% TODO: Make all [H] -> [t]
\chapter{Decoding under Detector Error Models}

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}.

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 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.
The main way this is addressed in the literature is \emph{sliding
window decoding}, which attempts to divide the overall decoding
problem into many smaller ones that can be solved more efficiently.

% TODO: This could potentially be abit more text (e.g., go into
% SC-LDPC like structure that serves as the inspiration for the
% warm-start decoding. Or just go into warm-start decoding)
We will start by briefly reviewing the existing work related to
sliding-window decoding,
before focusing on one specific incarnation.
We will then introduce a modification to the existing algorithm and
perform numerical simulations to evaluate it.

% 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}

% Spacetime codes

\ac{qec} codes are often viewed through the lenses of the
\emph{space} and \emph{time} dimensions.
Both directions add redundancy, but they do so in a different way and
guard against different defects.
The space dimension corresponds to the redundancy added through the
code itself, while the time dimension corresponds to the repetition
of the syndrome measurements \cite[Sec.~IV.B]{dennis_topological_2002}.

% Basic idea

The idea of sliding-window decoding is to exploit the time-like
structure by splitting the circuit into overlapping windows along the
time dimension.
Each of these windows is then decoded separately.

%%%%%%%%%%%%%%%%
\subsection{Existing Literature}
\label{subsec:Existing Literature}

% Review of existing literature

Research on this topic has been ongoing for some time, though mostly
for topological codes.
The literature on \ac{qldpc} codes is more limited. Figure
\Cref{fig:literature} gives an overview of the related body of work.

\red{
    \begin{itemize}
        \item \cite{huang_increasing_2024} use BP+OSD,
            \cite{gong_toward_2024} use BP+GDG
        \item \cite{huang_improved_2023} use phenomenological noise,
            \cite{gong_toward_2024} circuit-level noise
        \item Go into the way the parallel decoding approaches
            consolidate the overlap regions
        \item \cite{huang_improved_2023} use hypegraph and lifted
            product codes, \cite{gong_toward_2024} use BB codes
        \item \cite{kuo_fault-tolerant_2024} use toric codes, the
            rest of the topological papers surface codes
        \item \cite{dennis_topological_2002} call their scheme ``overlap-add''
        \item QUITS views sliding-window decoding more separately
        \item Reasons for latency improvement ()
    \end{itemize}
}

\begin{figure}[H]
    \centering

    \tikzset{
        literature/.append style={
            minimum width=6mm,
            minimum height=6mm,
            text width=18mm,
            align=left,
        }
    }
    \tikzset{
        heading/.append style={
            draw=black,
            minimum width=22mm,
            minimum height=6mm,
            align=left,
            rounded corners = 1mm,
        }
    }

    \begin{tikzpicture}[node distance = 0mm and 0mm]
        % tex-fmt: off
        \node[heading, minimum width=15mm, fill=gray!25]              (code)  {Code};
        \node[heading, below right=1mm and -5mm of code, fill=orange!20]  (top)   {Topological};
        \node[heading, below right=42mm and -5mm of code, fill=orange!20] (qldpc) {QLDPC};

        \node[literature, below right=0mm and -12mm of top] (dennis) {\cite{dennis_topological_2002}};
        \node[literature, below=of dennis]                  (tan)    {\cite{tan_scalable_2023}};
        \node[literature, below=of tan]                     (skoric) {\cite{skoric_parallel_2023}};
        \node[literature, below=of skoric]                  (bombin) {\cite{bombin_modular_2023}};
        \node[literature, below=of bombin]                  (kuo)    {\cite{kuo_fault-tolerant_2024}};

        \node[literature, below right=0mm and -12mm of qldpc] (huang) {\cite{huang_improved_2023},\cite{huang_increasing_2024}};
        \node[literature, below=of huang]                     (gong)  {\cite{gong_toward_2024}};

        \coordinate (code-anchor)  at ($(code.south) + (-2mm,0)$);
        \coordinate (top-anchor)   at ($(top.south) + (-5mm,0)$);
        \coordinate (qldpc-anchor) at ($(qldpc.south) + (-5mm,0)$);

        \draw (code-anchor) |- (top);
        \draw (code-anchor) |- (qldpc);

        \draw (top-anchor) |- (dennis);
        \draw (top-anchor) |- (tan);
        \draw (top-anchor) |- (skoric);
        \draw (top-anchor) |- (bombin);
        \draw (top-anchor) |- (kuo);

        \draw (qldpc-anchor) |- (huang);
        \draw (qldpc-anchor) |- (gong);

        \draw [
            line width=1pt,
            decorate,
            decoration={brace,amplitude=2mm,raise=5mm}
        ]
        (dennis.north east) -- (dennis.south east)
        node[midway,right,xshift=10mm]{Sequential};

        \draw [
            line width=1pt,
            decorate,
            decoration={brace,amplitude=2mm,raise=5mm}
        ]
        (tan.north east) -- (kuo.south east)
        node[midway,right,xshift=10mm]{Parallel};

        \draw [
            line width=1pt,
            decorate,
            decoration={brace,amplitude=2mm,raise=5mm}
        ]
        (huang.north east) -- (gong.south east)
        node[midway,right,xshift=10mm]{Sequential};
        % tex-fmt: on
    \end{tikzpicture}

    \caption{Overview of literature on sliding-window decoding.}
    \label{fig:literature}
\end{figure}

