% 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
comparable thresholds with substantially improved encoding rates
\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 in the presence 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.
Together, these two aspects lead to substantial convergence problems
of \ac{bp} for quantum codes, when it is used on its 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 measurements,
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} grew from $144$ to $9504$, and the
number of \acfp{cn} grew 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}, where 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 received less attention.
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 a bit 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)
Our own work will focus mostly on the the solution of the second
problem using sliding-window decoding.
We will start by briefly reviewing the existing work related to
sliding-window decoding,
before focusing on one specific realization.
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 to 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{Review of Existing Literature}
\label{subsec:Review of Existing Literature}

\begin{figure}[t]
    \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,
        }
    }

    \tikzexternaldisable
    \begin{tikzpicture}[node distance = 0mm and 0mm]
        % tex-fmt: off
        \node[heading, minimum width=15mm, fill=gray!25]                  (code)  {Code};
        \node[heading, below right=2mm and -5mm of code, fill=orange!20]  (top)   {Topological};
        \node[heading, below right=45mm and -5mm of code, fill=orange!20] (qldpc) {QLDPC};

        \node[literature, below right=1mm 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=1mm and -12mm of qldpc] (huang) {\cite{huang_improved_2023},\cite{huang_increasing_2024}};
        \node[literature, below=of huang]                     (gong)  {\cite{gong_toward_2024}};
        \node[literature, below=of gong]                      (kang)  {\cite{kang_quits_2025}};

        \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 (qldpc-anchor) |- (kang);

        \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) -- (kang.south east)
        node[midway,right,xshift=10mm]{Sequential};
        % tex-fmt: on
    \end{tikzpicture}
    \tikzexternalenable

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

% Some general notes

\Cref{fig:literature} gives an overview over the existing body of work
related to sliding-window decoding.
The papers \cite{huang_improved_2023} and \cite{huang_increasing_2024} are
lumped together, as they share the same content;
one is simply a preprint published earlier.
We will only refer to \cite{huang_increasing_2024} in the following.
\cite{kang_quits_2025} is somewhat special in that the authors focus
more on the introduction of a new simulator framework they call
QUITS, rather than the performance of sliding-window decoding itself.
\cite{gong_toward_2024} and \cite{kang_quits_2025} have made their
software freely available online%
\footnote{
    \url{https://github.com/mkangquantum/quits}
}%
\footnote{
    \url{https://github.com/gongaa/SlidingWindowDecoder}
}.
A final thing to note is that \cite{dennis_topological_2002} never
explicitly mentions sliding windows; the authors call their scheme
``overlapping recovery''.

% Topological vs QLDPC

Research has focused on two categories of \ac{qec} codes, topological
and \ac{qldpc} codes.
Most of the work on topological codes has treated surface codes,
with the exception of \cite{kuo_fault-tolerant_2024} where toric
codes were considered.
With regard to \ac{qldpc} codes, in \cite{huang_increasing_2024}
the authors examine \emph{hypergraph product} (\acs{hgp}) and
\emph{lifted-product} (\acs{lp}) codes.
HGP codes are constructed from the product of two classical codes,
while LP codes generalize this construction by additionally applying
a lift to reduce the qubit overhead.
In \cite{kang_quits_2025}, \emph{balanced product codes} (\acs{bpc})
are additionally considered.
Like HGP codes, BPC codes are derived from a product construction,
but exploit an additional symmetry to yield fewer physical qubits for
the same code parameters.
Finally, \cite{gong_toward_2024} explores \ac{bb} codes.

% Sequential vs parallel

After having divided the whole circuit into separate windows, the question
arises of how exactly to realize the decoding.
There are two main approaches, with differing mechanisms of reducing
the latency.
Some papers decode the sliding windows in a parallel fashion.
The benefit in this case is
is that classical hardware can be utilized more effectively.
Others choose a sequential approach.
Here, decoding can start earlier, as there is no need to wait for the
syndrome measurements of all windows before beginning with the decoding.
With the exception of \cite{dennis_topological_2002}, literature
treating topological codes has mostly focused on parallel decoding
while literature treating \ac{qldpc} codes has wholly considered
sequential decoding.

