\chapter{Conclusion and Outlook}
\label{ch:Conclusion}

% Recap of motivation

This thesis investigates decoding under \acp{dem} for fault-tolerant
\ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes.
The repetition of the syndrome measurements, especially under
consideration of circuit-level noise, leads to a significant increase
in decoding complexity: In our experiments on the $\llbracket
144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction
rounds, the check matrix grows from 144 \acp{vn} and 72
\acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}.

% Recap of research gap and own work

Sliding-window decoding addresses the latency constraint by
exploiting the time-like locality of the syndrome extraction circuit.
This manifests as a block-diagonal structure in the detector error
matrix when detectors are defined as the difference of consecutive
syndrome measurement rounds.
We draw a comparison to windowed decoding for \ac{sc}-\ac{ldpc}
codes, but note that the existing realizations of sliding-window
decoding discard the soft information produced inside one window
before moving to the next.
Building on this observation, we proposed warm-start sliding-window
decoding, in which the \ac{bp} messages on the edges crossing into
the overlap region of the previous window are reused to initialise
the corresponding messages of the next window in place of the
standard cold-start initialisation.

We formulate the warm start for standard \ac{bp} and for
\ac{bpgd}.
The latter is particularly attractive as an inner decoder because it
addresses the convergence problems caused by short cycles and
degeneracy in \ac{qldpc} Tanner graphs.
The decoders are evaluated by conducting Monte Carlo simulations on the
$\llbracket 144,12,12 \rrbracket$ \ac{bb} code over $12$ syndrome
extraction rounds under standard circuit-based depolarizing noise.
We focus on a qualitative analysis, refraining from further
optimizations such as introducing a normalization parameter for the
min-sum algorithm.

% Recap of experimental conclusions

For standard min-sum \ac{bp}, the warm start is consistently
beneficial to the cold start, across the considered parameter ranges.
The size of the gain depends on the overlap between consecutive
windows: Enlarging $W$ or shrinking $F$, both of which enlarge the
overlap, result in larger gains of the warm-start.
We observe that the underlying mechanism is an effective increase in
the number of \ac{bp} iterations spent on the \acp{vn} in the overlap
region: Each such \ac{vn} is processed by multiple consecutive window
invocations, and the warm start lets these invocations accumulate
iterations on the same \acp{vn} rather than restarting from scratch.
The gain was most pronounced at low numbers of maximum iterations, where
every additional iteration carries proportionally more information.

For \ac{bpgd}, we note that more information is available in the
overlap region of a window: In addition to the \ac{bp} messages,
there is information about which \acp{vn} were decimated and to what value.
Passing this decimation information to the next window in addition to
the messages turned out to worsen the performance considerably, which
we attributed to a premature hard decision of the \acp{vn} in the
overlap region.
Restricting the warm start to the \ac{bp} messages alone, removed this effect.
The resulting message-only warm start recovered a consistent
improvement over cold-start that followed the same qualitative
behaviour as for standard \ac{bp}: Larger overlap, achieved by larger
$W$ or smaller $F$, yielded a larger gain, and the
performance difference is most pronounced at low numbers of maximum iterations.

% Implications from experimental results

These observations imply that the warm-start modification to
sliding-window decoding can provide a consistent improvement, as long as
some care is taken with specifying the information to be passed to
the subsequent window.
Note that this comes at no additional cost to the decoding complexity,
since the only difference between warm- and cold-start sliding-window
decoding is the initialization of the \ac{bp} messages.
We expect similar behavior with other inner decoders that support
soft information initialization in the overlap region.

% Further research

Several directions for further research emerge from this work.
The most immediate is an extension of the evaluation to other
\ac{qldpc} code families, to other circuit-level noise models such as
SI1000 or EM3, and to a range of code sizes.
This would clarify the generality of the gain due to the warm-start
initialization.
We expect the qualitative findings to carry over, since the
underlying mechanism is structural rather than code-specific, but
quantifying the gain across code families and noise models is left to
future work.

A second direction is a systematic study of other inner decoders under the
warm-start framework, such as automorphism ensemble decoding
\cite{koutsioumpas_automorphism_2025} or neural \ac{bp}
\cite{miao_quaternary_2025}.

A final direction is suggested by the structural similarity between
sliding-window decoding for \acp{dem} and windowed decoding for
\ac{sc}-\ac{ldpc} codes.
The current approach to generating the syndrome extraction circuitry
necessarily leads to a coupling width of one between adjacent
syndrome measurement rounds.
A natural question is whether the coupling width could be
increased, e.g., by interleaving two separate realizations of the
syndrome measurement circuitry instead of always repeating the same one.
Work in this direction would also be a step toward bringing
sliding-window decoding under DEMs within the scope of the analytical
machinery developed for SC-LDPC codes.

