Add outline and first page of content for thesis

src/thesis/acronyms.tex

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+\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
+}

src/thesis/chapters/1_introduction.tex

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+\chapter{Introduction}

src/thesis/chapters/2_fundamentals.tex

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+\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}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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: Is an explanation of BP with guided decimation needed here?
+% TODO: Is an explanation of OSD needed here?
+
+\subsection{Binary Linear Block Codes}
+
+\red{
+    \begin{itemize}
+        \item Note that binary linear codes are not the only coding
+            scheme out there
+    \end{itemize}
+}
+
+Binary linear block codes split the information to be protected into
+separate blocks.
+Each block is encoded, transmitted, and decoded separately.
+The encoding step introduces redundancy by mapping \textit{data words} of
+length $k \in \mathbb{N}$ onto \textit{codewords} of length $n \in \mathbb{N}$
+with $n > k$.
+The number of words stays the same, but they are mapped into an
+expanded Hilbert space where they are ``further appart'', giving rise
+to the error correcting properties of the coding scheme.
+
+Figure \ref{fig:Diagram of a transmission system} visualizes the
+entire communication process.
+A data word $\bm{u}\in \mathbb{F}_2^k$ is mapped onto a codeword $\bm{x}
+\in \mathbb{F}_2^n$. This is passed on to a transmitter, which
+interacts with the physical channel.
+A receiver acquires the message $\bm{y} \in \mathbb{R}^n$ and
+forwards it to a decoder.
+Finally, the decoder is responsible for obtaining an estimate
+$\hat{\bm{u}} \in \mathbb{F}_2^k$ of the original codeword from the
+received message.
+
+\vspace{10mm}
+
+For linear codes, this encoding process can be described as%
+%
+\begin{align*}
+    \bm{x} = \bm{u}\bm{G},
+\end{align*}%
+%
+using a generator matrix $\bm{G} \in \mathbb{F}_2^{k\times n}$.
+
+% We call the set of all codewords the codespace $\mathcal{C}$.
+
+% $\mathcal{C} = \left\{ \bm{x} \in \mathbb{F}_2^n :
+% \bm{H}\bm{x}^\text{T} = \bm{0} \right\}$.
+
+\begin{figure}[H]
+    \centering
+
+    \begin{tikzpicture}
+        \node[]                                    (in)  {$\bm{u}$};
+        \node[rectangle, draw=black, right=of in]  (enc) {Encoder};
+        \node[rectangle, draw=black, right=of enc] (tra) {Transmitter};
+        \node[rectangle, draw=black, right=of tra] (cha) {Channel};
+        \node[rectangle, draw=black, right=of cha] (rec) {Receiver};
+        \node[rectangle, draw=black, right=of rec] (dec) {Decoder};
+        \node[right=of dec]                        (out) {$\hat{\bm{u}}$};
+
+        \draw[-{latex}] (in) -- (enc);
+        \draw[-{latex}] (enc) -- (tra);
+        \draw[-{latex}] (tra) -- (cha);
+        \draw[-{latex}] (cha) -- (rec);
+        \draw[-{latex}] (rec) -- (dec);
+        \draw[-{latex}] (dec) -- (out);
+    \end{tikzpicture}
+
+    \caption{Diagram of a transmission system}
+    \label{fig:Diagram of a transmission system}
+\end{figure}
+
+\red{
+    \textbf{Topics to cover}
+    \begin{itemize}
+        \item Parity checks and describing a code by $H$ rather than
+            $\mathcal{C}$
+        \item G, H, notation
+        \item Minimum distance, [] - notation
+        \item The syndrome
+        \item The decoding problem
+        \item Soft vs. Hard information
+    \end{itemize}
+}
+
+\red{
+    \textbf{General Notes:}
+    \begin{itemize}
+        \item Make sure all coding concepts used later on have been
+            introduced (e.g., code rate?)
+    \end{itemize}
+}
+
+\subsection{Low-Density Parity-Check Codes}
+
+\red{
+    \begin{itemize}
+        \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

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+\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

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+\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

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+\chapter{Conclusion and Outlook}

src/thesis/main.tex

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+\documentclass[dvipsnames]{report}
+
+\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}
+\addbibresource{src/thesis/MA.bib}
+
+%
+%
+% Content
+%
+%
+
+\title{Fault Tolerant Quantum Error Correction}
+% \subtitle{Master's Thesis}
+\author{Andreas Tsouchlos}
+
+\begin{document}
+
+\maketitle
+\tableofcontents
+\newpage
+
+\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}
+
+\printbibliography
+
+\end{document}
+