% Deep-dive into QLDPC methods

For this work, the publications treating \ac{qldpc} codes are
especially interesting.
The experimental conditions for these are summarized in
\Cref{table:experimental_conditions}.
As we noted above, \ac{hgp} and \ac{lp} codes are considered in
\cite{huang_increasing_2024},
\ac{hgp}, \ac{lp} and \ac{bpc} codes are considered in \cite{kang_quits_2025},
and \ac{bb} codes are considered in \cite{gong_toward_2024}.
The employed noise models also differ;
\cite{huang_increasing_2024} uses phenomenological noise, while
\cite{gong_toward_2024} and \cite{kang_quits_2025} use circuit-level noise.
Finally, in \cite{gong_toward_2024} the authors introduce their own variation of
\ac{bpgd}, \ac{bp} with \ac{gdg}, while \cite{huang_increasing_2024}
and \cite{kang_quits_2025} use \ac{bp} + \ac{osd}.
We would additionally like to note that only in
\cite{gong_toward_2024} and \cite{kang_quits_2025}
explicitly work with the \ac{dem} formalism.

\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{12pt}
\begin{table}[t]
    \centering
    \caption{Experimental conditions in the literature on
    sliding-window decoding for \ac{qldpc} codes.}
    \vspace*{3mm}
    \label{table:experimental_conditions}
    \begin{tabular}{l|ccc}
        % tex-fmt: off
        Publication                                                            &    Code                       & Noise Model                     & Decoder \\ \hline
        \hspace{-2.5mm}\cite{huang_improved_2023},\cite{huang_increasing_2024} & \acs{hgp}, \acs{lp}           & Phenomenological noise & \acs{bp} + \acs{osd} \\
        \hspace{-2.5mm}\cite{gong_toward_2024}                                 & \acs{bb}                      & Circuit-level noise             & \acs{bp} + \acs{gdg} \\
        \hspace{-2.5mm}\cite{kang_quits_2025}                                  & \acs{hgp}, \acs{lp}, \acs{bpc}  & Circuit-level noise             & \acs{bp} + \ac{osd}
        % tex-fmt: on
    \end{tabular}
\end{table}

% \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}
% }

%%%%%%%%%%%%%%%%
\subsection{Window Splitting and Sequential Sliding-Window Decoding}
\label{subsec:Window Splitting and Sequential Sliding-Window Decoding}

In this section, we will examine the methodology by which a detector
error matrix is divided into overlapping windows.
The algorithm detailed here follows \cite{kang_quits_2025}, which
is in turn based on \cite{huang_increasing_2024}.

% Very high-level overview

Sliding-window decoding is made possible by the time-like structure
of the syndrome extraction circuitry.
This is especially clearly visible under the \ac{dem} formalism, where
this manifests as a block-diagonal structure of the detector
error matrix $\bm{H}$.
Note that this presupposes a choice of detectors as seen in
\Cref{subsec:Detector Error Matrix}.
This block-diagonal structure introduces some locality in the
interdependence between \acp{vn} and \acp{cn}.
For each local set of \acp{vn}, there is only a local set of connected \acp{cn}.
We exploit this fact by partitioning the matrix into overlapping windows.
\Cref{fig:windowing_pcm} depicts this process using the $\llbracket
72, 6, 6 \rrbracket$ BB code as an example.

% High-level overview

How the locality is leveraged can be understood by considering the
decoding process.
After decoding a window, there is a subset of \acp{cn} that
no longer contribute to decoding, since none of their
neighboring \acp{vn} appear in subsequent windows.
We call the set of \acp{vn} connected to those \acp{cn} the
\emph{commit region} and we wish to commit them before moving to the
next window, i.e., fix the values we estimate for the corresponding bits.
As mentioned above, the benefit of this sequential sliding-window
decoding approach
is that the decoding process can begin as soon as the syndrome
measurements for the first window are complete.

