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Andreas Tsouchlos
2026-05-03 02:16:22 +02:00
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src/thesis/chapters/4_decoding_under_dems.tex
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@@ -1801,7 +1801,8 @@ previous experiments.
\end{axis}
\end{tikzpicture}
\caption{Comparison of window sizes for $F=1$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bp_f_over_p}
\end{subfigure}%
\hfill%
@@ -1909,7 +1910,8 @@ previous experiments.
\end{tikzpicture}
\vspace{-3.2mm}
\caption{Comparison of step sizes for $W=5$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bp_f_over_iter}
\end{subfigure}
@@ -2025,6 +2027,15 @@ This motivates the next subsection, in which we replace the inner
\subsection{Belief Propagation with Guided Decimation}
\label{subsec:Belief Propagation with Guided Decimation}
% [Thread] Intro to BPGD + Local experimental setup
We now turn to \ac{bpgd} as the inner decoder, in order to address
the convergence issues of plain \ac{bp} on \ac{qec} codes.
For the underlying \ac{bp} step we use the \ac{spa} variant rather
than the min-sum approximation employed in
\Cref{subsec:Belief Propagation}, since this made the implementation
of the guided decimation more straightforward.
\begin{figure}[t]
\centering
\hspace*{-6mm}
@@ -2053,7 +2064,7 @@ This motivates the next subsection, in which we replace the inner
ylabel={Per-round-LER},
% extra description/.code={
% \node[rotate=90, anchor=south]
% at ([xshift=10mm]current axis.east)
% at ()
% {Warm s. (---), Cold s. (- - -)};
% },
]
@@ -2093,7 +2104,9 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{Comparison of window sizes for $F=1$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bpgd_w}
\end{subfigure}%
\hfill%
\begin{subfigure}{0.5\textwidth}
@@ -2162,14 +2175,112 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{Comparison of step sizes for $W=5$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bpgd_f}
\end{subfigure}
\caption{
\red{\lipsum[2]}
}
\label{fig:bpgd_wf}
\end{figure}
% [Experimental parameters] Figure 4.10
\Cref{fig:bpgd_wf} shows the per-round \ac{ler} of \ac{bpgd}
sliding-window decoding as a function of the physical error rate.
In both panels the dashed curves correspond to cold-start
sliding-window decoding and the solid curves to the
corresponding warm-start decoding, where the warm start carries over both
the \ac{bp} messages and the decimation information of the overlap
region as described in
\Cref{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}.
The maximum number of inner \ac{bp} iterations was set to
$n_\text{iter} = 5000$.
This value was chosen to be at least as large as the number of
\acp{vn} in any single window, since with one \ac{bp} iteration
between consecutive decimations ($T = 1$ in the notation of
\Cref{alg:bpgd}) this is the maximum number of inner iterations
that can occur before every \ac{vn} in the window has been decimated.
A preliminary investigation showed that \ac{bpgd} only delivers its
intended performance gain once most \acp{vn} have actually been decimated,
which motivated this choice.
The physical error rate was swept from $p = 0.001$ to $p = 0.004$
in steps of $0.0005$.
\Cref{fig:bpgd_w} sweeps over the window size with
$W \in \{3, 4, 5\}$ at fixed step size $F = 1$, and
\Cref{fig:bpgd_f} sweeps over the step size with
$F \in \{1, 2, 3\}$ at fixed window size $W = 5$.
% [Description] Figure 4.10
In both panels, every curve again exhibits the expected monotonic
increase of the per-round \ac{ler} with the physical error rate.
Across both panels and across all parameter choices, the warm-start
curves lie above the corresponding cold-start curves, i.e.,
the warm-start variant performsworse than its cold-start counterpart.
This is the opposite of what we observed for plain \ac{bp}, where
warm-start improved upon cold-start at every parameter setting.
