Change cel slides submodule url

.gitmodules

@@ -1,3 +1,6 @@
 [submodule "lib/cel-slides-template-2025"]
 	path = lib/cel-slides-template-2025
-	url = git@gitlab.kit.edu:kit/cel/misc/cel-slides-template-2025.git
+	url = ssh://git@100.123.176.93:2222/an.tsouchlos/cel-slides-template-2025.git
+[submodule "lib/cel-thesis"]
+	path = lib/cel-thesis
+	url = ssh://git@100.123.176.93:2222/an.tsouchlos/cel-thesis.git

.tmux_session.sh

@@ -0,0 +1,9 @@
+#!/bin/bash
+SESSION=$1
+
+tmux send-keys -t "$SESSION:1" "cd ~/workspace/private/ma-thesis/" Enter
+tmux send-keys -t "$SESSION:1" "./.setup_local_env.sh" Enter
+# tmux send-keys -t "$SESSION:1" "export TEXINPUTS=./lib/cel-slides-template-2025:\$TEXINPUTS" C-m
+tmux send-keys -t "$SESSION:1" "trap './.clean_local_env.sh' EXIT" Enter
+tmux send-keys -t "$SESSION:1" "nvim" Enter
+tmux send-keys -t "$SESSION:1" "\\ll" Enter

Makefile

@@ -4,7 +4,8 @@ DOCUMENTS := $(patsubst src/%/main.tex,build/%.pdf,$(wildcard src/*/main.tex))
 all: $(DOCUMENTS)
 
 build/%.pdf: src/%/main.tex build/prepared
-	latexmk $<
+	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
+	latexmk -outdir=build/$* $<
 	mv build/main.pdf $@
 
 build/prepared:

src/fundamentals/main.tex

@@ -64,8 +64,7 @@
 \Ac{qec} is a field of research combining quantum mechanics and
 ``classical'' communications engineering.
 This chapter provides the relevant theoretical background on both of
-these topics and subsequently, building on top of this, introduces the
+these topics and subsequently introduces the the fundamentals of \ac{qec}.
-the fundamentals of \ac{qec}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % TODO: Is Quantum Information Theory the correct title here? Would someth

src/midterm_presentation/main.tex

@@ -53,7 +53,7 @@
 
 \title{Fault Tolerant Quantum Error Correction}
 \subtitle{Master's Thesis Midterm Presentation}
-\author[Andreas]{Andreas Tsouchlos}
+\author[ Andreas]{Andreas Tsouchlos}
 \date[]{}
 
 \DeclareFieldFormat{note}{}
@@ -148,7 +148,7 @@
 
 \DeclareAcronym{qldpc}{
     short=QLDPC,
-    long=quantum low - density parity - check,
+    long=quantum low-density parity-check,
 }
 
 \DeclareAcronym{scldpc}{
@@ -156,6 +156,11 @@
     long=spatially-coupled low-density parity-check
 }
 
+\DeclareAcronym{ler}{
+    short=LER,
+    long=logical error rate,
+}
+
 %
 %
 % Document body
@@ -401,7 +406,7 @@
                 \citereferencemanual{NC10}
             }
             \visible<3>{
-            \item Superposition $\rightarrow$ multiple solutions to the
+            \item Superposition $\rightarrow$ Multiple solutions to the
                 decoding problem
                 (\schlagwort{quantum degeneracy})
             \citereferencemanual{RWB$^+$20}}
@@ -520,7 +525,7 @@
         \begin{subfigure}{0.15\textwidth}
             \centering
             \begin{align*}
-                \bm{H} \bm{y}^\text{T} = \bm{s}
+                \bm{H} \bm{y}^\text{T} = \bm{H} \bm{e}^\text{T} = \bm{s}
             \end{align*}
             \vspace*{-5mm}
         \end{subfigure}
@@ -1568,6 +1573,7 @@
                     \citereferencemanual{HP23}
                     \citereferencemanual{GCR24}
             \end{itemize}
+            \visible<2>{
             \item Degraded \ac{bp} performance addressed with
                 modification or extension
                 \begin{itemize}
@@ -1580,6 +1586,7 @@
                     \item Ensemble decoding
                         \citereferencemanual{KSW$^+$25}
                 \end{itemize}
+            }
     \end{itemize}
 
     \vspace*{15mm}
@@ -1737,7 +1744,7 @@
 \begin{frame}
     \frametitle{Future Work}
 
-    \vspace*{-12mm}
+    \vspace*{-15mm}
 
     \begin{itemize}
         \item Completed work
@@ -1745,6 +1752,7 @@
                 \item Review literature on fault-tolerant \ac{qec}
                     using \acp{dem}
                 \item Identify research gap
+                \item Familiarize with software toolboxes
             \end{itemize}
             \vspace*{7mm}
         \item Research gap
@@ -1787,32 +1795,166 @@
 