% W and F and why we look at rows, not columns

There are two degrees of freedom in how we perform the windowing.
The \emph{window size} $W \in \mathbb{N}$ represents the number of
syndrome extraction rounds lumped into one window, while
the \emph{step size} $F \in \mathbb{N}$ represents the number of
syndrome extraction rounds skipped before starting the next window.
$W$ controls the size of the windows while $F$ controls the overlap
between them.
As illustrated in \Cref{fig:windowing_pcm}, $W$ and $F$ control the
window dimensions and locations by defining the related \acp{cn},
not the \acp{vn}.
This is because while the number of overall \acp{cn} is only affected
by the choice of the underlying code and the number of syndrome
measurement rounds, the number of \acp{vn} depends on the noise model
and is difficult to predict beforehand.

\begin{figure}[t]
    \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 a detector error
        matrix generated from the $\llbracket 72, 6, 6 \rrbracket$
        BB code under circuit-level noise.
        The block-diagonal structure reflects the time-like locality
        of the syndrome extraction circuit., with each block
        corresponding to one syndrome measurement round.
        Two consecutive windows are highlighted: the window size $W$
        controls the number of syndrome rounds included in each
        window, while the step size $F$ controls how many rounds
        separate the start of one window from the next.
        The bracketed region indicates the commit
        region of the first window, i.e., the \acp{vn} that are committed
        before moving to the second window.
        % Visualization of the windowing process on a detector
        % error matrix generated from the $\llbracket 72, 6, 6
        % \rrbracket$ BB code.
    }
    \label{fig:windowing_pcm}
\end{figure}

% Notation recap

We briefly reintroduce the notation important for the definition of the windows.
We use the variables $n,m \in \mathbb{N}$ to describe the number of
\acp{vn} and \acp{cn} respectively.
We index the \acp{vn} using the variable $i \in \mathcal{I} :=
[0:n-1]$ and the \acp{cn} using the variable $j \in \mathcal{J} := [ 0 : m-1]$.
Finally, we call $\mathcal{N}_\text{V}(i) = \left\{ j\in \mathcal{J}:
\bm{H}_{j,i} = 1 \right\}$ and $\mathcal{N}_\text{C}(j) := \left\{ i
\in \mathcal{I} : \bm{H}_{j,i} = 1 \right\}$ the neighborhoods of the
corresponding nodes.
In this case, we take $\bm{H} \in \mathbb{F}_2^{m\times n}$ to be the
check matrix of the underlying code, from which the \ac{dem} was generated.
We use $m_\text{DEM}, \mathcal{I}_\text{DEM}$, and $\mathcal{J}_\text{DEM}$
to refer to the respective values defined from the detector error matrix.

% How we get the corresponding rows

We begin by describing the sets of \acp{cn} relevant to each window.
For indexing, we use the variable $\ell \in [0:n_\text{win} - 1]$,
where $n_\text{win} \in \mathbb{N}$ is the number of windows.
Because we defined the step size $F$ as the number of syndrome
extraction rounds to skip, the first \ac{cn} of window $\ell$ should have index
$\ell F m$.
Similarly, because of the way we defined the window size $W$, the
number of \acp{cn} should be $Wm$ for all but the last window.
The number of \acp{cn} in the last window may differ if there are
not enough \acp{cn} left to completely fill it.
We thus define
\begin{align*}
    \mathcal{J}_\text{win}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~
        \ell F m \le j < \min \left\{m_\text{DEM}, (\ell F + W) m \right\}
    \right\} \\[2mm]
    & \hspace{30mm} \text{and} \\[2mm]
    \mathcal{J}_\text{commit}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~
        \ell F m \le j < \min \left\{m_\text{DEM}, (\ell + 1) F m \right\}
    \right\}
    .%
\end{align*}
$\mathcal{J}_\text{win}^{(\ell)}$ is the set of all \acp{cn} in the
window while $\mathcal{J}_\text{commit}^{(\ell)}$ is the set of \acp{cn}
that do not contribute to the next window and whose neighboring
\acp{vn} will thus be committed.
We can additionally define the set of \acp{cn} that are shared between windows
$\ell$ and $\ell + 1$ as $\mathcal{J}_\text{overlap}^{(\ell)} :=
\mathcal{J}_\text{win}^{(\ell)}\setminus \mathcal{J}_\text{commit}^{(\ell)}$.

