r \mid 2^n\)</span> holds.</li>
 <li>We measure the top wire and get one random multiple of <span class="math inline">\(\frac{2^n}{r}\)</span>, which we can denote as <span class="math inline">\(a\cdot\frac{2^n}{r}\)</span></li>
 </ol>
 <p>Since we get a multiple of <span class="math inline">\(\frac{2^n}{r}\)</span> on each run, we can simply run the algorithm multiple times to get different multiples and then compute <span class="math inline">\(\frac{2^n}{r}\)</span> by taking the gcd of those multiples. From that we compute <span class="math inline">\(r\)</span>.</p>
@@ -416,7 +416,7 @@ xy = (a^{\frac{r}{2}}+1) (a^{\frac{r}{2}}-1) = a^r - 1 \equiv 1-1 = 0 \pmod N
 <div id="thm-shor-post-process" class="therorem theorem">
 <p><span class="theorem-title"><strong>Theorem 9.3</strong></span> If <span class="math inline">\(\{0,1\}^n \rightarrow \{0,1\}^n\)</span> is <span class="math inline">\(r\)</span>-periodic with probability <span class="math inline">\(\Omega(1/\log\log r)\)</span> the following holds: <span class="math display">\[
 \frac{-r}{2} \leq rc\bmod 2^n \leq \frac{r}{2}
-\]</span> where <span class="math inline">\(c\)</span> is the output of the second measurement of the quantum circuit described in <a href="#sec-shor-algo" class="quarto-xref"><span class="quarto-unresolved-ref">sec-shor-algo</span></a>.</p>
+\]</span> where <span class="math inline">\(c\)</span> is the output of the second measurement of the quantum circuit described in <a href="#sec-shor-algo" class="quarto-xref"><span>Section 9.3</span></a>.</p>
 </div>
 </div>
 </div>
@@ -427,7 +427,7 @@ xy = (a^{\frac{r}{2}}+1) (a^{\frac{r}{2}}-1) = a^r - 1 \equiv 1-1 = 0 \pmod N
 &amp;\lvert rc - d2^n\rvert \leq \frac{r}{2} \\
 \Leftrightarrow&amp;\lvert \frac{c}{2^n} - \frac{d}{r}\rvert \leq \frac{1}{2^{n+1}}
 \end{aligned}
-\]</span> The fraction <span class="math inline">\(\frac{c}{2^n}\)</span> is know so the goal is to find a fraction <span class="math inline">\(\frac{d}{r}\)</span> that is <span class="math inline">\(\frac{1}{2^{n+1}}\)</span> close to <span class="math inline">\(\frac{c}{2^n}\)</span>.</p>
+\]</span> The fraction <span class="math inline">\(\frac{c}{2^n}\)</span> is known, so the goal is to find a fraction <span class="math inline">\(\frac{d}{r}\)</span> that is <span class="math inline">\(\frac{1}{2^{n+1}}\)</span> close to <span class="math inline">\(\frac{c}{2^n}\)</span>.</p>
 <p>The rest of postprocessing will be updated after the next lecture.</p>
 
 
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diff --git a/_book/sitemap.xml b/_book/sitemap.xml
new file mode 100644
index 0000000..b28a67e
--- /dev/null
+++ b/_book/sitemap.xml
@@ -0,0 +1,59 @@
+<?xml version="1.0" encoding="UTF-8"?>
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+  </url>
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+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/quantumBasics.html</loc>
+    <lastmod>2024-05-06T10:45:40.810Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/introduction.html</loc>
+    <lastmod>2024-05-16T20:35:45.192Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/probabilisticSystems.html</loc>
+    <lastmod>2024-05-28T14:06:17.846Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/quantumSystems.html</loc>
+    <lastmod>2024-05-31T17:19:26.978Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/observingSystems.html</loc>
+    <lastmod>2024-05-31T17:18:58.860Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/partialObserving.html</loc>
+    <lastmod>2024-05-28T15:34:11.850Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/compositeSystems.html</loc>
+    <lastmod>2024-05-28T15:40:53.028Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/quantumCircutsKetNotation.html</loc>
+    <lastmod>2024-05-28T15:41:23.573Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/quantumAlgorithms.html</loc>
+    <lastmod>2024-05-28T15:28:23.655Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/bernsteinVazirani.html</loc>
+    <lastmod>2024-05-28T15:32:36.617Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/shorsAlgorithm.html</loc>