 % TODO: Organize sections properly
 %%%%%%%%%%%%%%%%
-\section{Simulation Methodology}
+\section{Remarks on Evaluation}
-\label{sec:Simulation Methodology}
+\label{sec:Remarks on Evaluation}
 
 %%%%%%%%%%%%%%%%
-\subsection{The Code and Other Parameters}
+\subsection{Figures of Merit}
-\label{subsec:The Code and Other Parameters}
+\label{subsec:Figures of Merit}
 
 \begin{frame}
-    \frametitle{Noise Models and Figures of Merit}
+    \frametitle{Performance Evaluation}
 
+    % - Gong et al. don't actually analyze the latency ->
+    %   Benchmarking against other methods would be interesting
+
+    % \item For circuit-level noise, use same
+    % \schlagwort{physical error rate} for all error
+    % locations \citereferencemanual{FSG09}
+
+    \vspace*{-15mm}
+
+    \begin{minipage}{0.35\textwidth}
+        \only<1>{
             \begin{itemize}
-        \item \red{For circuit-level noise, often, all error probabilities
+                \item Independent variables
-                are set to the same value for simulations
+                    \begin{itemize}
-            \citereferencemanual{FSG09}}
+                        \item Physical error rate
-        \item \red{There are other approaches (e.g., SDMB noise, SI noise)
+                        \item CNOT infidelity
-            \citereferencemanual{DTB$^+$25}}
+                        \item Total qubit count
+                        \item \ldots
+                    \end{itemize}
+            \end{itemize}
+        }
+        \only<2->{
+            \begin{itemize}
+                \item Independent variables
+                    \begin{itemize}
+                        \item \textbf{Physical error rate}
+                        \item CNOT infidelity
+                        \item \textbf{Total qubit count}
+                        \item \ldots
+                    \end{itemize}
+            \end{itemize}
+        }
+    \end{minipage}%
+    \begin{minipage}{0.65\textwidth}
+        \begin{itemize}
+                \only<1>{
+                \item Noise models
+                    \begin{itemize}
+                        \item Standard circuit-based depolarizing noise
+                            \citereferencemanual{FSG09}
+                        \item Superconductor inspired (SI1000)
+                            \citereferencemanual{GNF$^+$21}
+                        \item Entangling Measurements (EM3)
+                            \citereferencemanual{GNF$^+$21}
+                        \item \ldots
+                    \end{itemize}
+                }
+                \only<2->{
+                \item Noise models
+                    \begin{itemize}
+                        \item \textbf{Standard circuit-based depolarizing noise}
+                            \citereferencemanual{FSG09}
+                        \item Superconductor inspired (SI1000)
+                            \citereferencemanual{GNF$^+$21}
+                        \item Entangling Measurements (EM3)
+                            \citereferencemanual{GNF$^+$21}
+                        \item \ldots
+                    \end{itemize}
+                }
+        \end{itemize}
+    \end{minipage}
+
+    \vspace{5mm}
+
+    \visible<3->{
+        \begin{itemize}
+            \item Degeneracy, information stored in correlations
+                $\rightarrow$ Consider \schlagwort{\acl{ler}} (LER)
+        \end{itemize}
+    }
+    \visible<4->{
+        \begin{itemize}
+            \item Types of benchmarking plots
         \end{itemize}
 
-    \vspace*{10mm}
+        \vspace*{5mm}
 
-    \begin{itemize}
+        \begin{figure}[H]
-        \item \red{Footprint plots}
+            \centering
-        \item \red{Other figure of merit (Look into ECCentric?)}
+            \begin{subfigure}{0.35\textwidth}
-    \end{itemize}
+                \centering
+                \begin{tikzpicture}
+                    \begin{axis}[
+                            domain=-5:5,
+                            width=7cm,
+                            height=5.5cm,
+                            xticklabels=\empty,
+                            yticklabels=\empty,
+                            xlabel={Physical error rate},
+                            xlabel style={yshift=5mm},
+                            ylabel={LER},
+                            ylabel style={yshift=-5mm},
+                            grid,
+                        ]
 
