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Complete text for first two figures

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Andreas Tsouchlos
2026-05-02 14:38:50 +02:00
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src/thesis/chapters/4_decoding_under_dems.tex
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@@ -1201,82 +1201,23 @@ generated by simulating at least $200$ logical error events.
\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.
Because we expected more global decoding to work better (the inner
decoder then has access to a larger portion of the long-range
correlations encoded in the detector error matrix before any commit
is made) we initially decided to use decoding on the whole detector
error matrix as a proxy for the attainable decoding performance.
\begin{figure}[t]
\centering
@@ -1329,33 +1270,64 @@ provides the best performance.
\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.
\red{\lipsum[2]}
}
\label{fig:whole_vs_cold}
\end{figure}
% Initial results of warm-start decoding
% [Experimental parameters] Figure 4.7
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.
\Cref{fig:whole_vs_cold} shows the simulation results for this initial
investigation.
The three colored curves correspond to cold-start sliding-window
decoding with window sizes $W \in \{3, 4, 5\}$, all with the step
size fixed to $F = 1$, while the black curve gives the per-round
\ac{ler} obtained when decoding on the whole detector error matrix at once.
In all cases, the inner \ac{bp} decoder was allowed a maximum of
$200$ iterations, and the physical error rate was swept from
$p = 0.001$ to $p = 0.004$ in steps of $0.0005$.
% Unexpected: Warm-start better than whole
% [Description] Figure 4.7
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.
Across the entire range of physical error rates, all curves exhibit
the expected monotonic increase in logical error rate with increasing
physical noise.
The $W = 3$ decoder consistently yields the highest LER, performing
roughly an order of magnitude worse than the baseline at low physical
error rates.
Increasing the window size to $W = 4$ substantially closes this gap,
and the $W = 5$ curve nearly coincides with the whole-block decoder
across the full range of physical error rates.
% [Interpretation] Figure 4.7
This behavior is consistent with the intuition behind sliding-window decoding.
The detector error matrix encodes correlations between detection
events that span the full syndrome extraction history, so errors
lying in the commit region of an early window are in general
constrained by check nodes that only become visible in subsequent windows.
Larger windows expose the inner decoder to more of these constraints
before any commit is made, leading to better-informed decisions and a
lower per-round \ac{ler}.
Decoding the whole matrix at once represents the limiting case of
this trend and, as expected, achieves the strongest performance.
The fact that the $W = 5$ curve is already very close to the
whole-block decoder indicates that the marginal benefit of enlarging
the window saturates after a certain point.
From a practical standpoint, the choice of $W$ thus represents a
trade-off between decoding latency and accuracy: larger windows
delay the start of decoding by requiring more syndrome extraction
rounds to be collected upfront, while the diminishing returns above
$W = 4$ suggest that growing the window much further yields little
additional accuracy in return.
% [Thread] First comparison with warm start
Next, we additionally generated error rate curves for warm-start
sliding-window decoding to assess how much of the gap between
cold-start and whole-block decoding can be recovered by our modification.
We chose the same window sizes as before, so that the warm- and
cold-start curves can be compared directly at matching values of $W$.
\begin{figure}[t]
\centering
@@ -1429,53 +1401,107 @@ though the process is less global.
\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.
\red{\lipsum[2]}
}
\label{fig:whole_vs_cold_vs_warm}
\end{figure}
% [Experimental parameters] Figure 4.8
\Cref{fig:whole_vs_cold_vs_warm} extends the previous comparison by
additionally including the warm-start variant of sliding-window decoding.
The dashed colored curves reproduce the cold-start results from
\Cref{fig:whole_vs_cold}, while the solid colored curves show the
corresponding warm-start runs for the same window sizes
$W \in \{3, 4, 5\}$.
The remaining experimental parameters are unchanged:
the step size is fixed to $F = 1$,
the inner \ac{bp} decoder is allowed up to $200$ iterations per
window invocation, the black curve again gives the whole-block
reference, and the physical error rate is swept from $p = 0.001$ to
$p = 0.004$ in steps of $0.0005$.
% [Description] Figure 4.8
For each window size, the warm-start variant consistently outperforms
its cold-start counterpart, with the dashed curves lying above the
corresponding solid curves across the entire range of physical error rates.
The performance gap between the two approaches is most pronounced for
the largest window ($W = 5$) and gradually narrows as the window size decreases.
Additionally, the gap between the cold- and warm-start curves
generally widens as the physical error rate decreases.
% [Interpretation] Figure 4.8
The improvement of warm-start over cold-start decoding matches the
motivation for the modification:
By reusing already existing messages from the previous window in the
overlap region, the next window invocation has additional information
at its disposal about the reliability of the \acp{vn} and \acp{cn}.
The widening of the gap towards larger window sizes is consistent
with this picture, since with $F$ fixed to $1$ the overlap between
consecutive windows spans $W - F = W - 1$ syndrome rounds, so larger
$W$ implies that more messages are carried over and a larger fraction