% How we get the corresponding columns

We can now turn our attention to defining the sets of \acp{vn} relevant
to each window.
We first introduce a helper function $i_\text{max} :
\mathcal{P}(\mathbb{N}) \to \mathbb{N}$, which takes a set of
\ac{cn} indices and returns the largest neighboring \ac{vn} index.
We define
\begin{align*}
    i_\text{max}\left( \mathcal{S} \right) := \max \left\{ i\in
    \mathcal{N}_\text{C}(j) : j\in \mathcal{S} \right\}
    ,
\end{align*}
where we set $i_\text{max} (\emptyset) = -1$ by convention%
\footnote{
    This has the effect of later automatically setting the lower
    bounds for the indices in $\mathcal{I}_\text{commit}^{(\ell)}$
    and $\mathcal{I}_\text{win}^{(\ell)}$ appropriately.
}%
.
The commit region of window $\ell$ should include all of the \acp{vn}
neighboring any of the \acp{cn} in $\mathcal{J}_\text{commit}^{(\ell)}$.
Consequently, the maximum index of the \acp{vn} we consider should be
$i_\text{max}(\mathcal{J}_\text{commit}^{(\ell)})$.
Additionally, the set of \acp{vn} committed in the next window should
start immediately afterwards.
We thus define
\begin{align*}
    \mathcal{I}_\text{commit}^{(\ell)}
    &:= \left\{i \in \mathcal{I}_\text{DEM} :~
        i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
        < i \le
        i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell)} \right)
    \right\}\\[2mm]
    & \hspace{39mm} \text{and} \\[2mm]
    \mathcal{I}_\text{win}^{(\ell)}
    &:= \left\{i \in \mathcal{I}_\text{DEM} :~
        i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
        < i \le
        i_\text{max}\left( \mathcal{J}_\text{win}^{(\ell)} \right)
    \right\}
    .%
\end{align*}
Note that we have
\begin{align*}
    \bigcup_{\ell=0}^{n_\text{win}-1}
    \mathcal{I}_\text{commit}^{(\ell)} = \mathcal{I}
\end{align*}
and after decoding all windows we will therefore have committed all \acp{vn}.

\begin{figure}[t]
    \centering

    \begin{tikzpicture}
        \def\sx{1.5}
        \def\sy{1.5}

        \coordinate (a00) at (0,0);
        \coordinate (a01) at (0,      3*\sy);
        \coordinate (a11) at (6*\sx,  3*\sy);
        \coordinate (a10) at (6*\sx,  0*\sy);

        \coordinate (b00) at (3.2*\sx, -1*\sy);
        \coordinate (b01) at (3.2*\sx,  2*\sy);
        \coordinate (b11) at (9.2*\sx,  2*\sy);
        \coordinate (b10) at (9.2*\sx, -1*\sy);

        \fill[gray!40] (a00) -- (a00 |- b01) -- (b01) -- (b01 |- a00) -- cycle;

        \draw (a00) -- (a01) -- (a11) -- (a10) -- cycle;
        \draw[densely dashed] (b00) -- (b01) -- (b11) -- (b10) -- cycle;

        \draw [
            decorate,
            decoration={brace,amplitude=3mm,raise=1mm}
        ]
        (a01) -- (a11)
        node[midway,above,yshift=4mm]{$\mathcal{I}_\text{win}^{(\ell)}$};

        \draw [
            decorate,
            decoration={brace,amplitude=3mm,raise=1mm}
        ]
        (a00 -| b00) -- (a00)
        node[midway,below,yshift=-4mm]{$\mathcal{I}_\text{commit}^{(\ell)}$};

        \draw [
            decorate,
            decoration={brace,amplitude=3mm,raise=1mm}
        ]
        (a00) -- (a01)
        node[midway,xshift=-3mm,left]{$\mathcal{J}_\text{win}^{(\ell)}$};

        \draw [
            decorate,
            decoration={brace,amplitude=3mm,raise=1mm}
        ]
        (a11) -- (a11 |- b11)
        node[midway,xshift=3mm,right]{$\mathcal{J}_\text{commit}^{(\ell)}$};