-    \vspace*{15mm}
+                        \addplot+[
+                            mark=none,
+                            kit-red,
+                            line width=2pt,
+                        ]
+                        table[row sep=crcr] {
+                            x y \\
+                            1.134800559068837 0.5575221183357257 \\
+                            2.0632737437615223 0.9764009116710485 \\
+                            2.861072612292603 1.7787608707489788 \\
+                            3.7551580964997053 2.8407080379684153 \\
+                            4.264098875196703 3.513274267363004 \\
+                            4.573589936760932 3.9911505302955272 \\
+                            4.903713970055305 4.268436552233389 \\
+                        };
+                    \end{axis}
+                \end{tikzpicture}
+            \end{subfigure}%
+            \begin{subfigure}{0.35\textwidth}
+                \centering
+                \begin{tikzpicture}
+                    \begin{axis}[
+                            domain=-5:5,
+                            width=7cm,
+                            height=5.5cm,
+                            xticklabels=\empty,
+                            yticklabels=\empty,
+                            xlabel={Total qubit count},
+                            xlabel style={yshift=5mm},
+                            ylabel={LER},
+                            ylabel style={yshift=-5mm},
+                            grid,
+                        ]
+
+                        \addplot+[
+                            mark=none,
+                            kit-blue,
+                            line width=2pt,
+                        ]
+                        table[row sep=crcr] {
+                            x y \\
+                            1.147643096789246 3.8430493581808607 \\
+                            1.7245658892318043 2.762331811591747 \\
+                            2.3573205843145306 2.3587443650766753 \\
+                            2.9156332708646624 1.560537992857378 \\
+                            3.6352360073136527 1.0403588210329737 \\
+                            4.392060012189421 0.7130042787942606 \\
+                        };
+                    \end{axis}
+                \end{tikzpicture}
+            \end{subfigure}
+        \end{figure}
+    }
+
+    \vspace*{4mm}
 
     \addreferencesmanual
     {FSG09}{
@@ -1820,30 +1962,44 @@
         ``High-threshold universal quantum computation on the surface
         code,'' \emph{Physical Review}, 2009.
     }
-    {DTB$^+$25}{
+    {GNF$^+$21}{
-        P.- J. H. S. Derks et al., ``Designing fault-tolerant
+        C. Gidney et al., ``A Fault-Tolerant Honeycomb Memory'',
-        circuits using detector error models,'' \emph{Quantum}, 2025.
+        \emph{Quantum}, 2021.
     }
     \stopreferencesmanual
 \end{frame}
 
 %%%%%%%%%%%%%%%%
-\subsection{Proposed Methodology}
+\subsection{Conclusion and Outlook}
-\label{subsec:Proposed Methodology}
+\label{subsec:Conclusion and Outlook}
 
 \begin{frame}
-    \frametitle{Conclusion}
+    \frametitle{Conclusion and Outlook}
 
-    \vspace*{-15mm}
+    \vspace*{-10mm}
 
     \begin{minipage}[c]{0.65\textwidth}
         \begin{itemize}
-            \item \red{Noise model}
+            \item Problem setting
-            \item \red{Memory or stability experiment}
+                \begin{itemize}
-            \item \red{Figure of merit: Footprint plot}
+                    \item Research area: Decoder design for \acp{dem}
-            \item \red{Comparison with BB code also simulated by
+                        under circuit-level noise
-                \citereferencemanual{GCR24}}
+                    \item Research gap: Consideration of \acp{dem} as
-            \item \red{Comparison with surface code}
+                        \ac{scldpc} codes
+                \end{itemize}
+                \vspace*{5mm}
+            \item Future work
+                \begin{itemize}
+                    \item Modify existing decoder to pass soft information
+                    \item Test different \ac{bp} variations
+                    \item \ldots
+                \end{itemize}
+                \vspace*{5mm}
+            \item Parameters
+                \begin{itemize}
+                    \item Use standard depolarizing noise for comparability
+                    \item Compare performance with other \ac{bb} code decoders
+                \end{itemize}
         \end{itemize}
     \end{minipage}%
     \begin{minipage}[c]{0.35\textwidth}
@@ -1852,9 +2008,13 @@
         \begin{figure}[H]
             \centering
 
-            \begin{tikzpicture}[every node/.style={scale=10}]
+            \vspace*{-25mm}
-                \node at (0, 0)
+            \begin{tikzpicture}
+                \node[scale=10] at (0, 0)
                 {\textcolor{kit-blue}{{\fontfamily{phv}\selectfont ?}}};
+
+                \node[align=center] at (0,-5) {Thank you for your
+                attention! \\ Any questions?};
             \end{tikzpicture}
         \end{figure}
     \end{minipage}
@@ -1868,7 +2028,7 @@
 \begin{frame}
     \frametitle{System Level Overview}
 
-    \vspace*{-15mm}
+    \vspace*{-10mm}
 
     \begin{figure}[H]
         \centering
@@ -1933,7 +2093,7 @@
         \end{subfigure}
     \end{figure}
 
