Minor changes to conclusion

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -3,24 +3,24 @@
 
 % Recap of motivation
 
-This thesis investigated decoding under \acp{dem} for fault-tolerant
+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 grew from 144 \acp{vn} and 72
+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,
-which manifests as a block-diagonal structure in the detector error
+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 drew a comparison to windowed decoding for \ac{sc}-\ac{ldpc}
-codes, but noted that the existing realizations of sliding-window
+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
@@ -29,25 +29,26 @@ 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 formulated the warm start first for plain \ac{bp} and then for
-\ac{bpgd}, the latter being attractive as an inner decoder because it
+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 were evaluated by Monte Carlo simulation on the
+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 focused on a qualitative analysis, refraining from further
+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 plain min-sum \ac{bp}, the warm start was consistently beneficial
-across the parameter ranges we examined. The size of the gain depended
-on the overlap between consecutive windows: enlarging $W$ or
-shrinking $F$, both of which enlarge the overlap, raised the
-warm-start performance increase.
-We argued that the underlying mechanism is an effective increase in
+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
@@ -55,7 +56,7 @@ 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 noted that more information is available in the
+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
@@ -65,14 +66,14 @@ 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 plain \ac{bp}: larger overlap, achieved by larger
+behaviour as for standard \ac{bp}: larger overlap, achieved by larger
 $W$ or smaller $F$, yielded a larger gain, and the
-performance difference was most pronounced at low numbers of maximum iterations.
+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 provides a consistent improvement, as long as
+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,
@@ -94,25 +95,10 @@ 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 inner decoders under the
-warm-start framework.
-We considered plain min-sum \ac{bp} and \ac{bpgd}, but other
-algorithms used for \ac{qldpc} decoding, such as automorphism
-ensemble decoding \cite{koutsioumpas_automorphism_2025} or neural
-\ac{bp} \cite{miao_quaternary_2025} may admit warm-start variants of their own.
-
-A third direction is a softer treatment of the decimation state in \ac{bpgd}.
-Rather than discarding the decimation information of the previous
-window entirely, as in the message-only warm start used here, one
-could encode the decimation decisions as strong but finite biases on
-the channel \acp{llr} of the next window, allowing the new window's parity
-checks to override them if the syndrome calls for it.
-This would interpolate between the two warm-start variants studied here and
-might combine the benefits of both.
-A related question is whether the decimation schedule itself should
-be aware of the window structure, for instance by deferring
-decimation of \acp{vn} in the overlap region until they have been
-visited by the next window.
+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

src/thesis/chapters/abstract.tex

@@ -4,6 +4,8 @@
 
 \Ac{qec} protects fragile quantum states against decoherence by
 encoding logical information into a larger number of physical qubits.
+To obtain parity information on an encoded state without disturbing it, a
+syndrome extraction is performed.
 Because the syndrome extraction circuitry is itself implemented on
 noisy quantum hardware, practical \ac{qec} must be fault-tolerant,
 accounting for errors introduced by the correction procedure itself.
@@ -19,31 +21,31 @@ can be decoded.
 Together, these factors pose a serious challenge for practical decoders.
 Sliding-window decoding addresses this challenge by exploiting the
 repeated structure of the syndrome extraction circuitry, partitioning
-the \ac{dem}'s check matrix into overlapping windows that can be
+the check matrix of the \ac{dem} into overlapping windows that can be
 decoded sequentially.
-This allows for an earlier start to the decoding process, before all
-syndrome measurements have been completed, thereby lowering the latency.
+Therefore, decoding can begin as soon as the syndrome components
+associated with the first window have been measured.
 
 % Our work: Identify research gap
 
 In this thesis, we perform a review of the existing literature on
 sliding-window decoding and draw an analogy to windowed
-decoding for classical spatially-coupled low-density parity-check
+decoding of classical spatially-coupled low-density parity-check
 (\acs{sc}-\acs{ldpc}) codes.
 We recognize that in contrast to the latter, existing realizations
 of sliding-window decoding for \ac{qec} discard the soft information
-produced inside one window before moving to the next.
+produced inside one window before moving to the subsequent window.
 
 % Our work: Warm-start
 
 % TODO: Quantify improvement. Also for conclusion
-We propose 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 initialize the corresponding messages of the
-next window.
-The warm start is formulated first for plain \ac{bp} and then extended to
+To take this information into account, we propose 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
+initialize the corresponding messages of the next window.
+The warm start is formulated first for standard \ac{bp} and then extended to
 \ac{bp} with guided decimation (\acs{bpgd}).
-For both plain min-sum \ac{bp} and \ac{bpgd} decoding, the warm-start
+For both standard \ac{bp} and \ac{bpgd} decoding, the warm-start
 initialization provides a consistent improvement across all examined
 parameter settings.
 We attribute this to an effective increase in \ac{bp} iterations on