        \draw [
            decorate,
            decoration={brace,amplitude=3mm,raise=1mm}
        ]
        (a11 |- b11) -- (a10)
        node[midway,xshift=3mm,right]{$\mathcal{J}_\text{overlap}^{(\ell)}
            := \mathcal{J}_\text{win}^{(\ell)} \setminus
        \mathcal{J}_\text{commit}^{(\ell)}$};

        \node[align=center] at ($(a00)!0.5!(b01)$)
        {%
            $\bm{H}_\text{overlap}^{(\ell)}$ \\[3mm]
            $=
            \left(\bm{H}_\text{DEM}\right)_{\mathcal{J}_\text{overlap}^{(\ell)},
            \mathcal{I}_\text{commit}^{(\ell)}}$%
        };
    \end{tikzpicture}

    \caption{
        Visual representation of the index sets used to define a sliding window.
        The solid box delimits the rows ($\mathcal{J}_\text{win}^{(\ell)}$)
        and columns ($\mathcal{I}_\text{win}^{(\ell)}$) of the detector
        error matrix considered when decoding window $\ell$, while the
        dashed box shows the analogous region for window $\ell + 1$.
        The shaded region marks the submatrix
        $\bm{H}_\text{overlap}^{(\ell)}$, whose rows correspond to the
        overlap CNs $\mathcal{J}_\text{overlap}^{(\ell)} =
        \mathcal{J}_\text{win}^{(\ell)} \setminus
        \mathcal{J}_\text{commit}^{(\ell)}$ shared with the next window,
        and whose columns correspond to the committed VNs
        $\mathcal{I}_\text{commit}^{(\ell)}$.
        After decoding window $\ell$, this submatrix is used to update
        the syndrome of the overlap CNs based on the committed bit estimates.
    }
    \label{fig:vis_rep}
\end{figure}

% Syndrome update

\Cref{fig:vis_rep} illustrates the meaning of the various sets of nodes.
We can also see a subtlety we must handle carefully when
moving on to decode the next window.
While the \acp{vn} in $\mathcal{J}_\text{commit}^{(\ell)}$ have no
bearing on the further decoding process, the values commited for the
\acp{vn} in $\mathcal{I}_\text{commit}^{(\ell)}$ do.
This is the case because these \acp{vn} have neighboring \acp{cn} in
the next window.
The part of the detector error matrix $\bm{H}_\text{DEM}$ describing
these connections is
$\bm{H}_\text{overlap}^{(\ell)} =
\left(\bm{H}_\text{DEM}\right)_{\mathcal{J}_\text{overlap}^{(\ell)},
\mathcal{I}_\text{commit}^{(\ell)}}$.
We have to account for this fact by updating the syndrome $\bm{s}$
based on the committed bit values.
Specifically, if $\bm{e}_\text{commit}^{(\ell)}$ describes the error
estimates committed after decoding window $\ell$, we have to set
\begin{align*}
    \bm{s}_{\mathcal{J}_\text{overlap}^{(\ell)}} =
    \bm{H}_\text{overlap}^{(\ell)}
    \left( \bm{e}_\text{commit}^{(\ell)} \right)^\text{T}
    .%
\end{align*}

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

% Intro

The sliding-window structure visible in \Cref{fig:windowing_pcm} is
highly reminiscent of windowed decoding for \ac{sc}-\ac{ldpc} codes.
Switching our viewpoint to the Tanner graph depicted in
\Cref{fig:windowing_tanner}, however, we can see an important
difference between \ac{sc}-\ac{ldpc} decoding and the
sliding-window decoding procedure detailed above.
While the windowing process is similar, the algorithm above
reinitializes the decoder to start from a clean state when moving to
the next window.
It therefore does not make use of the integral property of
\ac{sc}-\ac{ldpc} decoding of exploiting the spatially coupled
structure by passing soft information from earlier to later spatial positions.

We propose a modification to the procedure detailed in
\Cref{subsec:Window Splitting and Sequential Sliding-Window Decoding}:
Instead of zero-initializing the \ac{bp} messages of the next window,
we perform a \emph{warm start} by initializing the messages in the
overlapping region to the values last held during the decoding of the
previous window.

