116 lines
5.6 KiB
TeX
116 lines
5.6 KiB
TeX
\chapter{Conclusion and Outlook}
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\label{ch:Conclusion}
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% Recap of motivation
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This thesis investigates decoding under \acp{dem} for fault-tolerant
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\ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes.
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The repetition of the syndrome measurements, especially under
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consideration of circuit-level noise, leads to a significant increase
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in decoding complexity: in our experiments on the $\llbracket
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144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction
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rounds, the check matrix grows from 144 \acp{vn} and 72
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\acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}.
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% Recap of research gap and own work
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Sliding-window decoding addresses the latency constraint by
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exploiting the time-like locality of the syndrome extraction circuit.
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This manifests as a block-diagonal structure in the detector error
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matrix when detectors are defined as the difference of consecutive
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syndrome measurement rounds.
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We draw a comparison to windowed decoding for \ac{sc}-\ac{ldpc}
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codes, but note that the existing realizations of sliding-window
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decoding discard the soft information produced inside one window
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before moving to the next.
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Building on this observation, we proposed warm-start sliding-window
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decoding, in which the \ac{bp} messages on the edges crossing into
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the overlap region of the previous window are reused to initialise
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the corresponding messages of the next window in place of the
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standard cold-start initialisation.
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We formulate the warm start for standard \ac{bp} and for
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\ac{bpgd}.
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The latter is particularly attractive as an inner decoder because it
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addresses the convergence problems caused by short cycles and
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degeneracy in \ac{qldpc} Tanner graphs.
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The decoders are evaluated by conducting Monte Carlo simulations on the
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$\llbracket 144,12,12 \rrbracket$ \ac{bb} code over $12$ syndrome
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extraction rounds under standard circuit-based depolarizing noise.
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We focus on a qualitative analysis, refraining from further
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optimizations such as introducing a normalization parameter for the
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min-sum algorithm.
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% Recap of experimental conclusions
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For standard min-sum \ac{bp}, the warm start is consistently
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beneficial to the cold start, across the considered parameter ranges.
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The size of the gain depends on the overlap between consecutive
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windows: enlarging $W$ or shrinking $F$, both of which enlarge the
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overlap, result in larger gains of the warm-start.
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We observe that the underlying mechanism is an effective increase in
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the number of \ac{bp} iterations spent on the \acp{vn} in the overlap
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region: each such \ac{vn} is processed by multiple consecutive window
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invocations, and the warm start lets these invocations accumulate
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iterations on the same \acp{vn} rather than restarting from scratch.
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The gain was most pronounced at low numbers of maximum iterations, where
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every additional iteration carries proportionally more information.
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For \ac{bpgd}, we note that more information is available in the
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overlap region of a window: in addition to the \ac{bp} messages,
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there is information about which \acp{vn} were decimated and to what value.
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Passing this decimation information to the next window in addition to
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the messages turned out to worsen the performance considerably, which
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we attributed to a premature hard decision of the \acp{vn} in the
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overlap region.
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Restricting the warm start to the \ac{bp} messages alone, removed this effect.
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The resulting message-only warm start recovered a consistent
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improvement over cold-start that followed the same qualitative
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behaviour as for standard \ac{bp}: larger overlap, achieved by larger
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$W$ or smaller $F$, yielded a larger gain, and the
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performance difference is most pronounced at low numbers of maximum iterations.
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% Implications from experimental results
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These observations imply that the warm-start modification to
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sliding-window decoding can provide a consistent improvement, as long as
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some care is taken with specifying the information to be passed to
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the subsequent window.
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Note that this comes at no additional cost to the decoding complexity,
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since the only difference between warm- and cold-start sliding-window
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decoding is the initialization of the \ac{bp} messages.
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We expect similar behavior with other inner decoders that support
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soft information initialization in the overlap region.
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% Further research
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Several directions for further research emerge from this work.
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The most immediate is an extension of the evaluation to other
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\ac{qldpc} code families, to other circuit-level noise models such as
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SI1000 or EM3, and to a range of code sizes.
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This would clarify the generality of the gain due to the warm-start
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initialization.
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We expect the qualitative findings to carry over, since the
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underlying mechanism is structural rather than code-specific, but
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quantifying the gain across code families and noise models is left to
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future work.
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A second direction is a systematic study of other inner decoders under the
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warm-start framework, such as automorphism ensemble decoding
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\cite{koutsioumpas_automorphism_2025} or neural \ac{bp}
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\cite{miao_quaternary_2025}.
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A final direction is suggested by the structural similarity between
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sliding-window decoding for \acp{dem} and windowed decoding for
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\ac{sc}-\ac{ldpc} codes.
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The current approach to generating the syndrome extraction circuitry
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necessarily leads to a coupling width of one between adjacent
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syndrome measurement rounds.
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A natural question is whether the coupling width could be
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increased, e.g., by interleaving two separate realizations of the
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syndrome measurement circuitry instead of always repeating the same one.
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Work in this direction would also be a step toward bringing
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sliding-window decoding under DEMs within the scope of the analytical
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machinery developed for SC-LDPC codes.
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