+    <lastmod>2024-06-01T09:32:38.929Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/Introduction-to-Quantum-Computing.pdf</loc>
+    <lastmod>2024-06-01T09:33:36.308Z</lastmod>
+  </url>
+  <url>
+    <loc>https://qis.rwth-aachen.de/teaching/24ss/intro-quantum-computing/script/Introduction-to-Quantum-Computing.epub</loc>
+    <lastmod>2024-06-01T09:33:37.814Z</lastmod>
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-  pdftitle={Introduction to Quantum Computing},
-  pdfauthor={Jannik Hellenkamp; Dominique Unruh},
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-\title{Introduction to Quantum Computing}
-\usepackage{etoolbox}
-\makeatletter
-\providecommand{\subtitle}[1]{% add subtitle to \maketitle
-  \apptocmd{\@title}{\par {\large #1 \par}}{}{}
-}
-\makeatother
-\subtitle{Lecture notes for the summer term 2024}
-\author{Jannik Hellenkamp \and Dominique Unruh}
-\date{2024-05-31}
-
-\begin{document}
-\maketitle
-
-\renewcommand*\contentsname{Table of contents}
-{
-\hypersetup{linkcolor=}
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-\tableofcontents
-}
-\bookmarksetup{startatroot}
-
-\chapter*{Welcome}\label{welcome}
-\addcontentsline{toc}{chapter}{Welcome}
-
-\markboth{Welcome}{Welcome}
-
-These are the lecture notes for the ``Introduction to Quantum
-Computing'' lecture held by Dominique Unruh at RWTH Aachen in the summer
-term 2024. The lecture notes are updated throughout the semester and
-should be viewed as an addition to the handwritten notes and the lecture
-recordings.
-
-If you spot an error, please send Jannik Hellenkamp an e-mail. You can
-contact Jannik by sending an e-mail to firstname.lastname@rwth-aachen.de
-(please replace first and lastname with Jannik's full name). If you have
-a question of understanding, please ask it in the Moodle forum.
-
-These lecture notes are released under the CC BY-NC 4.0 license, which
-can be found
-\href{https://creativecommons.org/licenses/by-nc/4.0/}{here}.
-
-\section*{Changelog}\label{changelog}
-\addcontentsline{toc}{section}{Changelog}
-
-\markright{Changelog}
-
-\subsubsection*{Version 0.1.2
-(31.05.2024)}\label{version-0.1.2-31.05.2024}
-\addcontentsline{toc}{subsubsection}{Version 0.1.2 (31.05.2024)}
-
-\begin{itemize}
-\tightlist
-\item
-  minor changes to chapter 2
-\item
-  added chapter 9
-\end{itemize}
-
-\subsubsection*{Version 0.1.1
-(16.05.2024)}\label{version-0.1.1-16.05.2024}
-\addcontentsline{toc}{subsubsection}{Version 0.1.1 (16.05.2024)}
-
-\begin{itemize}
-\tightlist
-\item
-  Started the lecture notes.
-\end{itemize}
-
-\part{Quantum Basics}
-
-\chapter{Introduction}\label{introduction}
-
-\section{Double slit experiment}\label{double-slit-experiment}
-
-This section will be updated later on, since there is quite a lot of
-graphical stuff.
-
-\section{What is a quantum computer?}\label{what-is-a-quantum-computer}
-
-To start into the topic of quantum computing and to understand the
-differences from classical computers, we first need to look at some of
-the basics of such classical computers.
-
-In a classical computer the information is stored in \emph{bits} which
-can either be in the state \(0\) \emph{or} the state \(1\). These bits
-can be manipulated through different classical operations and we can
-look at these bits and read them, without interfering with the system or
-changing any states.
-
-In a quantum computer the information is stored in a \emph{qubit} which
-can be in a superposition \emph{between} the state \(0\) and \(1\). Just
-as with classical computers, we can construct variables from these
-qubits to store bigger numbers. For example a 64-\emph{qu}bit integer
-would be described by 64 qubits which are in a superposition between
-\(0\) and \(2^{64}-1\). This can be imagined best as a variable where
-the universe has not yet decided on its value and therefore the variable
-has all possible values at the same time.
-
-We can now use this superposition and manipulate it with different
-quantum operations. Contrary to a classical computer, in a quantum
-computer these operations are ``applied'' at all possible input values
-at the same time and the result is a superposition of all possible
-results of the operation. We call this effect \emph{quantum
-parallelism}.