% \red{
%     Existing work
%     \begin{itemize}
%         \item \cite{gong_toward_2024}
%             \begin{itemize}
%                 \item BB codes (QLDPC)
%                 \item Circuit-level noise
%                 \item Sequential
%                 \item Cites $\underbrace{\cite{dennis_topological_2002}
%                         \cite{tan_scalable_2023}
%                     \cite{skoric_parallel_2023}}_\text{Surface code}
%                     \underbrace{\cite{huang_improved_2023}}_\text{QLDPC,Phenomenological}$
%             \end{itemize}
%         \item \cite{huang_improved_2023}
%             \begin{itemize}
%                 \item Hypergraph product codes, Lifted product codes (QLDPC)
%                 \item Phenomenological noise
%                 \item Sequential
%                 \item Cites $\underbrace{\cite{dennis_topological_2002}
%                         [Huang, Brown, 2021]
%                         \cite{skoric_parallel_2023}
%                         \cite{tan_scalable_2023}
%                     \cite{bombin_modular_2023}}_\text{Surface code}$
%             \end{itemize}
%         \item \cite{dennis_topological_2002}
%             \begin{itemize}
%                 \item Surface code (Topological)
%                 \item No idea what noise, don't care either (Gong et
%                     al. say circuit-level noise)
%                 \item ``Overlapping recovery'' -> Sequential
%             \end{itemize}
%         \item \cite{tan_scalable_2023}
%             \begin{itemize}
%                 \item Surface code (Topological)
%                 \item Circuit-level noise
%                 \item Parallel
%                 \item Cites \cite{dennis_topological_2002}
%             \end{itemize}
%         \item \cite{skoric_parallel_2023}
%             \begin{itemize}
%                 \item Surface code (Topological)
%                 \item Circuit-level noise
%                 \item Parallel
%                 \item Cites \cite{dennis_topological_2002}
%             \end{itemize}
%         \item \cite{huang_increasing_2024}
%             \begin{itemize}
%                 \item Same as \cite{huang_improved_2023}
%             \end{itemize}
%         \item \cite{kuo_fault-tolerant_2024}
%             \begin{itemize}
%                 \item Toric codes (Topological)
%                 \item Circuit-level noise
%                 \item Parallel
%                 \item Cites \cite{dennis_topological_2002}
%                     \cite{tan_scalable_2023}
%                     \cite{skoric_parallel_2023} \cite{gong_toward_2024}
%             \end{itemize}
%         \item \cite{bombin_modular_2023}
%             \begin{itemize}
%                 \item Surface codes (Topological)
%                 \item No idea if it's even fault-tolerant
%                 \item Parallel
%                 \item Cites \cite{dennis_topological_2002}
%                     \cite{tan_scalable_2023}
%                     \cite{skoric_parallel_2023} \cite{leverrier_decoding_2022}
%             \end{itemize}
%             % This is not BP and not parallelization over the time dimension
%             % \item \cite{leverrier_decoding_2022}
%             %     \begin{itemize}
%             %         \item Quantum tanner codes (QLDPC)
%             %         \item Parallel
%             %         \item No idea if it's even fault-tolerant
%             %         \item Cites [don't care]
%             %     \end{itemize}
%         \item \cite{kang_quits_2025}
%             \begin{itemize}
%                 \item Cites \cite{huang_increasing_2024} \ldots
%             \end{itemize}
%     \end{itemize}
% }

\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. Combine this with QUITS modular structure
for sliding window decoding}

%%%%%%%%%%%%%%%%
\subsection{Implementation of Sliding-Window Decoding}
\label{subsec:Implementation of Sliding-Window Decoding}

We build on the approach taken by \cite{huang_increasing_2024} and
\cite{gong_toward_2024}.

% High-level overview of Sliding-Window decoding

\content{Benefits of sliding-window decoding (lower latency due to
earlier decoding start)}
\content{Why it works (block diagonal structure $\rightarrow$ ``Done
with processing'' some VNs)}

% Detailed explanation of sliding-window decoding

\content{We look at rows not columns}
\content{Define W}
\content{Define F}
\content{Explain how we get the columns once we know the rows}
\content{\textbf{General note}: Mathematical definitions where possible}

% Syndrome update

\content{Explain commit region}
\content{Why we need to update the syndrome}
\content{How we update the syndrome}
\content{\textbf{General note}: Mathematical definitions where possible}

% Complete process

\content{1. Decode window}
\content{2. Commit VN estimates}
\content{3. Update syndrome}
\content{4. Decode next window}
\content{(?) Explicitly mention we don't reuse existing messages?}

\begin{figure}[H]
    \centering

    \hspace*{-114mm}%
    \begin{tikzpicture}
        \draw[decorate, decoration={brace, amplitude=10pt}, line width=1pt]
        (0,0) -- (3.1,0) node[midway, above=4mm] {Commit region};
    \end{tikzpicture}

    \centering
    \includegraphics[scale=0.75]{res/72_bb_dem.pdf}

    \vspace*{-25.3mm}

    \hspace*{-98mm}%
    \begin{tikzpicture}
        \draw[{Latex}-{Latex}, line width=.7pt] (0,   -0.75mm) -- (0,  5mm);
        \draw[line width=1pt]                   (-1mm,-0.75mm) --
(3mm,-0.75mm);
        \draw[line width=1pt]                   (-1mm,5mm)     -- (3mm,5mm);
        \node[left] at                          (-2mm,2.125mm) {$\sim W$};

        \draw[{Latex}-{Latex}, line width=.3pt] (6.5cm,1.6mm)  -- (6.5cm,5mm);
        \draw[line width=1pt]                   (6.5cm,4.9mm)  -- (6.5cm,7mm);
        \node[above] at                         (6.5cm,7mm) {$\sim F$};
    \end{tikzpicture}

    \vspace*{10mm}

    \caption{
        Visualization of the windowing process on the detector
        error matrix.
    }
    \label{fig:windowing_pcm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Warm-Start Sliding-Window Decoding}
\label{sec:warm_start_bp}

% Intro

\content{Change view from PCM to Tanner graph}
\content{Call attention to SC-LDPC-like structure}
\content{High-level overview of modification}

% Warm-Start decoding for BP

\content{Pass messages to next window}
\content{(?) Explicitly mention initialization using only CN->VN
messages + swapping of CN and VN update?}
\content{(?) Algorithm}

