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src/thesis/chapters/4_decoding_under_dems.tex
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@@ -72,7 +72,7 @@ problem into many smaller ones that can be solved more efficiently.
% warm-start decoding. Or just go into warm-start decoding)
We will start by briefly reviewing the existing work related to
sliding-window decoding,
before focusing on one specific incarnation.
before focusing on one specific realization.
We will then introduce a modification to the existing algorithm and
perform numerical simulations to evaluate it.
@@ -110,100 +110,6 @@ Each of these windows is then decoded separately.
\subsection{Review of Existing Literature}
\label{subsec:Review of Existing Literature}
% Description of the figure
\Cref{fig:literature} gives an overview over the existing body of work
related to sliding-window decoding.
The papers \cite{huang_improved_2023} and \cite{huang_increasing_2024} are
lumped together, as they share the same content;
one is simply preprint published earlier.
We will only refer to \cite{huang_increasing_2024} in the following.
\cite{kang_quits_2025} is somewhat special in that the authors focus
more on the introduction of a new simluator framework they call
QUITS, rather than the performance of sliding-window decoding itself.
\cite{gong_toward_2024} and \cite{kang_quits_2025} have made their
software freely available online%
\footnote{
https://github.com/mkangquantum/quits
}%
\footnote{
https://github.com/gongaa/SlidingWindowDecoder
}.
A final thing to note is that \cite{dennis_topological_2002} never
explicitly mention sliding windows, they call their scheme
``overlapping recovery''.
% Topological vs QLDPC
Research has focused on two categories of \ac{qec} codes, topological
and \ac{qldpc} codes.
Most of the work on topological codes has treated surface codes,
with the exception of \cite{kuo_fault-tolerant_2024} where toric
codes were considered.
With regard to \ac{qldpc} codes, in \cite{huang_increasing_2024}
they examine \emph{hypergraph product} (\acs{hgp}) and
\emph{lifted-product} (\acs{lp}) codes.
HGP codes are constructed from the product of two classical codes,
while LP codes generalize this construction by additionally applying
a lift to reduce the qubit overhead.
In \cite{kang_quits_2025}, \emph{balanced product codes} (\acs{bpc})
are additionally considered.
Finally, in \cite{gong_toward_2024} the authors explore \ac{bb} codes.
% Sequential vs parallel
After having divided the whole circuit into separate windows, the question
arises of how exactly to realize the decoding.
There are two main approaches, with differing mechanisms of reducing
the latency.
Some papers decode the sliding windows in a parallel fashion.
The benefit in this case is the option to more effectively utilize
classical hardware for decoding.
Others choose a sequential approach.
Here, decoding can start earlier, as there is no need to wait for the
syndrome measurements of all windows before beginning with the decoding.
With the exception of \cite{dennis_topological_2002}, literature
treating topological codes has mostly focused on parallel decoding
while literature treating \ac{qldpc} codes has wholely considered
sequential decoding.
% Deep-dive into QLDPC methods
\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{12pt}
\begin{table}[t]
\centering
\caption{Experimental conditions for papers related to \ac{qldpc} codes.}
\vspace*{3mm}
\label{table:experimental_conditions}
\begin{tabular}{l|ccc}
% tex-fmt: off
Publication & Code & Noise Model & Decoder \\ \hline
\hspace{-2.5mm}\cite{huang_improved_2023},\cite{huang_increasing_2024} & \acs{hgp}, \acs{lp} & Phenomenological noise & \acs{bp} + \acs{osd} \\
\hspace{-2.5mm}\cite{gong_toward_2024} & \acs{bb} & Circuit-level noise & \acs{bp} + \acs{gdg} \\
\hspace{-2.5mm}\cite{kang_quits_2025} & \acs{hgp}, \acs{lp}, \acs{bpc} & Circuit-level noise & \acs{bp} + \ac{osd}
% tex-fmt: on
\end{tabular}
\end{table}
For this work, the publications treating \ac{qldpc} codes are
especially interesting.
The experimental conditions for these are summarized in
\Cref{table:experimental_conditions}.
As we noted above, \ac{hgp} and \ac{lp} codes are considered in
\cite{huang_increasing_2024},
\ac{hgp}, \ac{lp} and \ac{bpc} codes are considered in \cite{kang_quits_2025},
and \ac{bb} codes are considered in \cite{gong_toward_2024}.
The employed noise models also differ;
\cite{huang_increasing_2024} use phenomenological noise, while
\cite{gong_toward_2024} and \cite{kang_quits_2025} use circuit-level noise.
Finally, \cite{gong_toward_2024} introduce their own variation of
\ac{bpgd}, \ac{bp} with \ac{gdg}, while \cite{huang_increasing_2024}
and \cite{kang_quits_2025} use \ac{bp} + \ac{osd}.
We would additionally like to note that only in
\cite{gong_toward_2024} and \cite{kang_quits_2025} do the authors
explicitly work with the \ac{dem} formalism.
\begin{figure}[t]
\centering
@@ -290,6 +196,103 @@ explicitly work with the \ac{dem} formalism.
\label{fig:literature}
\end{figure}
% Some general notes
\Cref{fig:literature} gives an overview over the existing body of work
related to sliding-window decoding.
The papers \cite{huang_improved_2023} and \cite{huang_increasing_2024} are
lumped together, as they share the same content;
one is simply preprint published earlier.