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example: Quantum parallelism}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-Let's say you have a quantum variable \(x\) in a superposition of
-numbers between \(0\) and \(2^{64}-1\) (all possible 64-bit values) and
-some function \(f(x)\). You program a quantum computer to compute
-\(f(x)\).
-
-The quantum computer would compute \(f(x)\) for \(x=0,x=1,x=2,...\) at
-the the same time and the result of this computation is a superposition
-of all possible values \(f(x)\).
-
-\end{tcolorbox}
-
-Reading this, one might be tempted to utilize quantum parallelism to run
-any algorithm on a quantum computer in order to optimize runtime.
-Unfortunately there is a big catch with quantum computers: If we try to
-look at the state of a qubit (also called \emph{measuring}), the
-universe decides randomly on an outcome and therefore when measuring we
-only get the result of one computation and all the rest of the
-information is lost.
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example (continued): Quantum parallelism}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-After your quantum computer has calculated a superposition of all
-possible values \(f(x)\), you want to get some information on the output
-and therefore you do a measurement on the resulting quantum state.
-
-You will receive one random \(f(x)\) and all the other possible
-solutions are lost.
-
-\end{tcolorbox}
-
-Due to this restriction, naively running established algorithms on a
-quantum computer will not work. Fortunately there are some clever tricks
-to create some ``interference'' between different computations before
-measuring. This will give us useful information in some cases.
-
-\chapter{Probabilistic systems}\label{sec-prob}
-
-To describe a quantum computer mathematically, we can do math similar to
-the known topic of probabilistic systems. We therefore first look into
-describing a probabilistic system.
-
-\section{Deterministic possibilities}\label{deterministic-possibilities}
-
-At first we need to define all the different possible outcomes of our
-system. For example, for a coin flip this could be \emph{heads} or
-\emph{tails} and for a dice this could be the labels of the different
-sides. We call these possibilities \emph{deterministic possibilities}.
-Note that we will only be using a \emph{finite} number of possibilities.
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example: Random 2-bit number}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-Imagine you have a random number generator, which outputs 2-bit numbers.
-The deterministic possibilities of this generator are \(00\), \(01\),
-\(10\) and \(11\).
-
-\end{tcolorbox}
-
-\section{Probability distribution}\label{probability-distribution}
-
-Next, we need to assign each possibility a probability. We write this as
-\(\Pr[x]=p\) where \(p \in [0,1]\) is the probability of the
-deterministic possibility \(x\).
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example: Coin flip}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-For a coin flip the probability of heads would be
-\(\Pr[\text{heads}] = \frac{1}{2}\) and the probability for tails would
-be \(\Pr[\text{tails}] = \frac{1}{2}\).
-
-\end{tcolorbox}
-
-If we combine all probabilities for all the possible outcomes and write
-them as a vector, we get a \emph{probability distribution}.
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{definition}[Probability
-distribution]\protect\hypertarget{def-prob-distribution}{}\label{def-prob-distribution}
-
-A vector \(d \in \mathbb{R}^n\) is a valid probability distribution iff
-\(\sum d_i = 1\) and \(\forall i\) \(d_i \geq 0\).
-
-\end{definition}
-
-\end{tcolorbox}
-
-This vector has \(n\) entries, where each entry corresponds to a
-deterministic possibility \(X\) and the probability of \(X\) is
-\(\Pr[X] = d_i\). The sum over all probabilities has to be \(1\) and
-each entry needs to be nonnegative in order to be a valid probability.
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example (continued): Coin flip}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-For a coin flip the probability distribution would be
-\(d_{\text{coin}} \in \mathbb{R}^2\) with
-\(d = \begin{pmatrix}\frac{1}{2}\\ \frac{1}{2} \end{pmatrix}\)
-
-\end{tcolorbox}
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example (continued): Random 2-bit number}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-Recall your random 2-bit number generator from above. Imagine your
-generator outputs each deterministic possibility with equal probability,
-except for the possibility \(00\), which is never generated. The
-corresponding probability distribution would be \[
-d_{\text{2-bit}} = \begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3} \end{pmatrix}
-\]
-
-\end{tcolorbox}
-
-\section{Probabilistic processes}\label{sec-prob-apply}
-
-With a probability distribution, we can only describe the probabilities
-of possibilities without any knowledge of a prior state. We therefore
-add another element to our toolbox of probabilistic systems called a
-\emph{probabilistic process}.