% Warm-Start decoding for BPGD

\content{Modified structure of BPGD $\rightarrow$ In addition to
messages, pass decimation info}
\content{(?) Explicitly mention decimation info = channel llrs?}
\content{(?) Algorithm}

\begin{figure}[H]
    \centering

    \tikzset{
        VN/.style={
            circle, fill=KITgreen, minimum width=1mm, minimum height=1mm,
        },
        CN/.style={
            rectangle, fill=KITblue, minimum width=1mm, minimum height=1mm,
        },
    }

    \begin{tikzpicture}[node distance = 5mm]
        \node[VN]                  (vn00) {};
        \node[VN, below = of vn00] (vn01) {};
        \node[VN, below = of vn01] (vn02) {};
        \node[VN, below = of vn02] (vn03) {};
        \node[VN, below = of vn03] (vn04) {};

        \coordinate (temp) at ($(vn01)!0.5!(vn02)$);

        \node[CN, left =10mm of temp] (cn00) {};
        \node[CN, below = of cn00] (cn01) {};

        \draw (vn00) -- (cn00);
        \draw (vn01) -- (cn00);
        \draw (vn03) -- (cn00);
        \draw (vn01) -- (cn01);
        \draw (vn02) -- (cn01);
        \draw (vn04) -- (cn01);

        \foreach \i in {1,2,3,4} {
            \pgfmathtruncatemacro{\prev}{\i-1}

            \node[VN, right = 25mm of vn\prev 0] (vn\i0) {};
            \node[VN, below = of vn\i0]          (vn\i1) {};
            \node[VN, below = of vn\i1]          (vn\i2) {};
            \node[VN, below = of vn\i2]          (vn\i3) {};
            \node[VN, below = of vn\i3]          (vn\i4) {};

            \coordinate (temp) at ($(vn\i1)!0.5!(vn\i2)$);

            \node[CN, left = 10mm of temp] (cn\i0) {};
            \node[CN, below = of cn\i0]     (cn\i1) {};

            \draw (vn\i0) -- (cn\i0);
            \draw (vn\i1) -- (cn\i0);
            \draw (vn\i3) -- (cn\i0);
            \draw (vn\i1) -- (cn\i1);
            \draw (vn\i2) -- (cn\i1);
            \draw (vn\i4) -- (cn\i1);
        }

        \foreach \i in {1,2,3,4} {
            \pgfmathtruncatemacro{\prev}{\i-1}

            \draw (vn\prev 3) -- (cn\i 0);
            \draw (vn\prev 4) -- (cn\i 1);
        }

        \node[
            draw, inner sep=5mm,line width=1pt,
            fit=(vn00)(vn04)(cn00)(cn01)(vn20)(vn24)(cn20)(cn21)
        ]
        (box1) {};
        \node[
            draw, dashed, inner sep=5mm, inner ysep=8mm,line width=1pt,
            fit=(vn10)(vn14)(cn10)(cn11)(vn30)(vn34)(cn30)(cn31)
        ]
        (box2) {};

        % Marker for W on the bottom
        \draw[line width=1pt]
        ([yshift=-5mm, line width=1pt]box1.south west) -- ++(0,-4mm)
        coordinate (dim1l);
        \draw[line width=1pt]
        ([yshift=-5mm]box1.south east) -- ++(0,-4mm)
        coordinate (dim1r);
        \draw[{Latex}-{Latex}, line width=1pt]
        ([yshift=1mm]dim1l) -- ([yshift=1mm]dim1r)
        node[midway, below=2pt] {$W$};

        % Marker for F on top
        \draw[line width=1pt]
        ([yshift=3mm]box2.north west) -- ++(0,4mm)
        coordinate (dim3l);
        \draw[line width=1pt]
        ([yshift=3mm]box2.north west -| box1.north west) -- ++(0,4mm)
        coordinate (dim3r);
        \draw[{Latex}-{Latex}, line width=1pt]
        ([yshift=-1mm]dim3l) -- ([yshift=-1mm]dim3r)
        node[midway, above=2pt] {$F$};

        % Arrow on the top right
        \draw[-{Latex}, line width=1pt]
        ([yshift=8mm] box1.north east) -- ++(28mm,0);
    \end{tikzpicture}

    \caption{Visualization of the windowing process on the Tanner graph.}
    \label{fig:windowing_tanner}
\end{figure}

\begin{figure}[H]
    \centering

    \tikzset{
        VN/.style={
            circle, fill=KITgreen, minimum width=1mm, minimum height=1mm,
        },
        CN/.style={
            rectangle, fill=KITblue, minimum width=1mm, minimum height=1mm,
        },
    }

    \begin{tikzpicture}[node distance = 5mm]
        \node[VN]                  (vn00) {};
        \node[VN, below = of vn00] (vn01) {};
        \node[VN, below = of vn01] (vn02) {};
        \node[VN, below = of vn02] (vn03) {};
        \node[VN, below = of vn03] (vn04) {};

        \coordinate (temp) at ($(vn01)!0.5!(vn02)$);

        \node[CN, left =10mm of temp] (cn00) {};
        \node[CN, below = of cn00] (cn01) {};

        \draw (vn00) -- (cn00);
        \draw (vn01) -- (cn00);
        \draw (vn03) -- (cn00);
        \draw (vn01) -- (cn01);
        \draw (vn02) -- (cn01);
        \draw (vn04) -- (cn01);

        \foreach \i in {1,2,3,4} {
            \pgfmathtruncatemacro{\prev}{\i-1}

            \node[VN, right = 25mm of vn\prev 0] (vn\i0) {};
            \node[VN, below = of vn\i0]          (vn\i1) {};
            \node[VN, below = of vn\i1]          (vn\i2) {};
            \node[VN, below = of vn\i2]          (vn\i3) {};
            \node[VN, below = of vn\i3]          (vn\i4) {};

            \coordinate (temp) at ($(vn\i1)!0.5!(vn\i2)$);

            \node[CN, left = 10mm of temp] (cn\i0) {};
            \node[CN, below = of cn\i0]     (cn\i1) {};

