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Write numerical results intro

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Andreas Tsouchlos
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src/thesis/chapters/4_decoding_under_dems.tex
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@@ -628,12 +628,13 @@ and after decoding all windows we will therefore have committed all \acp{vn}.
\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)},
$=
\left(\bm{H}_\text{DEM}\right)_{\mathcal{J}_\text{overlap}^{(\ell)},
\mathcal{I}_\text{commit}^{(\ell)}}$%
};
\end{tikzpicture}
\caption{Visual representation of notation for window splitting.}
\caption{Visual representation of notation used for window splitting.}
\label{fig:vis_rep}
\end{figure}
@@ -669,25 +670,27 @@ estimates commited after decoding window $\ell$, we have to set
% Intro
\content{Change view from PCM to Tanner graph}
\content{Call attention to SC-LDPC-like structure}
\content{High-level overview of modification}
The sliding-window structure visible in \Cref{fig:windowing_pcm} is
highly reminicent of the way \ac{sc}-\ac{ldpc} codes are decoded.
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.
% Warm-Start decoding for BP
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{Pass messages to next window}
\content{(?) Explicitly mention initialization using only CN->VN
messages + swapping of CN and VN update?}
\content{(?) Algorithm}
% Warm-Start decoding for BPGD
\content{Modified structure of BPGD $\rightarrow$ In addition to
messages, pass decimation info}
\content{(?) Explicitly mention decimation info = channel llrs?}
\content{(?) Algorithm}
\begin{figure}[H]
\begin{figure}[t]
\centering
\tikzset{
@@ -789,7 +792,18 @@ messages, pass decimation info}
\label{fig:windowing_tanner}
\end{figure}
\begin{figure}[H]
%%%%%%%%%%%%%%%%
\subsection{Warm-Start Belief Propagation Decoding}
\label{subsec:Warm-Start Belief Propagation Decoding}
% Warm-Start decoding for BP
\content{Pass messages to next window}
\content{(?) Explicitly mention initialization using only CN->VN
messages + swapping of CN and VN update?}
\content{(?) Algorithm}
\begin{figure}[t]
\centering
\tikzset{
@@ -911,7 +925,18 @@ messages, pass decimation info}
\label{fig:messages_tanner}
\end{figure}
\begin{figure}[H]
%%%%%%%%%%%%%%%%
\subsection{Warm-Start Belief Propagation with Guided Decimation Decoding}
\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{
@@ -1045,16 +1070,122 @@ messages, pass decimation info}
\label{fig:messages_decimation_tanner}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical results}
\label{sec:warm_start_bpgd}
\section{Numerical Results}
\label{sec:Numerical Results}
% Intro
\content{Some info on used code (what it is, why it was chosen)}
\content{Some info on simulation setup (Stim, circuit-level noise,
standard circuit-based depolarizing noise model, etc.)}
\content{All datapoints generated with at least 100 error frames}
In this section, we perform numerical experiments to evaluate the
modification to sliding-window decoding we introduced in
\Cref{sec:warm_start_bp}.
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 \cite{gong_toward_2024}.
We employ standard circuit-based depolarizing noise as described in
\Cref{subsec:Choice of Noise Model}, and report performance in terms
of the per-round \ac{ler} as defined in
\Cref{subsec:Per-Round Logical Error Rate}.
All datapoints have been generated by simulating at least $200$
logical error events.
For the practical aspects of implementation, several layers of
abstraction must be considered.
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.
Moving one level of abstraction higher, the syndrome extraction
circuit itself must be generated.
This entails defining the gate schedule for the ancilla measurements
and specifying the error locations introduced by the chosen noise
model, both of which depend on the code and noise model in question.
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.
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.
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.
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{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}.
% We chose to carry out our simulations on \ac{bb} codes, as they
% \red{[something about qldpc codes being a hot topic in the literature
% currently because of some promising properties]}.
% Specifically, we chose the $\llbracket 144, 12, 12 \rrbracket$ BB
% code, as this \red{[something something]}.\\
% \red{[Circuit-level noise]} \\
% \red{[Per-round LER]} \\
% All datapoints have been generated by simulating at least $200$
% logical error events.
%
% For the practical aspects of implementation, several layers of
% abstraction must be considered.
% The lowest layer is the circuit-level simulator.
% This serves as the backbone of all further simulations.
% It takes care of the quantum mechanical aspects of the system.
% \red{It is, for example, responsible for the introduction of noise
% [rephrase this]}.
%
% Moving one level of abstraction higher, aside from the circuit
% simulation itself, the circuit also has to be generated.
% E.g., the syndrome extraction circuitry must be defined and possible
% sources of noise must be modeled.
% This heavily depends on the code in question and the chosen noise model.
% \red{[Find something more to say]}
%
% Even further up, we have already defined syndrome extraction
% circuitry and built the \acf{dem}.
% We must now split the detector error matrix into separate windows and
% manage the interplay of the inner decoders acting on the individual
% windows themselves.
% \red{[Find something more to say]}
%
% Finally, we require the decoder itself.
% This simply gets a \acf{pcm} and a syndrome with no regard
% \red{[Rephrase this] for the complexity in the rest of the system}.
%
% In our implementations, Stim \cite{gidney_stim_2021} served as the
% circuit-level simulator \red{[Possibly mention why stim was chosen]}.
% For the circuit generation, we employed utilities from QUITS
% \cite{kang_quits_2025}, where syndrome extraction circuitry
% generation is implemented for a number of different \ac{qldpc} codes.
% An initial Python implementation 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.
%%%%%%%%%%%%%%%%
\subsection{Belief Propagation}