-
-A probabilistic process is a collection of \(n\) probability
-distributions, where for each deterministic possibility \(i\) there is a
-probability distribution \(a_i\). This means, that if the system is in
-deterministic possibility \(i\) before the process is applied, the
-system will afterwards be distributed according to \(a_i\). We can write
-this as a matrix, where each column is a probability distribution
-\(a_i\).
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{definition}[Probabilistic
-process]\protect\hypertarget{def-prob-process}{}\label{def-prob-process}
-
-A matrix \(A \in \mathbb{R}^{n\times n}\) is a valid probabilistic
-process iff for every column \(a\) of \(A\), \(a\) is a valid
-probability distribution.
-
-\end{definition}
-
-\end{tcolorbox}
-
-From Definition~\ref{def-prob-distribution} we know that a valid
-probability distribution \(a\) has the properties \(\sum a_i = 1\) and
-\(\forall i\) \(a_i \geq 0\), therefore a matrix \(A\) is a
-probabilistic process iff \(A \in \mathbb{R}^{n \times n}\) with
-\(\sum a_i = 1\) and \(\forall i\) \(a_i \geq 0\) . Such a matrix is
-also called a \emph{stochastic matrix}.
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example (continued): Random 2-bit number}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-Imagine a second device, which receives a 2-bit number as an input and
-flips both bits at the same time with a probability of \(\frac{1}{3}\).
-The probability distributions for each of the deterministic possibility
-would then be \[
-a_{00} = \begin{pmatrix} \frac{2}{3} \\ 0 \\ 0 \\ \frac{1}{3} \end{pmatrix}, a_{01} =\begin{pmatrix} 0 \\ \frac{2}{3} \\ \frac{1}{3} \\ 0  \end{pmatrix}, a_{10} =\begin{pmatrix} 0 \\ \frac{1}{3} \\ \frac{2}{3} \\ 0  \end{pmatrix}, a_{11} = \begin{pmatrix} \frac{1}{3} \\ 0 \\ 0 \\ \frac{2}{3} \end{pmatrix}
-\] From this we can construct the process as a matrix from these
-processes as follows: \[
-A_{\text{flip}} = \begin{pmatrix} a_{00} & a_{01} & a_{10} & a_{11} \end{pmatrix} = \begin{pmatrix} \frac{2}{3} & 0 & 0 &  \frac{1}{3} \\ 0 &  \frac{2}{3} &  \frac{1}{3} & 0 \\ 0 &  \frac{1}{3} &  \frac{2}{3} & 0 \\  \frac{1}{3} & 0 & 0 &  \frac{2}{3} \end{pmatrix}
-\]
-
-\end{tcolorbox}
-
-\subsection*{Applying a probabilistic
-process}\label{applying-a-probabilistic-process}
-\addcontentsline{toc}{subsection}{Applying a probabilistic process}
-
-Having defined probability distributions and probabilistic processes, we
-can now combine these two elements and apply a probabilistic process on
-a probability distribution.