            \draw (vn\i0) -- (cn\i0);
            \draw (vn\i1) -- (cn\i0);
            \draw (vn\i3) -- (cn\i0);
            \draw (vn\i1) -- (cn\i1);
            \draw (vn\i2) -- (cn\i1);
            \draw (vn\i4) -- (cn\i1);
        }

        \foreach \i in {1,2,3,4} {
            \pgfmathtruncatemacro{\prev}{\i-1}

            \draw (vn\prev 3) -- (cn\i 0);
            \draw (vn\prev 4) -- (cn\i 1);
        }

        \node[
            draw, inner sep=5mm,line width=1pt,
            fit=(vn00)(vn04)(cn00)(cn01)(vn20)(vn24)(cn20)(cn21)
        ]
        (box1) {};
        \node[
            draw, dashed, inner sep=5mm, inner ysep=8mm,line width=1pt,
            fit=(vn10)(vn14)(cn10)(cn11)(vn30)(vn34)(cn30)(cn31)
        ]
        (box2) {};

        \draw[KITorange, line width=2pt] (cn10) -- (vn10);
        \draw[KITorange, line width=2pt] (cn10) -- (vn11);
        \draw[KITorange, line width=2pt] (cn10) -- (vn13);
        \draw[KITorange, line width=2pt] (cn11) -- (vn11);
        \draw[KITorange, line width=2pt] (cn11) -- (vn12);
        \draw[KITorange, line width=2pt] (cn11) -- (vn14);

        \draw[KITorange, line width=2pt] (vn13) -- (cn20);
        \draw[KITorange, line width=2pt] (vn14) -- (cn21);

        \draw[KITorange, line width=2pt] (cn20) -- (vn20);
        \draw[KITorange, line width=2pt] (cn20) -- (vn21);
        \draw[KITorange, line width=2pt] (cn20) -- (vn23);
        \draw[KITorange, line width=2pt] (cn21) -- (vn21);
        \draw[KITorange, line width=2pt] (cn21) -- (vn22);
        \draw[KITorange, line width=2pt] (cn21) -- (vn24);

        % Marker for W on the bottom
        \draw[line width=1pt]
        ([yshift=-5mm, line width=1pt]box1.south west) -- ++(0,-4mm)
        coordinate (dim1l);
        \draw[line width=1pt]
        ([yshift=-5mm]box1.south east) -- ++(0,-4mm)
        coordinate (dim1r);
        \draw[{Latex}-{Latex}, line width=1pt]
        ([yshift=1mm]dim1l) -- ([yshift=1mm]dim1r)
        node[midway, below=2pt] {$W$};

        % Marker for F on top
        \draw[line width=1pt]
        ([yshift=3mm]box2.north west) -- ++(0,4mm)
        coordinate (dim3l);
        \draw[line width=1pt]
        ([yshift=3mm]box2.north west -| box1.north west) -- ++(0,4mm)
        coordinate (dim3r);
        \draw[{Latex}-{Latex}, line width=1pt]
        ([yshift=-1mm]dim3l) -- ([yshift=-1mm]dim3r)
        node[midway, above=2pt] {$F$};

        % Arrow on the top right
        \draw[-{Latex}, line width=1pt]
        ([yshift=8mm] box1.north east) -- ++(28mm,0);
    \end{tikzpicture}

    \caption{
        Visualization of the messages used for the
        initialization of the next window.
    }
    \label{fig:messages_tanner}
\end{figure}

\begin{figure}[H]
    \centering

    \tikzset{
        VN/.style={
            circle, fill=KITgreen, minimum width=1mm, minimum height=1mm,
        },
        CN/.style={
            rectangle, fill=KITblue, minimum width=1mm, minimum height=1mm,
        },
    }

    \begin{tikzpicture}[node distance = 5mm]
        \node[VN]                  (vn00) {};
        \node[VN, below = of vn00] (vn01) {};
        \node[VN, below = of vn01] (vn02) {};
        \node[VN, below = of vn02] (vn03) {};
        \node[VN, below = of vn03] (vn04) {};

        \coordinate (temp) at ($(vn01)!0.5!(vn02)$);

        \node[CN, left =10mm of temp] (cn00) {};
        \node[CN, below = of cn00] (cn01) {};

        \draw (vn00) -- (cn00);
        \draw (vn01) -- (cn00);
        \draw (vn03) -- (cn00);
        \draw (vn01) -- (cn01);
        \draw (vn02) -- (cn01);
        \draw (vn04) -- (cn01);

        \foreach \i in {1,2,3,4} {
            \pgfmathtruncatemacro{\prev}{\i-1}

            \node[VN, right = 25mm of vn\prev 0] (vn\i0) {};
            \node[VN, below = of vn\i0]          (vn\i1) {};
            \node[VN, below = of vn\i1]          (vn\i2) {};
            \node[VN, below = of vn\i2]          (vn\i3) {};
            \node[VN, below = of vn\i3]          (vn\i4) {};

            \coordinate (temp) at ($(vn\i1)!0.5!(vn\i2)$);

            \node[CN, left = 10mm of temp] (cn\i0) {};
            \node[CN, below = of cn\i0]     (cn\i1) {};

            \draw (vn\i0) -- (cn\i0);
            \draw (vn\i1) -- (cn\i0);
            \draw (vn\i3) -- (cn\i0);
            \draw (vn\i1) -- (cn\i1);
            \draw (vn\i2) -- (cn\i1);
            \draw (vn\i4) -- (cn\i1);
        }

        \foreach \i in {1,2,3,4} {
            \pgfmathtruncatemacro{\prev}{\i-1}

            \draw (vn\prev 3) -- (cn\i 0);
            \draw (vn\prev 4) -- (cn\i 1);
        }

        \node[
            draw, inner sep=5mm,line width=1pt,
            fit=(vn00)(vn04)(cn00)(cn01)(vn20)(vn24)(cn20)(cn21)
        ]
        (box1) {};
        \node[
            draw, dashed, inner sep=5mm, inner ysep=8mm,line width=1pt,
            fit=(vn10)(vn14)(cn10)(cn11)(vn30)(vn34)(cn30)(cn31)
        ]
        (box2) {};