We will only refer to \cite{huang_increasing_2024} in the following.
\cite{kang_quits_2025} is somewhat special in that the authors focus
more on the introduction of a new simluator framework they call
QUITS, rather than the performance of sliding-window decoding itself.
\cite{gong_toward_2024} and \cite{kang_quits_2025} have made their
software freely available online%
\footnote{
https://github.com/mkangquantum/quits
}%
\footnote{
https://github.com/gongaa/SlidingWindowDecoder
}.
A final thing to note is that \cite{dennis_topological_2002} never
explicitly mention sliding windows, they call their scheme
``overlapping recovery''.
% Topological vs QLDPC
Research has focused on two categories of \ac{qec} codes, topological
and \ac{qldpc} codes.
Most of the work on topological codes has treated surface codes,
with the exception of \cite{kuo_fault-tolerant_2024} where toric
codes were considered.
With regard to \ac{qldpc} codes, in \cite{huang_increasing_2024}
they examine \emph{hypergraph product} (\acs{hgp}) and
\emph{lifted-product} (\acs{lp}) codes.
HGP codes are constructed from the product of two classical codes,
while LP codes generalize this construction by additionally applying
a lift to reduce the qubit overhead.
In \cite{kang_quits_2025}, \emph{balanced product codes} (\acs{bpc})
are additionally considered.
Like HGP codes, BPC codes are derived from a product construction,
but exploit an additional symmetry to yield fewer physical qubits for
the same code parameters.
Finally, in \cite{gong_toward_2024} the authors explore \ac{bb} codes.
% Sequential vs parallel
After having divided the whole circuit into separate windows, the question
arises of how exactly to realize the decoding.
There are two main approaches, with differing mechanisms of reducing
the latency.
Some papers decode the sliding windows in a parallel fashion.
The benefit in this case is the option to more effectively utilize
classical hardware for decoding.
Others choose a sequential approach.
Here, decoding can start earlier, as there is no need to wait for the
syndrome measurements of all windows before beginning with the decoding.
With the exception of \cite{dennis_topological_2002}, literature
treating topological codes has mostly focused on parallel decoding
while literature treating \ac{qldpc} codes has wholely considered
sequential decoding.
% Deep-dive into QLDPC methods
For this work, the publications treating \ac{qldpc} codes are
especially interesting.
The experimental conditions for these are summarized in
\Cref{table:experimental_conditions}.
As we noted above, \ac{hgp} and \ac{lp} codes are considered in
\cite{huang_increasing_2024},
\ac{hgp}, \ac{lp} and \ac{bpc} codes are considered in \cite{kang_quits_2025},
and \ac{bb} codes are considered in \cite{gong_toward_2024}.
The employed noise models also differ;
\cite{huang_increasing_2024} use phenomenological noise, while
\cite{gong_toward_2024} and \cite{kang_quits_2025} use circuit-level noise.
Finally, \cite{gong_toward_2024} introduce their own variation of
\ac{bpgd}, \ac{bp} with \ac{gdg}, while \cite{huang_increasing_2024}
and \cite{kang_quits_2025} use \ac{bp} + \ac{osd}.
We would additionally like to note that only in
\cite{gong_toward_2024} and \cite{kang_quits_2025} do the authors
explicitly work with the \ac{dem} formalism.
\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{12pt}
\begin{table}[t]
\centering
\caption{Experimental conditions for papers related to \ac{qldpc} codes.}
\vspace*{3mm}
\label{table:experimental_conditions}
\begin{tabular}{l|ccc}
% tex-fmt: off
Publication & Code & Noise Model & Decoder \\ \hline
\hspace{-2.5mm}\cite{huang_improved_2023},\cite{huang_increasing_2024} & \acs{hgp}, \acs{lp} & Phenomenological noise & \acs{bp} + \acs{osd} \\
\hspace{-2.5mm}\cite{gong_toward_2024} & \acs{bb} & Circuit-level noise & \acs{bp} + \acs{gdg} \\
\hspace{-2.5mm}\cite{kang_quits_2025} & \acs{hgp}, \acs{lp}, \acs{bpc} & Circuit-level noise & \acs{bp} + \ac{osd}
% tex-fmt: on
\end{tabular}
\end{table}
% \red{
% Existing work
% \begin{itemize}
@@ -381,12 +384,6 @@ error matrix is divided into overlapping windows.
The algorithm detailed here follows \cite{kang_quits_2025}, whose
work is in turn based on \cite{huang_increasing_2024}.
\red{
\begin{itemize}
\item QUITS views sliding-window decoding more separately
\end{itemize}
}
\content{Possibly go into the fact that current sliding-window
approaches don't differentiate clearly between the sliding-window
part and the decoder part. This work aims to extend the
@@ -394,9 +391,6 @@ work is in turn based on \cite{huang_increasing_2024}.
different decoder parts. Combine this with QUITS modular structure
for sliding window decoding}
We build on the approach taken by \cite{huang_increasing_2024} and
\cite{gong_toward_2024}.
% High-level overview of Sliding-Window decoding
\content{Benefits of sliding-window decoding (lower latency due to
@@ -427,7 +421,7 @@ with processing'' some VNs)}
\content{4. Decode next window}
\content{(?) Explicitly mention we don't reuse existing messages?}
\begin{figure}[H]
\begin{figure}[t]
\centering
\hspace*{-114mm}%