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{definition}[Applying a probabilistic
-process]\protect\hypertarget{def-prob-apply}{}\label{def-prob-apply}
-
-Given an initial probability distribution \(x \in \mathbb{R}^n\) and a
-probabilistic process \(A \in \mathbb{R}^{n\times n}\), the result
-\(y \in \mathbb{R}^n\) of applying the process \(A\) is defined as \[
-y = Ax 
-\]
-
-\end{definition}
-
-\end{tcolorbox}
-
-\begin{tcolorbox}[enhanced jigsaw, leftrule=.75mm, title={Example (continued): Random 2-bit number}, opacityback=0, arc=.35mm, bottomtitle=1mm, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, left=2mm, titlerule=0mm, coltitle=black, rightrule=.15mm, breakable, bottomrule=.15mm, colback=white, colframe=quarto-callout-tip-color-frame, colbacktitle=quarto-callout-tip-color!10!white]
-
-Recall the 2-bit number generator and the bit flip from above. Imagine
-you would first draw a random 2-bit number from the generator and then
-run the bit flip device. We already know that the probability
-distribution of the generator is \(d_\text{2-bit}\). Using
-\(A_\text{flip}\) we can calculate the final probability distribution:
-\[
-A_\text{flip} \cdot d_\text{2-bit} = \begin{pmatrix} \frac{2}{3} & 0 & 0 &  \frac{1}{3} \\ 0 &  \frac{2}{3} &  \frac{1}{3} & 0 \\ 0 &  \frac{1}{3} &  \frac{2}{3} & 0 \\  \frac{1}{3} & 0 & 0 &  \frac{2}{3} \end{pmatrix}\begin{pmatrix} 0 \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{1}{3} \end{pmatrix} = \begin{pmatrix} \frac{1}{9} \\ \frac{1}{3}\\ \frac{1}{3} \\ \frac{2}{9} \end{pmatrix}
-\]
-
-\end{tcolorbox}
-
-\chapter{Quantum systems}\label{quantum-systems}
-
-\chapter{Observing probabilistic and measuring quantum
-systems}\label{observing-probabilistic-and-measuring-quantum-systems}
-
-\chapter{Partial observing and measuring
-systems}\label{partial-observing-and-measuring-systems}
-
-\chapter{Composite Systems}\label{composite-systems}
-
-\chapter{Quantum Circuits}\label{quantum-circuits}
-
-\chapter{Ket Notation}\label{ket-notation}
-
-\part{Quantum Algorithms}
-
-\chapter{Bernstein-Vazirani
-Algorithm}\label{bernstein-vazirani-algorithm}
-
-\chapter{Shor's Algorithm}\label{shors-algorithm}
-
-One of the best known quantum algorithm is Shor's algorithm for finding
-the prime factors of an integer. It was developed by Peter Shor in 1994.
-
-\section{Discrete Fourier
-Transformation}\label{discrete-fourier-transformation}
-
-One of the tools required for Shor's algorithm is the Discrete Fourier
-Transformation (DFT). Generally, a Fourier transformation is a
-mathematical technique that decomposes a function into its constituent
-frequencies. We use the DFT to find the period of a vector.
-
-The DFT is defined as follows:
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{definition}[Discrete Fourier Transformation
-(DFT)]\protect\hypertarget{def-shor-dft}{}\label{def-shor-dft}
-
-The discrete Fourier transform (DFT) is a linear transformation on
-\(\mathbb{C}^N\) represented by the matrix \[
-\operatorname{DFT}_N = \frac{1}{\sqrt{N}} (\omega^{kl})_{kl} \in \mathbb{C}^{N\times N}
-\] with \(\omega = e^{2i\pi/N}\), which is the \(N\)-th root of unity.
-
-\end{definition}
-
-\end{tcolorbox}
-
-This transformation is best imagined as a process, which takes a
-periodic vector as an input and outputs the period of that vector. The
-DFT has some important properties, which help us later on.
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{theorem}[Properties of the
-DFT]\protect\hypertarget{thm-dft-properties}{}\label{thm-dft-properties}
-
-Here are some properties of the DFT which can be used without further
-proof.
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  The DFT is unitary.
-\item
-  \(\omega^t = \omega^{t\mod N}\) for all \(t \in \mathbb{Z}\).
-\item
-  Given a quantum state \(\psi \in \mathbb{C}^N\) which is
-  \(r\)-periodic and where \(r\mid N\), \(\operatorname{DFT}_N \psi\)
-  will compute a quantum state \(\phi \in \mathbb{C}^N\), which has
-  non-zero values on the multiples of \(\frac{N}{r}\). Note that
-  \(\frac{N}{r}\) intuitively represents the frequency of \(\psi\). This
-  means, that \[
-  |\phi_i| = \begin{cases} \frac{1}{\sqrt{t}}, & \text{if}\ \frac{N}{t}\mid i \\ 0, & \text{otherwise} \end{cases}
-  \]
-\end{enumerate}
-
-\end{theorem}
-
-\end{tcolorbox}
-
-\section{Reducing factoring to period
-finding}\label{reducing-factoring-to-period-finding}
-
-With the DFT, we have seen, that we can use a unitary to find the period
-of a quantum state. We now look into using period finding to factor
-integers. We first look at the definition of the two problems:
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{definition}[Factoring
-problem]\protect\hypertarget{def-shor-factoring}{}\label{def-shor-factoring}
-
-Given integer \(N\) with two prime factors \(p,q\) such that \(pq=N\)
-and \(p \neq q\), find \(p\) and \(q\).