        \draw[KITorange, line width=2pt] (cn10) -- (vn10);
        \draw[KITorange, line width=2pt] (cn10) -- (vn11);
        \draw[KITorange, line width=2pt] (cn10) -- (vn13);
        \draw[KITorange, line width=2pt] (cn11) -- (vn11);
        \draw[KITorange, line width=2pt] (cn11) -- (vn12);
        \draw[KITorange, line width=2pt] (cn11) -- (vn14);

        \draw[KITorange, line width=2pt] (vn13) -- (cn20);
        \draw[KITorange, line width=2pt] (vn14) -- (cn21);

        \draw[KITorange, line width=2pt] (cn20) -- (vn20);
        \draw[KITorange, line width=2pt] (cn20) -- (vn21);
        \draw[KITorange, line width=2pt] (cn20) -- (vn23);
        \draw[KITorange, line width=2pt] (cn21) -- (vn21);
        \draw[KITorange, line width=2pt] (cn21) -- (vn22);
        \draw[KITorange, line width=2pt] (cn21) -- (vn24);

        \node[VN, draw=KITorange, fill=KITorange] at (vn10) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn11) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn12) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn13) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn14) {};

        \node[VN, draw=KITorange, fill=KITorange] at (vn20) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn21) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn22) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn23) {};
        \node[VN, draw=KITorange, fill=KITorange] at (vn24) {};

        % Marker for W on the bottom
        \draw[line width=1pt]
        ([yshift=-5mm, line width=1pt]box1.south west) -- ++(0,-4mm)
        coordinate (dim1l);
        \draw[line width=1pt]
        ([yshift=-5mm]box1.south east) -- ++(0,-4mm)
        coordinate (dim1r);
        \draw[{Latex}-{Latex}, line width=1pt]
        ([yshift=1mm]dim1l) -- ([yshift=1mm]dim1r)
        node[midway, below=2pt] {$W$};

        % Marker for F on top
        \draw[line width=1pt]
        ([yshift=3mm]box2.north west) -- ++(0,4mm)
        coordinate (dim3l);
        \draw[line width=1pt]
        ([yshift=3mm]box2.north west -| box1.north west) -- ++(0,4mm)
        coordinate (dim3r);
        \draw[{Latex}-{Latex}, line width=1pt]
        ([yshift=-1mm]dim3l) -- ([yshift=-1mm]dim3r)
        node[midway, above=2pt] {$F$};

        % Arrow on the top right
        \draw[-{Latex}, line width=1pt]
        ([yshift=8mm] box1.north east) -- ++(28mm,0);
    \end{tikzpicture}

    \caption{
        Visualization of the messages and decimation information used for the
        initialization of the next window.
    }
    \label{fig:messages_decimation_tanner}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical results}
\label{sec:warm_start_bpgd}

% Intro

\content{Some info on used code (what it is, why it was chosen)}
\content{Some info on simulation setup (Stim, circuit-level noise,
standard circuit-based depolarizing noise model, etc.)}
\content{All datapoints generated with at least 100 error frames}

%%%%%%%%%%%%%%%%
\subsection{Belief Propagation}
\label{subsec:Belief Propagation}

% Simulation setup

\content{Use min-sum}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
                width=\figwidth,
                height=\figheight,
                ymode=log,
                legend style={
                    cells={anchor=west},
                    cells={align=left},
                },
                enlargelimits=false,
                ymin=1e-5, ymax=2e-1,
                grid=both,
                legend pos = south east,
                xtick={0.001,0.0015,...,0.004},
                xticklabel style={/pgf/number format/fixed},
                xticklabel style={/pgf/number format/precision=4},
                scaled x ticks=false,
                xlabel={Physical error rate},
                ylabel={Per-round-LER},
            ]

            \foreach \W/\col/\mark in
            {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                \edef\temp{\noexpand
                    \addplot+[mark=\mark, solid, mark
options={fill=\col}, \col]
                    table[
                        col sep=comma, x=physical_p,
                        y=LER_per_round,
                    ]
                    {res/sim/WF/WindowingSyndromeMinSumDecoder/max_iter_200/pass_soft_info_False/F_1/W_\W/LERs.csv};
                }
                \temp

                \addlegendentryexpanded{$W = \W$}
            }

            \addplot+[mark=*, solid, mark options={fill=black}, black]
            table[
                col sep=comma, x=physical_p,
                y=LER_per_round,
            ]
            {res/sim/whole/SyndromeMinSumDecoder/max_iter_200/LERs.csv};

            \addlegendentry{Whole}
        \end{axis}
    \end{tikzpicture}

    \caption{
        Comparison of the decoding performance of the $\llbracket
        144,12,12 \rrbracket$ \ac{bb} code under min-sum decoding
        ($200$ iterations) for different window sizes.
        The step size was fixed to $F=1$, $12$ rounds of syndrome
        extraction were performed and the noise model is
        standard circuit-based depolarizing noise.
    }
\end{figure}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
                width=\figwidth,
                height=\figheight,
                ymode=log,
                legend style={
                    cells={anchor=west},
                    cells={align=left},
                },
                enlargelimits=false,
                ymin=1e-5, ymax=2e-1,
                grid=both,
                legend pos = south east,
                xtick={0.001,0.0015,...,0.004},
                xticklabel style={/pgf/number format/fixed},
                xticklabel style={/pgf/number format/precision=4},
                scaled x ticks=false,
                xlabel={Physical error rate},
                ylabel={Per-round-LER},
                extra description/.code={
                    \node[rotate=90, anchor=south]
                    at ([xshift=10mm]current axis.east)
                    {Warm s. (---), Cold s. (- - -)};
                },
            ]