\content{callback to intro: explain why we consider BPGD instead of,
e.g., BP+OSD}
\content{Explain why we expect a warm start to be beneficial}
\content{Mention that our own work ties into the bottom category in
\Cref{fig:literature}}
\content{Explicitly state that $\mathcal{I}_\text{win}^{(\ell)}$
    overlaps with $\mathcal{I}_\text{win}^{(\ell + 1)}$, and that this is
where the warm start applies}

\begin{figure}[t]
    \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}

%%%%%%%%%%%%%%%%
\subsection{Belief Propagation}
\label{subsec:Warm-Start Belief Propagation}

% Warm-Start decoding for BP

\content{Explicitly name messages passed (${L_{j\leftarrow i} : i \in
\ldots, j\in \ldots}$)}
\content{Pass messages to next window}
\content{(?) Explicitly mention initialization using only CN->VN
messages + swapping of CN and VN update?}
\content{(?) Algorithm}

\begin{figure}[t]
    \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}

%%%%%%%%%%%%%%%%
\subsection{Belief Propagation with Guided Decimation}
\label{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}

% 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}[t]
    \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:Numerical Results}

% Intro

In this section, we perform numerical experiments to evaluate the
modification to sliding-window decoding we introduced in
\Cref{sec:warm_start_bp}.
For the practical aspects of implementation, several layers of
abstraction must be considered.

% Software stack: Layer 1

The lowest layer is the circuit-level simulator.
This serves as the backbone of all further simulations, handling the
quantum mechanical aspects of the system, including the modeling of
noise on gates, idling qubits, and measurements according to the
chosen noise model.

% Software stack: Layer 2

Moving one level of abstraction higher, the syndrome extraction
circuit itself must be generated.
This entails constructing the full circuit, including the ancilla
measurements and the error locations introduced by the chosen noise
model, both of which depend on the code and noise model in question.

% Software stack: Layer 3

Even further up, given an already constructed syndrome extraction
circuit and the resulting \acf{dem}, we must split the detector error
matrix into separate windows and manage the interplay between the
inner decoders acting on those individual windows.

% Software stack: Layer 4

Finally, we require the decoder itself, which operates on a
\acf{pcm} and a syndrome, with no dependence on the complexity of the
layers below.

% Software stack: Tools

In our implementation, Stim \cite{gidney_stim_2021} served as the
circuit-level simulator, chosen for its efficiency and native support
for the \ac{dem} formalism.
For the circuit generation, we employed utilities from QUITS
\cite{kang_quits_2025}, which provides syndrome extraction circuitry
generation for a number of different \ac{qldpc} codes.
We initially created a Python implementation, which used QUITS for the window
splitting and subsequent sliding-window decoding as well.
The \ac{bp} and \ac{bpgd} decoders were also initially implemented in Python.
After a preliminary investigation, we opted for a complete
reimplementation in Rust to achieve higher simulation speeds due to
the compiled nature of the language.
We reimplemented both the window splitting and the decoders themselves.

% Global experimental setup
%   - Code
%   - # SE rounds
%   - Noise model
%   - Per-round LER as figure of merit
%   - Detector definition
%   - # simulated error frames

We chose to carry out our simulations on \ac{bb} codes, as they have
recently emerged as particularly promising candidates for practical
\ac{qec}, offering high encoding rates and large minimum distances
while admitting short-depth syndrome extraction circuits
\cite[Sec.~1]{bravyi_high-threshold_2024}.
Specifically, we chose the $\llbracket 144, 12, 12 \rrbracket$ BB
code, as it represents a good trade-off between code size and
simulation tractability.
For the generation of the \ac{dem} we set the number of syndrome
extraction rounds to $12$, similarly to \cite{gong_toward_2024}, and
we defined our detectors as in the example in
\Cref{subsec:Detector Error Matrix}.
We employed circuit-lose noise as described in
\Cref{subsec:Choice of Noise Model} as our noise model, specifically standard
ciruit-based depolarizing noise \cite[Sec.~VIII]{fowler_high-threshold_2009},
i.e., all error locations in the circuit get assigned the same
physical error probability.
We report performance in terms of the per-round \ac{ler} as defined
in \Cref{subsec:Per-Round Logical Error Rate} and all datapoints were
generated by simulating at least $200$ logical error events.

