Write chapter 4 intro

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -1,4 +1,5 @@
 \chapter{Fault-Tolerant Quantum Error Correction}
+\label{ch:Fault tolerance}
 
 % Intro
 

src/thesis/chapters/4_decoding_under_dems.tex

@@ -1,14 +1,87 @@
 % TODO: Make all [H] -> [t]
 \chapter{Decoding under Detector Error Models}
 
-% Intro
+In \Cref{ch:Fundamentals} we introduced the fundamentals of classical
+error correction, before moving on to quantum information science and
+finally combining the two in \acf{qec}.
+In \Cref{ch:Fault tolerance} we then turned to fault-tolerance, with
+a focus on a specific way of implementing it, called \acfp{dem}.
+In this chapter, we move on from the fundamental concepts and examine
+how to apply them in practice.
+Specifically, we concern ourselves with the practical aspects of decoding
+under \acp{dem}.
 
-\content{Intro}
+We investigate decoding \acf{qldpc} codes under \acp{dem} in particular.
+We focus on \ac{qldpc} codes, as they have emerged as leading
+candidates for practical quantum error correction, offering the
+ability to encode more logical qubits per physical qubit than surface
+codes while maintaining favorable threshold properties
+\cite[Sec.~1]{bravyi_high-threshold_2024}.
+Because of this, the decoding algorithms we consider will all be
+related to \acf{bp} in some way.
+Our aim is to build a fault-tolerant \ac{qec} system that works well
+even under consideration of circuit-level noise.
+We must overcome two main challenges to achieve this.
+
+First, recall the problems related to degeneracy, which is inherent
+to quantum codes.
+Because multiple minimum-weight codewords exist, the \ac{bp}
+algorithm becomes uncertain of the direction to proceed in.
+Additionally, the commutativity conditions of the stabilizers
+necessitate the existence of short cycles.
+These two aspects together lead to substantial convergence problems
+of \ac{bp} for quantum codes, when it is used on it's own.
+
+Second, the consideration of circuit-level noise introduces many more
+error locations into the circuit.
+Using \acp{dem}, we construct a new circuit code and model each of
+these error locations as a new \acf{vn}.
+We also perform multiple rounds of syndrome measuremetns,
+exacerbating the problem.
+This leads to a massively increased computational complexity and
+latency of the decoding process.
+In our experiments using the $\llbracket 144,12,12 \rrbracket$
+\acf{bb} code with $12$ syndrome measurement rounds, for example, the
+number of \acp{vn} was increased from $144$ to $9504$, and the
+number of \acfp{cn} was increased from $72$ to $1008$.
+
+The first problem is not inherent to \acp{dem} or fault-tolerance,
+but rather quantum codes in general.
+Many different approaches to solving it exist, usually centered
+around somehow modifying \ac{bp}.
+The most popular approach by far is combining a few initial
+iterations of \ac{bp} with a second decoding algorithm, \ac{osd}
+\cite{roffe_decoding_2020}.
+Other approaches exist, such as \ac{aed}
+\cite{koutsioumpas_automorphism_2025}, were multiple variations of
+the code are decoded simultaneously to increase the chances of convergence.
+Here, we will focus on the \acf{bpgd} algorithm
+\cite{yao_belief_2024} we already introduced in \Cref{ch:Fundamentals},
+for reasons that will become clear later in the chapter.
+
+The second problem is inherent to decoding using \acp{dem}.
+This is an area that has been less studied.
+As we saw in \Cref{sec:Quantum Error Correction}, for \ac{qec},
+latency is the main constraint, not raw computational complexity,
+and reducing latency is the main goal of the existing literature.
+This is generally done using windowing approaches; either
+sliding-window based, where the latency is reduced due an earlier
+start to the decoding process \cite{kuo_fault-tolerant_2024}%
+\cite{huang_improved_2023}\cite{huang_increasing_2024}\cite{gong_toward_2024},
+or by decoding multiple windows in parallel
+\cite{skoric_parallel_2023}\cite{tan_scalable_2023}.
+This work is based on the sliding-window method.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Sliding-Window Decoding}
 \label{sec:Sliding-Window Decoding}
 
+\content{Possibly go into the fact that current sliding-window
+    approaches don't differentiate clearly between the sliding-window
+    part and the decoder part. This work aims to extend the
+    sliding-window part in a general fashion that is compatible with many
+different decoder parts.}
+
 % Intro
 
 \content{Callback to previous chapter}

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -1 +1,5 @@
 \chapter{Conclusion and Outlook}
+
+\content{\textbf{Ideas for further research}}
+\content{Softer way of decimating VNs}
+