            \foreach \W/\col/\mark in
            {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                \edef\temp{\noexpand
                    \addplot+[
                        mark=\mark, densely dashed, mark options={fill=\col},
                        \col, forget plot
                    ]
                    table[
                        col sep=comma, x=physical_p,
                        y=LER_per_round,
                    ]
                    {res/sim/WF/WindowingSyndromeMinSumDecoder/max_iter_200/pass_soft_info_False/F_1/W_\W/LERs.csv};
                }
                \temp
            }

            \foreach \W/\col/\mark in
            {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                \edef\temp{\noexpand
                    \addplot+[mark=\mark, solid, mark
options={fill=\col}, \col]
                    table[
                        col sep=comma, x=physical_p,
                        y=LER_per_round,
                    ]
                    {res/sim/WF/WindowingSyndromeMinSumDecoder/max_iter_200/pass_soft_info_True/F_1/W_\W/LERs.csv};
                }
                \temp

                \addlegendentryexpanded{$W = \W$}
            }

            \addplot+[mark=*, solid, mark options={fill=black}, black]
            table[
                col sep=comma, x=physical_p,
                y=LER_per_round,
            ]
            {res/sim/whole/SyndromeMinSumDecoder/max_iter_200/LERs.csv};

            \addlegendentry{Whole}
        \end{axis}
    \end{tikzpicture}

    \caption{
        Comparison of the decoding performance of cold and warm-start
        decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
        code for different window sizes.
        Decoding was performed using the min-sum algorithm ($200$
        iterations).
        The step size was fixed to $F=1$, $12$ rounds of syndrome
        extraction were performed and the noise model is
        standard circuit-based depolarizing noise.
    }
\end{figure}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
                width=\figwidth,
                height=\figheight,
                ymode=log,
                legend style={
                    cells={anchor=west},
                    cells={align=left},
                },
                enlargelimits=false,
                ymin=1e-5, ymax=2e-1,
                grid=both,
                legend pos = south east,
                xtick={0.001,0.0015,...,0.004},
                xticklabel style={/pgf/number format/fixed},
                xticklabel style={/pgf/number format/precision=4},
                scaled x ticks=false,
                xlabel={Physical error rate},
                ylabel={Per-round-LER},
                extra description/.code={
                    \node[rotate=90, anchor=south]
                    at ([xshift=10mm]current axis.east)
                    {Warm s. (---), Cold s. (- - -)};
                },
            ]

            \foreach \F/\col/\mark in
            {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                \edef\temp{\noexpand
                    \addplot+[
                        mark=\mark, densely dashed, mark options={fill=\col},
                        \col, forget plot
                    ]
                    table[
                        col sep=comma, x=physical_p,
                        y=LER_per_round,
                    ]
                    {res/sim/WF/WindowingSyndromeMinSumDecoder/max_iter_200/pass_soft_info_False/F_\F/W_5/LERs.csv};
                }
                \temp
            }

            \foreach \F/\col/\mark in
            {3/KITred/triangle*,2/KITblue/diamond*,1/KITorange/square*} {
                \edef\temp{\noexpand
                    \addplot+[mark=\mark, solid, mark
options={fill=\col}, \col]
                    table[
                        col sep=comma, x=physical_p,
                        y=LER_per_round,
                    ]
                    {res/sim/WF/WindowingSyndromeMinSumDecoder/max_iter_200/pass_soft_info_True/F_\F/W_5/LERs.csv};
                }
                \temp

                \addlegendentryexpanded{$F = \F$}
            }
        \end{axis}
    \end{tikzpicture}

    \caption{
        Comparison of the decoding performance of cold and warm-start
        decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
        code for different step sizes.
        Decoding was performed using the min-sum algorithm ($200$
        iterations).
        The window size was fixed to $W=5$, $12$ rounds of syndrome
        extraction were performed and the noise model is
        standard circuit-based depolarizing noise.
    }
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}{0.48\textwidth}
        \centering
        \hspace*{-6mm}
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    % xmode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-3, ymax=1e-1,
                    grid=both,
                    legend pos = north east,
                    xtick={32,512,1024,2048,4096},
                    % xtick={0.001,0.0015,...,0.004},
                    xticklabels =
                    {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
                    xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Number of BP iterations},
                    ylabel={Per-round-LER},
                    % extra description/.code={
                    %     \node[rotate=90, anchor=south]
                    %     at ([xshift=10mm]current axis.east)
                    %     {Warm s. (---), Cold s. (- - -)};
                    % },
                ]

                \foreach \W/\col/\mark in
                {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, densely dashed,
forget plot, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_False/F_1/W_\W/LERs.csv};
                    }
                    \temp
                }

                \foreach \W/\col/\mark in
                {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_True/F_1/W_\W/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$W = \W$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of window sizes for $F=1$.}
    \end{subfigure}%
    \hfill%
    \begin{subfigure}{0.48\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    % xmode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-3, ymax=1e-1,
                    grid=both,
                    legend pos = north east,
                    xtick={32,512,1024,2048,4096},
                    % xtick={0.001,0.0015,...,0.004},
                    xticklabels =
                    {$32$, $512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
                    xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Number of BP iterations},
                    yticklabels={\empty},
                    % ylabel={Per-round-LER},
                    extra description/.code={
                        \node[rotate=90, anchor=south]
                        at ([xshift=10mm]current axis.east)
                        {Warm s. (---), Cold s. (- - -)};
                    },
                ]

                \foreach \F/\col/\mark in
                {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, densely dashed,
forget plot, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_False/F_\F/W_5/LERs.csv};
                    }
                    \temp
                }

                \foreach \F/\col/\mark in
                {3/KITred/triangle*,2/KITblue/diamond*,1/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_True/F_\F/W_5/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$F = \F$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of step sizes for $W=5$.}
    \end{subfigure}

    \caption{
        Comparison of cold and warm-start sliding-window
        min-sum decoding for the $\llbracket 144, 12, 12 \rrbracket$
        \ac{bb} code
        under circuit-level noise.
        $12$ rounds of syndrome extraction were performed and
        standard circuit-based depolarizing noise was chosen as the
noise model.
        The physical error probabilty was fixed to $0.0025$.
    }
\end{figure}