-    % \vspace*{-2mm}
+    \vspace*{5mm}
 
     \addreferencesmanual
     {ZZC$^+$23}{
@@ -1952,11 +2112,30 @@
 \begin{frame}
     \frametitle{Guided Decimation Guessing Decoding}
 
+    \begin{minipage}{0.57\textwidth}
         \begin{itemize}
-        \item \red{Explain paper}
+            \item BP guided decimation (BPGD) \\
+                $\rightarrow$ Iteratively fix most reliable variable node (VN)
+                \vspace*{10mm}
+            \item \schlagwort{Guided decimation guessing} (GDG)
+                \citereferencemanual{GCR24}
+                \begin{itemize}
+                    \item Choose VN with \schlagwort{lowest}
+                        log-likelihood ratio
+                    \item Choose VN to fix based on \schlagwort{LLR history}
+                    \item Explore both VN values in parallel
+                        (\schlagwort{guessing})
                 \end{itemize}
+        \end{itemize}
+    \end{minipage}%
+    \begin{minipage}{0.43\textwidth}
+        \begin{figure}[H]
+            \centering
+            \includegraphics[scale=1.3]{res/gdg.pdf}
+        \end{figure}
+    \end{minipage}%
 
-    \vspace*{25mm}
+    \vspace*{30mm}
 
     \addreferencesmanual
     {GCR24}{
@@ -1967,6 +2146,52 @@
     \stopreferencesmanual
 \end{frame}
 
+\begin{frame}
+    \frametitle{The Quantum Error Correcting Landscape}
+
+    \vspace*{-10mm}
+
+    \begin{itemize}
+        \item Taxonomy of main QEC code families \citereferencemanual{SPG$^+$25}
+    \end{itemize}
+
+    \vspace*{2mm}
+
+    \begin{figure}[H]
+        \centering
+        \includegraphics[scale=2.5]{res/taxonomy.pdf}
+    \end{figure}
+
+    \begin{itemize}
+        \item Surface code is the industry standard for
+            experimental implementations, but has poor encoding
+            efficiency \citereferencemanual{BCG$^+$24}
+        \item \Ac{qldpc} codes particularly interesting because of
+            \begin{itemize}
+                \item Constant overhead scaling \citereferencemanual{Got14}
+                \item Linear distance scaling \citereferencemanual{BCG$^+$24}
+            \end{itemize}
+    \end{itemize}
+
+    \vspace*{15mm}
+
+    \addreferencesmanual
+    {SPG$^+$25}{
+        A. Swierkowska et al., ``ECCentric: An Empirical
+        Analysis of Quantum Error Correction Codes'',
+        \emph{arXiv:2511.01062v1}, 2025.
+    }
+    {BCG$^+$24}{
+        S. Bravyi et al., ``High-threshold and low-overhead
+        fault-tolerant quantum memory,'' \emph{Nature}, 2024.
+    }
+    {Got14}{
+        D. Gottesman, ``Fault-Tolerant Quantum Computation with
+        Constant Overhead'', \emph{arXiv:1310.2984}, 2014.
+    }
+    \stopreferencesendmanual
+\end{frame}
+
 % TODO: Is this really necessary?
 % \begin{frame}
 %     \frametitle{The Quantum Error Correction Landscape}

src/thesis/acronyms.tex

@@ -0,0 +1,29 @@
+\DeclareAcronym{qec}{
+    short=QEC,
+    long=quantum error correction
+}
+
+\DeclareAcronym{bp}{
+    short=BP,
+    long=belief propagation
+}
+
+\DeclareAcronym{sc}{
+    short=SC,
+    long=spatially coupled
+}
+
+\DeclareAcronym{ldpc}{
+    short=LDPC,
+    long=low-density parity-check
+}
+
+\DeclareAcronym{ml}{
+    short=ML,
+    long=maximum likelihood
+}
+
+\DeclareAcronym{pcm}{
+    short=PCM,
+    long=parity-check matrix
+}