The gap between the warm- and cold-start curves additionally widens
as the physical error rate decreases:
at the lowest sampled rate $p = 0.001$, the per-round \ac{ler} of the
warm-start runs is more than two orders of magnitude above that of
the corresponding cold-start runs.
In \Cref{fig:bpgd_w}, larger window sizes yield lower per-round
\acp{ler} for both warm- and cold-start, and the spacing between the
cold-start curves shrinks as $W$ grows.
In \Cref{fig:bpgd_f}, the cold-start curves follow the previously seen
ordering with $F = 1$ at the bottom and $F = 3$ at the top.
The warm-start curves, however, exhibit the opposite ordering:
$F = 1$ now yields the highest per-round \ac{ler}, $F = 2$ lies below
it, and $F = 3$ is the lowest of the three warm-start curves.
% [Interpretation] Figure 4.10
The fact that warm-start sliding-window decoding now performs worse
than its cold-start counterpart is surprising in light of the results
for plain \ac{bp}, where the warm-start modification was uniformly beneficial.
The dependence on the window size in \Cref{fig:bpgd_w} is, on its own,
consistent with the same explanation that we gave for
\Cref{fig:whole_vs_cold}: larger windows expose the inner decoder to
a larger fraction of the constraints encoded in the detector error
matrix at the time of decoding, and this benefits both warm- and
cold-start decoding.
The dependence on the step size in \Cref{fig:bpgd_f}, however, is the
opposite of the corresponding dependence under plain \ac{bp}
(\Cref{fig:bp_f_over_p}): for warm-start, smaller $F$ now hurts
rather than helps, even though smaller $F$ implies a larger overlap
in both cases.
This inversion provides the clue to what is going wrong.
Recall from
\Cref{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}
that the warm start for \ac{bpgd} carries over not only the \ac{bp}
messages on the edges of the overlap region but also the decimation
information.
Because we run with an iteration budget large enough to decimate
every \ac{vn} in a window, by the time window $\ell$ ends, all
of its \acp{vn} have already been hard-decided.
For the \acp{vn} that lie in the overlap region with window $\ell + 1$
this hard decision is then carried into the next window through the
warm-start initialization, and the next window thus begins decoding
with a substantial fraction of its \acp{vn} already frozen, before
its own parity checks have had any chance to influence the
corresponding bit estimates.
This identifies one of two competing effects on the warm-start performance.
The larger the overlap, the more such prematurely frozen \acp{vn} the
next window inherits, which hurts performance.
On the other hand, a larger window still exposes the inner decoder to
a larger set of constraints, which helps performance.
The two effects together are consistent with what we observe in
\Cref{fig:bpgd_wf}.
Increasing $W$ at fixed $F$ enlarges both the overlap and the window
itself, and the benefit due to the larger $W$ dominates.
Decreasing $F$ at fixed $W$, by contrast, enlarges only the overlap
without enlarging the window, so the freezing effect is no longer
offset and warm-start performance worsens with smaller $F$.
\begin{figure}[t]
\centering
\hspace*{-6mm}
@@ -2240,7 +2351,8 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{Comparison of window sizes for $F=1$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\end{subfigure}%
\hfill%
\begin{subfigure}{0.48\textwidth}
@@ -2311,9 +2423,8 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{
Comparison of step sizes for $W=5$.
}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\end{subfigure}
\caption{
@@ -2389,7 +2500,8 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{Comparison of window sizes for $F=1$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\end{subfigure}%
\hfill%
\begin{subfigure}{0.5\textwidth}
@@ -2458,7 +2570,8 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{Comparison of step sizes for $W=5$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\end{subfigure}
\caption{
@@ -2536,7 +2649,8 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{Comparison of window sizes for $F=1$.}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\end{subfigure}%
\hfill%
\begin{subfigure}{0.48\textwidth}
@@ -2607,9 +2721,8 @@ This motivates the next subsection, in which we replace the inner
\end{axis}
\end{tikzpicture}
\caption{
Comparison of step sizes for $W=5$.
}
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\end{subfigure}
\caption{