%%%%%%%%%%%%%%%%
\subsection{Belief Propagation with Guided Decimation}
\label{subsec:Belief Propagation with Guided Decimation}

\begin{figure}[H]
    \centering
    \hspace*{-6mm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-5, ymax=2e-1,
                    grid=both,
                    legend pos = south east,
                    xtick={0.001,0.0015,...,0.004},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Physical error rate},
                    ylabel={Per-round-LER},
                    % extra description/.code={
                    %     \node[rotate=90, anchor=south]
                    %     at ([xshift=10mm]current axis.east)
                    %     {Warm s. (---), Cold s. (- - -)};
                    % },
                ]

                \foreach \W/\col/\mark in
                {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[
                            mark=\mark, densely dashed, mark
                            options={fill=\col},
                            \col, forget plot
                        ]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoder/max_iter_5000/pass_soft_info_False/F_1/W_\W/LERs.csv};
                    }
                    \temp
                }

                \foreach \W/\col/\mark in
                {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoderPassDecimation/max_iter_5000/pass_soft_info_True/F_1/W_\W/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$W = \W$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of window sizes for $F=1$.}
    \end{subfigure}%
    \hfill%
    \begin{subfigure}{0.5\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-5, ymax=2e-1,
                    grid=both,
                    legend pos = south east,
                    xtick={0.001,0.0015,0.002,...,0.004},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Physical error rate},
                    yticklabels={\empty},
                    % ylabel={Per-round-LER},
                    extra description/.code={
                        \node[rotate=90, anchor=south]
                        at ([xshift=10mm]current axis.east)
                        {Warm s. (---), Cold s. (- - -)};
                    },
                ]

                \foreach \F/\col/\mark in
                {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[
                            mark=\mark, densely dashed, mark
                            options={fill=\col},
                            \col, forget plot
                        ]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoder/max_iter_5000/pass_soft_info_False/F_\F/W_5/LERs.csv};
                    }
                    \temp
                }

                \foreach \F/\col/\mark in
                {3/KITred/triangle*,2/KITblue/diamond*,1/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoderPassDecimation/max_iter_5000/pass_soft_info_True/F_\F/W_5/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$F = \F$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of step sizes for $W=5$.}
    \end{subfigure}

    \caption{
        Comparison of the decoding performance of cold and warm-start
        decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}.
        Decoding was performed using the \ac{bpgd} algorithm with
        $T=1$ and no limit on the number of outer iterations.
        The information used for the warm-start intialization
        included both the messages on the Tanner graph and decimation
        information.
        $12$ rounds of syndrome extraction were performed and
        standard circuit-based depolarizing noise was chosen as the
noise model.
    }
\end{figure}

\begin{figure}[H]
    \centering
    \hspace*{-6mm}
    \begin{subfigure}{0.48\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    % xmode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-3, ymax=1e-1,
                    grid=both,
                    legend pos = south west,
                    xtick={32,512,1024,2048,4096},
                    % xtick={0.001,0.0015,...,0.004},
                    xticklabels =
                    {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
                    xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Number of BP iterations},
                    ylabel={Per-round-LER},
                    % extra description/.code={
                    %     \node[rotate=90, anchor=south]
                    %     at ([xshift=10mm]current axis.east)
                    %     {Warm s. (---), Cold s. (- - -)};
                    % },
                ]

                \foreach \W/\col/\mark in
                {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, densely dashed,
forget plot, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoderPassDecimation/p_0.0025/pass_soft_info_False/F_1/W_\W/LERs.csv};
                    }
                    \temp
                }

                \foreach \W/\col/\mark in
                {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoderPassDecimation/p_0.0025/pass_soft_info_True/F_1/W_\W/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$W = \W$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of window sizes for $F=1$.}
    \end{subfigure}%
    \hfill%
    \begin{subfigure}{0.48\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    % xmode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-3, ymax=1e-1,
                    grid=both,
                    legend pos = south west,
                    xtick={32,512,1024,2048,4096},
                    % xtick={0.001,0.0015,...,0.004},
                    xticklabels =
                    {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
                    xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Number of BP iterations},
                    % ylabel={Per-round-LER},
                    yticklabels={\empty},
                    extra description/.code={
                        \node[rotate=90, anchor=south]
                        at ([xshift=10mm]current axis.east)
                        {Warm s. (---), Cold s. (- - -)};
                    },
                ]

                \foreach \F/\col/\mark in
                {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, densely dashed,
forget plot, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoderPassDecimation/p_0.0025/pass_soft_info_False/F_\F/W_5/LERs.csv};
                    }
                    \temp
                }

                \foreach \F/\col/\mark in
                {3/KITred/triangle*,2/KITblue/diamond*,1/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoderPassDecimation/p_0.0025/pass_soft_info_True/F_\F/W_5/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$F = \F$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{
            Comparison of step sizes for $W=5$.
        }
    \end{subfigure}

    \caption{
        Comparison of the decoding performance of cold and warm-start
        decoding for the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
        under circuit-level noise.
        Decoding was performed using the \ac{bpgd} algorithm with
        $T=1$.
        The number of iterations refers to the outer \ac{bpgd}
        iterations, i.e., the number of decimations.
        The information used for the warm-start intialization
        included only the messages on the Tanner graph.
        $12$ rounds of syndrome extraction were performed and
        standard circuit-based depolarizing noise was chosen as the
noise model.
        The physical error probabilty was fixed to $0.0025$.
    }
\end{figure}

\begin{figure}[H]
    \centering
    \hspace*{-6mm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-5, ymax=2e-1,
                    grid=both,
                    legend pos = south east,
                    xtick={0.001,0.0015,...,0.004},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Physical error rate},
                    ylabel={Per-round-LER},
                    % extra description/.code={
                    %     \node[rotate=90, anchor=south]
                    %     at ([xshift=10mm]current axis.east)
                    %     {Warm s. (---), Cold s. (- - -)};
                    % },
                ]