src/thesis/chapters/1_introduction.tex

@@ -0,0 +1 @@
+\chapter{Introduction}

src/thesis/chapters/2_fundamentals.tex

@@ -0,0 +1,370 @@
+\chapter{Fundamentals}
+\label{ch:Fundamentals}
+
+\Ac{qec} is a field of research combining ``classical''
+communications engineering and quantum information science.
+This chapter provides the relevant theoretical background on both of
+these topics and subsequently introduces the the fundamentals of \ac{qec}.
+
+% TODO: Is an explanation of BP with guided decimation needed in this chapter?
+% TODO: Is an explanation of OSD needed chapter?
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Classical Error Correction}
+\label{sec:Classical Error Correction}
+
+% TODO: Maybe rephrase: The core concept is not the realization, its's the
+% thing itself
+The core concept underpinning error correcting codes is the
+realization that a finite amount of redundancy, introduced
+deliberately and systematically to information before its
+tranmission, can be utilized to reduce the error rate of a
+communications system considerably.
+This idea has been expanded upon significantly since it was first
+brought forward by Claude Shannon in 1948 \cite{shannon_mathematical_1948}.
+
+In this section, we explore the concepts of ``classical'' (as in non-quantum)
+error correction that are central to this work.
+We start by looking at different ways of encoding information,
+first considering binary linear block codes in general and then \ac{ldpc} and
+\ac{sc}-\ac{ldpc} codes.
+Finally, we pivot to the decoding process, specifically the \ac{bp}
+algorithm.
+
+% TODO: Use subsubsections?
+\subsection{Binary Linear Block Codes}
+
+%
+% Codewords, n, k, rate
+%
+
+% TODO: Do I need a specific reference for the expanded Hilbert space thing?
+One particularly important class of coding schemes is that of binary
+linear block codes.
+The information to be protected takes the form of a sequence of of
+binary symbols, which is split into separate blocks.
+Each block is encoded, transmitted, and decoded separately.
+The encoding step introduces redundancy by mapping input messages
+$\bm{u} \in \mathbb{F}_2^k$ of length $k \in \mathbb{N}$ (called the
+\textit{information length}) onto \textit{codewords} $\bm{x} \in
+\mathbb{F}_2^n$ of length $n \in \mathbb{N}$ (called the
+\textit{block length}) with $n > k$.
+A measure of the amount of introduced redundancy is the \textit{code
+rate} $R = k/n$.
+We call the set of all codewords $\mathcal{C}$ the \textit{code}
+\cite[Sec. 3.1]{ryan_channel_2009}.
+
+%
+% d_min and the [] Notation
+%
+
+During the encoding process, a mapping from $\mathbb{F}_2^k$
+onto $\mathcal{C} \subset \mathbb{F}_2^n$ takes place.
+The input messages are mapped onto an expanded vector space, where
+they are ``further appart'', giving rise to the error correcting
+properties of the code.
+This notion of the distance between two codewords $\bm{x}_1$ and
+$\bm{x}_2$ can be expressed using the \textit{Hamming distance} $d(\bm{x}_1,
+\bm{x}_2)$, which is defined as the number of positions in which they differ.
+We define the \textit{minimum distance} of a code $\mathcal{C}$ as
+\begin{align*}
+    d_\text{min} = \min \left\{ d(\bm{x}_1, \bm{x}_2) : \bm{x}_1,
+    \bm{x}_2 \in \mathcal{C}, \bm{x}_1 \neq \bm{x}_2 \right\}
+    .
+\end{align*}
+We can signify that a binary linear block code has information length
+$k$, block length $n$ and minimum distance $d_\text{min}$ using the
+notation $[n,k,d_\text{dmin}]$ \cite[Sec. 1.3]{macwilliams_theory_1977}.
+
+%
+% Parity checks, H, and the syndrome
+%
+
+A particularly elegant way of describing the subspace $C$ of
+$\mathbb{F}_2^n$ that the codewords make up is the notion of
+\textit{parity checks}.
+Since $\lvert \mathcal{C} \rvert = 2^k$ and $\lvert \mathbb{F}_2^n
+\rvert = 2^n$, we could introduce $n-k$ conditions to constrain the
+additional degrees of freedom.
+These conditions, called parity checks, take the form of equations
+over $\mathbb{F}_2^n$, linking the individual positions of each codeword.
+We can arrange the coefficients of these equations in the
+\textit{parity-check matrix} (\acs{pcm}) $\bm{H} \in
+\mathbb{F}_2^{(n-k) \times n}$ and equivalently define the code as
+\cite[Sec. 3.1]{ryan_channel_2009}
+\begin{align*}
+    \mathcal{C} = \left\{ \bm{x} \in \mathbb{F}_2^n :
+    \bm{H}\bm{x}^\text{T} = \bm{0} \right\}
+    .%
+\end{align*}
+The \textit{syndrome} $\bm{s} = \bm{H} \bm{v}^\text{T}$ describes
+which parity checks a candidate codeword $\bm{v} \in \mathbb{F}_2^n$ violates.
+The representation using the \ac{pcm} has the benefit of providing a
+description of the code, the memory complexity of which doesn't grow
+exponentially with $n$, in contrast to keeping track of all codewords directly.
+
+%
+% The decoding problem
+%
+