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

% Local experimental setup
%   - BP variant

We began our investigation by using \ac{bp} with no further
modifications as the inner decoder.
We chose the min-sum variant of \ac{bp} due to its low computational complexity.

% [Thread] Get impression for max gain
%   - More global = better -> Compare windowed vs. whole

% [Description] Figure 4.8
%   - Parameters
%     - # BP iterations
%     - W,F
%     - Physical error rates
%     - Windowed (cold start) vs whole decoding
%     - (?) Semilog y axis
%   - Figure description
%     - TODO:

% [Interpretation] Figure 4.8
%   - Larger window -> better, because more global decoding
%   - Diminishing returns as the window becomes larger
%   - As expected, whole works best

% [Thread] First comparison with warm start
%    - Compare performance of warm start to cold start

% [Description] Figure 4.9
%   - Parameters
%     - # BP iterations
%     - W,F
%     - Physical error rates
%     - Warm vs cold start
%   - Figure description
%     - TODO:

% [Interpretation] Figure 4.9
%   - Generally better performance with warm start, as expected
%   - It is surprising that warm start performs better than whole

% [Thread] Warm start is better than whole due to more effective iterations

% [Description] Figure 4.10
%   - Parameters
%     - # BP iterations
%     - W,F
%     - Physical error rates
%     - Warm vs cold start
%   - Figure description
%     - TODO:

% [Interpretation] Figure 4.10
%   -

We initially wanted to gain an impression for the performance gain we could
expect from a modification to the sliding-window decoding procedure.
To this end, we began by analyzing the decoding performance of the
original process, without our warm-start modification.
We will call this \emph{cold-start} decoding in the following.
We examined the decoding performance for different window sizes $W$
and compared it against the performance when decoding on the whole
detector error matrix at once, i.e., without windowing.

\Cref{fig:whole_vs_cold} depicts the results of this analysis.
\red{[Write more about the experimental setup (200 BP iterations,
fixed step size, what else?)]}
\red{[Describe the plot (whole decoding in black, (?) list different
window sizes and colors/markers, what else?)]}
We can see that a larger window results in a lower overall error rate.
This seems sensible, because the lower the window size, the more
locally the decoding is performed.
While this allows us to leverage the time-like structure of the
circuitry more strongly and further reduce the latency, it is
expected to lower the performance, since \red{[find something to say here]}.
Decoding the whole detector error matrix globally with no windowing
provides the best performance.

\begin{figure}[t]
    \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.
    }
    \label{fig:whole_vs_cold}
\end{figure}

% Initial results of warm-start decoding

As a next step, we additionally generated error rate curves using our
warm-start modification.
\red{[Again 200 BP iterations, etc.]}
\Cref{fig:whole_vs_cold_vs_warm} shows the numerical results from
this experiment.
The cold-start results from the previous graph are now plotted in
dashed lines, while the new warm-start results are plotted with solid lines.
We can see that the decoding performance has been improved overall.

% Unexpected: Warm-start better than whole

Additionally, we can see some initially unexpected behavior: The warm-start
sliding window decoding with $W=5$ performs better than decoding
under consideration of the whole detector error matrix at once, even
though the process is less global.

\begin{figure}[t]
    \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.
    }
    \label{fig:whole_vs_cold_vs_warm}
\end{figure}

\begin{figure}[t]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
                width=\figwidth,
                height=\figheight,
                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{$W = \W$}
            }

            \addplot+[mark=*, solid, mark options={fill=black}, black]
            table[
                col sep=comma, x=max_iter,
                y=LER_per_round,
            ]
            {res/sim/max_iter/SyndromeMinSumDecoder/p_0.0025/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 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}[t]
    \centering
    \begin{subfigure}{0.48\textwidth}
        \centering
        \hspace*{-7mm}
        \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 \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 window sizes for $F=1$.}
    \end{subfigure}%
    \hfill%
    \begin{subfigure}{0.48\textwidth}
        \centering
        \hspace*{-3mm}
        \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}
        \vspace{-3.2mm}

        \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}[t]
    \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}[t]
    \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}[t]
    \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}[t]
    \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}