-
-\end{definition}
-
-\end{tcolorbox}
-
-Note that this definition of the factoring problem is a simplified
-version of the factoring problem, where \(N\) has only 2 prime factors.
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{definition}[Period finding
-problem]\protect\hypertarget{def-shor-period}{}\label{def-shor-period}
-
-Given \(f: \mathbb{Z} \to X\) with \(f(x) = f(y)\) iff
-\(x \equiv y \bmod r\) for some fixed secret \(r\). \(r\) is called the
-\emph{period} of \(f\). Find \(r\).
-
-\end{definition}
-
-\end{tcolorbox}
-
-To start the reduction, we need a special case of the period finding
-problem called order finding:
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{definition}[Order finding
-problem]\protect\hypertarget{def-shor-order}{}\label{def-shor-order}
-
-For known \(a\) and \(N\) which are relatively prime, find the period
-\(r\) of \(f(i) = a^i \bmod n\). We call \(r\) the order of \(a\)
-written \(r = \text{ ord } a\). (This is similar to finding the smallest
-\(i > 0\) with \(f(i) = 1\)).
-
-\end{definition}
-
-\end{tcolorbox}
-
-Since the order finding problem is just the period finding problem for a
-specific \(f(x)\), we know that if we can solve the period finding
-problem within reasonable runtime, we can also solve the order finding
-problem within reasonable runtime. We now reduce the factoring problem
-to the order finding problem:
-
-We have a integer \(N\) as an input for the factoring problem.
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  Pick an \(a \in \{1,...,N-1\}\) with \(a\) relatively prime to \(n\).
-\item
-  Compute the order of \(a\), so that \(r = \text{ ord } a\) (using the
-  solver for the order finding problem).
-\item
-  If the order \(r\) is odd, we abort.
-\item
-  Calculate \(x:= a^{\frac{r}{2}}+1 \bmod N\) and
-  \(y:= a^{\frac{r}{2}}-1 \bmod N\).
-\item
-  If \(\gcd(x,N) \in \{1,N\}\), we abort.
-\item
-  We compute \(p = \gcd(x,N)\) and \(q = \gcd(y,N)\).
-\end{enumerate}
-
-The output of the reduction are \(p,q\), such that \(pq = N\). This
-holds, since \[
-xy = (a^{\frac{r}{2}}+1) (a^{\frac{r}{2}}-1) = a^r - 1 \equiv 1-1 = 0 \pmod N 
-\]
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{theorem}[Probability of an
-abort]\protect\hypertarget{thm-shor-abort}{}\label{thm-shor-abort}
-
-If \(N\) has at least two different prime factors and \(N\) is odd, then
-the probability to abort is \(\leq \frac{1}{2}\).
-
-\end{theorem}
-
-\end{tcolorbox}
-
-All in all this reduction shows, that if we have an oracle which can
-solve the period finding problem within reasonable runtime, we can also
-solve the factoring problem within reasonable runtime (since all other
-operations are classically fast to compute).
-
-\section{The quantum algorithm for period finding}\label{sec-shor-algo}
-
-We now look into an quantum algorithm that solves the period finding
-problem within reasonable runtime. The quantum circuit for Shor's
-algorithm requires a \(f:\{0,1\}^n\rightarrow\{0,1\}^m\) which is
-\(r\)-periodic and is show in this figure:
-
-\begin{figure}[H]
-
-{\centering \includegraphics[width=1\textwidth,height=\textheight]{shor.pdf}
-
-}
-
-\caption{Shor's algorithm (quantum part)}
-
-\end{figure}%
-
-The algorithm works as follows:
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  We start with a \(\ket{0}\) entry on every wire.
-\item
-  We bring the top wire into the superposition over all entries. The
-  quantum state is then
-  \(2^\frac{-n}{2}\sum_x \ket{x} \otimes \ket{0^m}\).
-\item
-  We apply \(U_f\), which is the unitary of
-  \(f:\{0,1\}^n\rightarrow\{0,1\}^m\). This calculates the superposition
-  over all possible values \(f(x)\) on the bottom wire. The resulting
-  quantum state is \(\frac{-n}{2}\sum_x \ket{x,f(x)}\).
-\item
-  To understand the algorithm better, we measure the bottom wire at this
-  point. This will give us one random value \(f(x_0)\) for some \(x_0\).