+Figure \ref{fig:Diagram of a transmission system} visualizes the
+entire communication process \cite[Sec. 1.1]{ryan_channel_2009}.
+An input message $\bm{u}\in \mathbb{F}_2^k$ is mapped onto a codeword $\bm{x}
+\in \mathbb{F}_2^n$. This is passed on to a modulator, which
+interacts with the physical channel.
+A demodulator processes the received message and forwards the result
+$\bm{y} \in \mathbb{R}^n$ to a decoder.
+Finally, the decoder is responsible for obtaining an estimate
+$\hat{\bm{u}} \in \mathbb{F}_2^k$ of the original input message from the
+received message.
+This is done by first finding an estimate $\hat{\bm{x}}$ of the sent
+codeword and undoing the encoding.
+The decoding problem that we generally attempt to solve thus consists
+in finding the best estimate $\hat{\bm{x}}$ given $\bm{y}$.
+One approach is to use the \ac{ml} criterion \cite[Sec.
+1.4]{ryan_channel_2009}
+\begin{align*}
+    \hat{\bm{u}}_\text{ML} = \arg\max_{\bm{x} \in \mathcal{C}}
+    P(\bm{Y} = \bm{y} \vert \bm{X} = \bm{x})
+    .
+\end{align*}
+Finally, we differentiate between \textit{soft decision} decoding, where
+$\bm{y} \in \mathbb{R}^n$, and \textit{hard decision} decoding, where
+$\bm{y} \in \mathbb{F}_2^n$ \cite[Sec. 1.5.1.3]{ryan_channel_2009}.
+%
+\begin{figure}[h]
+    \centering
+
+    \tikzset{
+        box/.style={
+            rectangle, draw=black, minimum width=17mm, minimum height=8mm,
+        },
+    }
+
+    \begin{tikzpicture}
+        [
+            node distance = 2mm and 7mm,
+        ]
+        \node                                              (in)  {};
+        \node[box, right=of in]                            (enc) {Encoder};
+        \node[box, minimum width=25mm, right=of enc]       (mod) {Modulator};
+        \node[box, below right=of mod]                     (cha) {Channel};
+        \node[box, minimum width=25mm, below left=of cha]  (dem) {Demodulator};
+        \node[box, left=of dem]                            (dec) {Decoder};
+        \node[left=of dec]                                 (out) {};
+
+        \draw[-{latex}] (in)  -- (enc) node[midway, above] {$\bm{u}$};
+        \draw[-{latex}] (enc) -- (mod) node[midway, above] {$\bm{x}$};
+        \draw[-{latex}] (mod) -| (cha);
+        \draw[-{latex}] (cha) |- (dem);
+        \draw[-{latex}] (dem) -- (dec) node[midway, above] {$\bm{y}$};
+        \draw[-{latex}] (dec) -- (out) node[midway, above] {$\hat{\bm{u}}$};
+    \end{tikzpicture}
+
+    \caption{Overview of a transmission system.}
+    \label{fig:Diagram of a transmission system}
+\end{figure}
+%
+
+%
+% Hard vs. soft information
+%
+
+\subsection{Low-Density Parity-Check Codes}
+
+\red{
+    \begin{itemize}
+        \item Use \cite[Ch. 5]{ryan_channel_2009} as a reference
+        \item Core concept (Large $n$ with manageable complexity)
+        \item Tanner graphs, VNs and CNs
+    \end{itemize}
+}
+
+\subsection{Spatially-Coupled LDPC Codes}
+
+\red{
+    \begin{itemize}
+        \item Core idea
+        \item Mathematical description (H)
+    \end{itemize}
+}
+
+\subsection{Belief Propagation}
+
+\red{
+    \begin{itemize}
+        \item Core idea
+        \item BP for SC-LDPC codes
+    \end{itemize}
+}
+
+\section{Quantum Mechanics and Quantum Information Science}
+\label{sec:Quantum Mechanics and Quantum Information Science}
+
+% TODO: Should the brief intro to QC be made later on or here?
+%%%%%%%%%%%%%%%%
+\subsection{Core Concepts and Notation}
+\label{subsec:Notation}
+
+\ldots can be very elegantly expressed using the language of
+linear algebra.
+\todo{Mention that we model the state of a quantum mechanical system
+as a vector}
+The so called Bra-ket or Dirac notation is especially appropriate,
+having been proposed by Paul Dirac in 1939 for the express purpose
+of simplifying quantum mechanical notation \cite{dirac_new_1939}.
+Two new symbols are defined, \emph{bra}s $\bra{\cdot}$ and
+\emph{ket}s $\ket{\cdot}$.
+Kets denote ordinary vectors, while bras denote their Hermitian conjugates.
+For example, two vectors specified by the labels $a$ and $b$
+respectively are written as $\ket{a}$ and $\ket{b}$.
+Their inner product is $\braket{a\vert b}$.
+
+\red{\textbf{Tensor product}}
+\red{\ldots
+    \todo{Introduce determinate state or use a different word?}
+    Take for example two systems with the determinate states $\ket{0}$
+    and $\ket{1}$. In general, the state of each can be written as the
+    superposition%
+    %
+    \begin{align*}
+        \alpha \ket{0} + \beta \ket{1}
+        .%
+    \end{align*}
+    %
+    Combining these two sytems into one, the overall state becomes%
+    %
+    \begin{align*}
+        &\mleft( \alpha_1 \ket{0} + \beta_1 \ket{1} \mright) \otimes
+        \mleft( \alpha_2 \ket{0} + \beta_2 \ket{1} \mright) \\
+        = &\alpha_1 \alpha_2 \ket{0} \ket{0}