of the next window starts in a warm state.
% TODO: Possibly insert explanation for higher gain at lowre error rates
A perhaps surprising observation is that the warm-start curve for
$W = 5$ actually lies below the whole-block reference across the
entire range of physical error rates, even though warm-start
sliding-window decoding is, by construction, more local than
whole-block decoding.
A possible explanation for this effect is discussed in the following.
% [Thread] Warm start is better than whole due to more effective iterations
A possible explanation for this surprising behavior lies in the
number of \ac{bp} iterations effectively spent on the \acp{vn}
inside the overlap region.
Each \ac{vn} in such an overlap is processed by multiple consecutive
window invocations, and because every new window resumes from the
messages left over by its predecessor, these invocations effectively
accumulate iterations on the same \acp{vn} rather than restarting
from scratch.
The whole-block decoder, by contrast, performs only a single run of
at most $200$ iterations on the entire detector error matrix, so
each of its \acp{vn} receives at most that many iterations.
It seems this larger effective iteration budget on the overlap
regions can outweigh the loss of globality incurred by windowing.
A natural way to test this hypothesis is to raise the maximum number
of \ac{bp} iterations of the whole-block decoder until its per-round
\ac{ler} saturates.
If the above interpretation is correct, the resulting saturation
level should constitute a lower bound that no windowed scheme,
irrespective of the initialization, can beat, since by construction
whole-block decoding has access to the full set of constraints
available to any window.
% [Description] Figure 4.9
% - Parameters
% - # BP iterations
% - W,F
% - Physical error rates
% - Warm vs cold start
% - Figure description
% - TODO:
% [Interpretation] Figure 4.9
% -
% At some later point
\content{When looking at max iterations: Callback to diminishing
returns with growing window size: More iterations more beneficial
than larger window (+1 for warm-start)}
\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. (- - -)};
},
]
\def\spyxmin{32}
\def\spyxmax{512}
\def\spyymin{5e-3}
\def\spyymax{7e-2}
\newcommand{\plotcurves}{%
\foreach \W/\col/\mark in
{3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
\edef\temp{\noexpand
@@ -1489,12 +1515,11 @@ though the process is less global.
}
\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]
options={fill=\col}, \col, forget plot]
table[
col sep=comma, x=max_iter,
y=LER_per_round,
@@ -1502,33 +1527,176 @@ though the process is less global.
{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,
forget plot]
table[col sep=comma, x=max_iter, y=LER_per_round]
{res/sim/max_iter/SyndromeMinSumDecoder/p_0.0025/LERs.csv};
}
\addplot+[mark=*, solid, mark options={fill=black}, black]
table[
col sep=comma, x=max_iter,
y=LER_per_round,
\begin{axis}[
name=main,
width=\figwidth,
height=\figheight,
ymode=log,
enlargelimits=false,
ymin=1e-3, ymax=1e-1,
grid=both,
legend pos=north east,
xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
xticklabels={$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
xticklabel style={/pgf/number format/fixed},
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. (- - -)};
},
]
{res/sim/max_iter/SyndromeMinSumDecoder/p_0.0025/LERs.csv};
\plotcurves
\addlegendimage{KITred, mark=triangle*}
\addlegendentry{$W = 3$}
\addlegendimage{KITblue, mark=diamond*}
\addlegendentry{$W = 4$}
\addlegendimage{KITorange, mark=square*}
\addlegendentry{$W = 5$}
\addlegendimage{black, mark=*}
\addlegendentry{Whole}
\node[draw=black, fit={(axis cs:\spyxmin,\spyymin) (axis
cs:\spyxmax,\spyymax)}, inner sep=0pt, name=spybox] {};
\end{axis}
\begin{axis}[
name=inset,
at={(main.north)},
anchor=south,
xshift=0mm, yshift=6mm,
width=6.5cm, height=4.875cm,
ymode=log,
enlargelimits=false,
xmin=\spyxmin, xmax=\spyxmax,
ymin=\spyymin, ymax=\spyymax,
xtick={32,128,256, 512},
yticklabels={\empty},
xticklabels={\empty},
grid=both,
axis background/.style={fill=white},
]
\plotcurves
\end{axis}
\draw (spybox.north east) -- (inset.south west);
\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.
\red{\lipsum[2]}
}
\end{figure}
% \begin{figure}[t]
% \centering
% \begin{tikzpicture}[spy using outlines={circle, magnification=2,
% connect spies}]
%
% \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}
%
% \coordinate (spypoint) at (axis cs:250,1e-2);
% \coordinate (magnifyglass) at (axis cs:2048,0.5);
%
% \end{axis}
% \spy [black, size=4cm] on (spypoint)
% in node[fill=white] at (magnifyglass);
%
% \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}
@@ -1674,14 +1842,7 @@ though the process is less global.
\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 $0.0025$.
\red{\lipsum[2]}
}
\end{figure}
@@ -1830,16 +1991,7 @@ though the process is less global.
\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.
\red{\lipsum[2]}
}
\end{figure}
@@ -1990,19 +2142,7 @@ though the process is less global.
\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 $0.0025$.
\red{\lipsum[2]}
}
\end{figure}
@@ -2147,16 +2287,7 @@ though the process is less global.
\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.
\red{\lipsum[2]}
}
\end{figure}
@@ -2307,19 +2438,7 @@ though the process is less global.
\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 $0.0025$.
\red{\lipsum[2]}
}
\end{figure}

1
src/thesis/chapters/5_conclusion_and_outlook.tex
View File

@@ -4,4 +4,5 @@
\content{Softer way of decimating VNs}
\content{Systematic study on using different inner decoders (AED,
SED, BPGD, ...)}
\content{Investigate SC-LDPC window decoding wave-like effects}

1
src/thesis/main.tex
View File

@@ -28,6 +28,7 @@
\usepackage{nicematrix}
\usepackage{colortbl}
\usepackage{cleveref}
\usepackage{lipsum}
\usetikzlibrary{calc, positioning, arrows, fit}
\usetikzlibrary{external}