                \foreach \W/\col/\mark in
                {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[
                            mark=\mark, densely dashed, mark
                            options={fill=\col},
                            \col, forget plot
                        ]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoder/max_iter_5000/pass_soft_info_False/F_1/W_\W/LERs.csv};
                    }
                    \temp
                }

                \foreach \W/\col/\mark in
                {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoder/max_iter_5000/pass_soft_info_True/F_1/W_\W/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$W = \W$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of window sizes for $F=1$.}
    \end{subfigure}%
    \hfill%
    \begin{subfigure}{0.5\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-5, ymax=2e-1,
                    grid=both,
                    legend pos = south east,
                    xtick={0.001,0.0015,0.002,...,0.004},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Physical error rate},
                    yticklabels={\empty},
                    % ylabel={Per-round-LER},
                    extra description/.code={
                        \node[rotate=90, anchor=south]
                        at ([xshift=10mm]current axis.east)
                        {Warm s. (---), Cold s. (- - -)};
                    },
                ]

                \foreach \F/\col/\mark in
                {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[
                            mark=\mark, densely dashed, mark
                            options={fill=\col},
                            \col, forget plot
                        ]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoder/max_iter_5000/pass_soft_info_False/F_\F/W_5/LERs.csv};
                    }
                    \temp
                }

                \foreach \F/\col/\mark in
                {3/KITred/triangle*,2/KITblue/diamond*,1/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=physical_p,
                            y=LER_per_round,
                        ]
                        {res/sim/WF/WindowingSyndromeSpaGdDecoder/max_iter_5000/pass_soft_info_True/F_\F/W_5/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$F = \F$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of step sizes for $W=5$.}
    \end{subfigure}

    \caption{
        Comparison of the decoding performance of cold and warm-start
        decoding for the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
        under circuit-level noise.
        Decoding was performed using the \ac{bpgd} algorithm with
        $T=1$ and no limit on the number of outer iterations.
        The information used for the warm-start intialization
        included only the messages on the Tanner graph.
        $12$ rounds of syndrome extraction were performed and
        standard circuit-based depolarizing noise was chosen as the
noise model.
    }
\end{figure}

\begin{figure}[H]
    \centering
    \hspace*{-6mm}
    \begin{subfigure}{0.48\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    % xmode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-3, ymax=1e-1,
                    grid=both,
                    legend pos = north east,
                    xtick={32,512,1024,2048,4096},
                    % xtick={0.001,0.0015,...,0.004},
                    xticklabels =
                    {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
                    xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Number of BP iterations},
                    ylabel={Per-round-LER},
                    % extra description/.code={
                    %     \node[rotate=90, anchor=south]
                    %     at ([xshift=10mm]current axis.east)
                    %     {Warm s. (---), Cold s. (- - -)};
                    % },
                ]

                \foreach \W/\col/\mark in
                {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, densely dashed,
forget plot, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoder/p_0.0025/pass_soft_info_False/F_1/W_\W/LERs.csv};
                    }
                    \temp
                }

                \foreach \W/\col/\mark in
                {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoder/p_0.0025/pass_soft_info_True/F_1/W_\W/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$W = \W$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{Comparison of window sizes for $F=1$.}
    \end{subfigure}%
    \hfill%
    \begin{subfigure}{0.48\textwidth}
        \centering
        \begin{tikzpicture}
            \begin{axis}[
                    width=8cm,
                    height=6cm,
                    ymode=log,
                    % xmode=log,
                    legend style={
                        cells={anchor=west},
                        cells={align=left},
                    },
                    enlargelimits=false,
                    ymin=1e-3, ymax=1e-1,
                    grid=both,
                    legend pos = north east,
                    xtick={32,512,1024,2048,4096},
                    % xtick={0.001,0.0015,...,0.004},
                    xticklabels =
                    {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
                    xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
                    xticklabel style={/pgf/number format/fixed},
                    xticklabel style={/pgf/number format/precision=4},
                    x tick label style={rotate=45, anchor=north east,
                    inner sep=1mm},
                    scaled x ticks=false,
                    xlabel={Number of BP iterations},
                    % ylabel={Per-round-LER},
                    yticklabels={\empty},
                    extra description/.code={
                        \node[rotate=90, anchor=south]
                        at ([xshift=10mm]current axis.east)
                        {Warm s. (---), Cold s. (- - -)};
                    },
                ]

                \foreach \F/\col/\mark in
                {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, densely dashed,
forget plot, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoder/p_0.0025/pass_soft_info_False/F_\F/W_5/LERs.csv};
                    }
                    \temp
                }

                \foreach \F/\col/\mark in
                {3/KITred/triangle*,2/KITblue/diamond*,1/KITorange/square*} {
                    \edef\temp{\noexpand
                        \addplot+[mark=\mark, solid, mark
                        options={fill=\col}, \col]
                        table[
                            col sep=comma, x=max_iter,
                            y=LER_per_round,
                        ]
                        {res/sim/max_iter/WindowingSyndromeSpaGdDecoder/p_0.0025/pass_soft_info_True/F_\F/W_5/LERs.csv};
                    }
                    \temp

                    \addlegendentryexpanded{$F = \F$}
                }
            \end{axis}
        \end{tikzpicture}

        \caption{
            Comparison of step sizes for $W=5$.
        }
    \end{subfigure}

    \caption{
        Comparison of the decoding performance of cold and warm-start
        decoding for the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
        under circuit-level noise.
        Decoding was performed using the \ac{bpgd} algorithm with
        $T=1$.
        The number of iterations refers to the outer \ac{bpgd}
        iterations, i.e., the number of decimations.
        The information used for the warm-start intialization
        included only the messages on the Tanner graph.
        $12$ rounds of syndrome extraction were performed and
        standard circuit-based depolarizing noise was chosen as the
noise model.
        The physical error probabilty was fixed to $0.0025$.
    }
\end{figure}