+        + \alpha_1 \alpha_2 \ket{0} \ket{1}
+        + \beta_1 \alpha_2 \ket{1} \ket{0}
+        + \beta_1 \beta_2 \ket{1} \ket{1}
+        % =: &\alpha_{00} \ket{00}
+        % + \alpha_{01} \ket{01}
+        % + \alpha_{10} \ket{10}
+        % + \alpha_{11} \ket{11}
+        .%
+    \end{align*}%
+    %
+    \ldots When not ambiguous in the context, the tensor product
+    symbol may be omitted, e.g.,
+    \begin{align*}
+        \ket{0} \otimes \ket{0} = \ket{0}\ket{0}
+        .%
+    \end{align*}
+}
+
+As we will see, the core concept that gives quantum computing its
+power is entanglement. When two quantum mechanical systems are
+entangled, measuring the state of one will collapse that of the other.
+Take for example two subsystems with the overall state
+%
+\begin{align*}
+    \ket{\psi} = \frac{1}{\sqrt{2}} \mleft( \ket{0}\ket{0} +
+    \ket{1}\ket{1} \mright)
+    .%
+\end{align*}
+%
+If we measure the first subsystem as being in $\ket{0}$, we can
+be certain that a measurement of the second subsystem will also yield $\ket{0}$.
+Introducing a new notation for entangled states, we can write%
+%
+\begin{align*}
+    \ket{\psi} = \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right)
+    .%
+\end{align*}
+%
+
+\subsection{Projective Measurements}
+\label{subsec:Projective Measurements}
+
+% TODO: Write
+
+%%%%%%%%%%%%%%%%
+\subsection{Quantum Gates}
+\label{subsec:Quantum Gates}
+
+\red{
+    \textbf{Content:}
+    \begin{itemize}
+        \item Bra-ket notation
+        \item The tensor product
+        \item Projective measurements (the related operators,
+            eigenvalues/eigenspaces, etc.)
+            \begin{itemize}
+                \item First explain what an operator is
+            \end{itemize}
+        \item Abstract intro to QC: Use gates to process qubit
+            states, similar to classical case
+        \item X, Z, Y operators/gates
+        \item Hadamard gate (+ X and Z are the same thing in differt bases)
+        \item Notation of operators on multi-qubit states
+        \item The Pauli, Clifford and Magic groups
+    \end{itemize}
+}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Quantum Error Correction}
+\label{sec:Quantum Error Correction}
+
+\red{
+    \textbf{Content:}
+    \begin{itemize}
+        \item  General context
+            \begin{itemize}
+                \item Why we want QC
+                \item Why we need QEC (correcting errors due to noisy gates)
+                \item Main challenges of QEC compared to classical
+                    error correction
+            \end{itemize}
+        \item Stabilizer codes
+            \begin{itemize}
+                \item Definition of a stabilizer code
+                \item The stabilizer its generators (note somewhere
+                        that the generators have to commute to be able to
+                    be measured without disturbing each other)
+                \item syndrome extraction circuit
+                \item Stabilizer codes are effectively the QM
+                    % TODO: Actually binary linear codes or just linear codes?
+                    equivalent of binary linear codes (e.g.,
+                    expressible via check matrix)
+            \end{itemize}
+        \item Digitization of errors
+        \item CSS codes
+        \item Color codes?
+        \item Surface codes?
+        \item Fault tolerant error correction (gates with which we do
+            error correction are also noisy)
+            \begin{itemize}
+                \item Transversal operations
+                \item \dots
+            \end{itemize}
+        \item Circuit level noise
+        \item Detector error model
+            \begin{itemize}
+                \item Columns of the check matrix represent different
+                    possible error patterns $\rightarrow$ Check matrix
+                    doesn't quite correspond to the codewords we used
+                    initially anymore, but some similar structure ist
+                    still there (compare with syndrome)
+            \end{itemize}
+    \end{itemize}
+    \textbf{General Notes:}
+    \begin{itemize}
+        \item Give a brief overview of the history of QEC
+        \item Note (and research if this is actually correct) that QC
+            was developed on an abstract level before thinking of
+            what hardware to use
+        \item Note that there are other codes than stabilizer codes
+            (and research and give some examples), but only
+            stabilizer codes are considered in this work
+        \item Degeneracy
+        \item The QEC decoding problem (considering degeneracy)
+    \end{itemize}
+}
+
+\subsection{Stabilizer Codes}
+\subsection{CSS Codes}
+\subsection{Quantum Low-Density Parity-Check Codes}
+