-  The top wire will then contain a superposition over all values \(x\)
-  where \(f(x) = f(x_0)\). Since \(f\) is know to be \(r\)-periodic, we
-  know, that \(f(x) = f(x_0)\) iff \(x \equiv x_0 \bmod r\). This means,
-  that on the resulting quantum state on the top wire is periodic and
-  can be written as
-  \(\frac{1}{\sqrt{2^\frac{n}{r}}} \sum_{x\equiv x_0 \bmod r} \ket{x} \otimes \ket{f(x_0)}\).
-\item
-  We apply the Discrete Fourier Transform on the top wire. This will
-  ``analyze'' the top wire for the period and output a vector with
-  entries at multiples of \(\frac{2^n}{r}\) as seen in
-  Theorem~\ref{thm-dft-properties}. For simplicity we assume, that
-  \(r \mid 2^n\) holds.
-\item
-  We measure the top wire and get one random multiple of
-  \(\frac{2^n}{r}\), which we can denote as \(a\cdot\frac{2^n}{r}\)
-\end{enumerate}
-
-Since we get a multiple of \(\frac{2^n}{r}\) on each run, we can simply
-run the algorithm multiple times to get different multiples and then
-compute \(\frac{2^n}{r}\) by taking the gcd of those multiples. From
-that we compute \(r\).
-
-Unfortunately this only works because we assumed \(r \mid 2^n\). Since
-this does usually not hold, we only get approximate multiples of
-\(\frac{2^n}{r}\) (which is not even an integer) and thus post
-processing is a bit more complex.
-
-\section{Post processing}\label{post-processing}
-
-So far we have seen the DFT to analyze the period of a quantum state, we
-have seen a way to reduce the factoring problem to the period finding
-and we have seen a quantum algorithm for finding an approximate multiple
-of such a period of a function. We just need one final step to find
-\(r\). For this we start with a theorem:
-
-\begin{tcolorbox}[enhanced jigsaw, toprule=.15mm, left=2mm, leftrule=.75mm, opacityback=0, arc=.35mm, breakable, rightrule=.15mm, bottomrule=.15mm, colframe=quarto-callout-note-color-frame, colback=white]
-
-\begin{theorem}[]\protect\hypertarget{thm-shor-post-process}{}\label{thm-shor-post-process}
-
-If \(\{0,1\}^n \rightarrow \{0,1\}^n\) is \(r\)-periodic with
-probability \(\Omega(1/\log\log r)\) the following holds: \[
-\frac{-r}{2} \leq rc\bmod 2^n \leq \frac{r}{2}
-\] where \(c\) is the output of the second measurement of the quantum
-circuit described in Section~\ref{sec-shor-algo}.
-
-\end{theorem}
-
-\end{tcolorbox}
-
-We assume that the theorem holds for our outcome of the second
-measurement (If that is not the case, our result will be wrong and we
-can just run the quantum algorithm again to get a different outcome):
-
-Then exists a \(d\) such that: \[
-\begin{aligned}
-&\lvert rc - d2^n\rvert \leq \frac{r}{2} \\
-\Leftrightarrow&\lvert \frac{c}{2^n} - \frac{d}{r}\rvert \leq \frac{1}{2^{n+1}}
-\end{aligned}
-\] The fraction \(\frac{c}{2^n}\) is know so the goal is to find a
-fraction \(\frac{d}{r}\) that is \(\frac{1}{2^{n+1}}\) close to
-\(\frac{c}{2^n}\).
-
-The rest of postprocessing will be updated after the next lecture.
-
-
-
-\end{document}
diff --git a/shorsAlgorithm.qmd b/shorsAlgorithm.qmd
index ad5a432..a1cbe2d 100644
--- a/shorsAlgorithm.qmd
+++ b/shorsAlgorithm.qmd
@@ -148,7 +148,7 @@ $$
 \Leftrightarrow&\lvert \frac{c}{2^n} - \frac{d}{r}\rvert \leq \frac{1}{2^{n+1}}
 \end{aligned}
 $$
-The fraction $\frac{c}{2^n}$ is know so the goal is to find a fraction $\frac{d}{r}$ that is $\frac{1}{2^{n+1}}$ close to $\frac{c}{2^n}$. 
+The fraction $\frac{c}{2^n}$ is known, so the goal is to find a fraction $\frac{d}{r}$ that is $\frac{1}{2^{n+1}}$ close to $\frac{c}{2^n}$. 
 
 The rest of postprocessing will be updated after the next lecture. 
 
-- 
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