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -0,0 +1,13 @@
+\chapter{Fault Tolerant QEC}
+\section{Fault Tolerance}
+\section{Noise Models}
+\subsection{Depolarizing Channel}
+\subsection{Phenomenological Noise}
+\subsection{Circuit-Level Noise}
+\section{Detector Error Models}
+\subsection{Measurement Syndrome Matrix}
+\subsection{Detector Error Matrix}
+\subsection{Detector Error Models}
+\section{Practical Considerations}
+\subsection{Practical Methodology}
+\subsection{Stim}

src/thesis/chapters/4_decoding_under_dems.tex

@@ -0,0 +1,5 @@
+\chapter{Decoding under Detector Error Models}
+\section{Sliding-Window Decoding}
+\section{Treating Detector Error Matrices like SC-LDPC Codes}
+\section{Soft-Information Aware Sliding-Window Decoding}
+\section{Numerical Results and Analysis}

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -0,0 +1 @@
+\chapter{Conclusion and Outlook}

src/thesis/main.tex

@@ -0,0 +1,98 @@
+\documentclass{lib/cel-thesis/cel-thesis}
+
+\usepackage[a4paper,left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry}
+\usepackage{float}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{mleftright}
+\usepackage{bm}
+\usepackage{tikz}
+\usepackage{xcolor}
+\usepackage{pgfplots}
+\pgfplotsset{compat=newest}
+\usepackage{acro}
+\usepackage{braket}
+% \usepackage[
+%     backend=biber,
+%     style=ieee,
+%     sorting=nty,
+% ]{biblatex}
+\usepackage{todonotes}
+
+\usetikzlibrary{calc, positioning, arrows}
+
+%
+%
+% Custom commands
+%
+%
+
+\newcommand{\red}[1]{\textcolor{red}{#1}}
+
+%
+%
+% Acronyms
+%
+%
+
+\input{acronyms.tex}
+
+\usepackage{babelbib}
+\setlanguage
+
+\usepackage{caption}
+\usepackage{bm}
+\usepackage{subcaption}
+\usepackage{todonotes} % great for draft annotations
+\DeclareCaptionLabelFormat{bf-nodot}{\textbf{#1}~\textbf{#2}}
+\captionsetup{labelformat=bf-nodot,labelsep=colon}
+
+%
+%
+% Content
+%
+%
+
+\thesisTitle{Fault Tolerant Quantum Error Correction}
+\thesisType{Master's Thesis}
+\thesisAuthor{Andreas Tsouchlos}
+\thesisAdvisor{Prof. Dr.-Ing. Laurent Schmalen}
+\thesisHeadOfInstitute{Prof. Dr.-Ing. Laurent Schmalen}
+% \thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost}
+%\thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost\\Prof. Dr.-Ing.
+% Laurent Schmalen}
+\thesisSupervisor{Jonathan Mandelbaum}
+\thesisStartDate{01.11.2025}
+\thesisEndDate{04.05.2026}
+\thesisSignatureDate{Signature date}
+\thesisLanguage{english}
+
+\begin{document}
+\pagenumbering{roman}  % all the preliminaries should be counted roman style
+
+\maketitle
+\newpage
+
+% \include{chapters/abstract}
+
+\cleardoublepage
+\pagenumbering{arabic}
+
+\tableofcontents
+\cleardoublepage
+
+\input{chapters/1_introduction.tex}
+\input{chapters/2_fundamentals.tex}
+\input{chapters/3_fault_tolerant_qec.tex}
+\input{chapters/4_decoding_under_dems.tex}
+\input{chapters/5_conclusion_and_outlook.tex}
+
+% \appendix
+% \listoffigures
+% \listoftables
+% \include{abbreviations}
+
+\bibliography{lib/cel-thesis/IEEEabrv,bibliography}
+
+\end{document}
+