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Finish first draft of text for fundamentals

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Andreas Tsouchlos
2026-04-24 17:44:16 +02:00
parent e59120b683
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5
src/thesis/acronyms.tex
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@@ -8,6 +8,11 @@
long=belief propagation
}
\DeclareAcronym{bpgd}{
short=BPGD,
long=belief propagation with guided decimation
}
\DeclareAcronym{nms}{
short=NMS,
long=normalized min-sum

211
src/thesis/chapters/2_fundamentals.tex
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@@ -517,6 +517,7 @@ This is precisely the effect that leads to the good performance of
\ac{sc}-\ac{ldpc} codes in the waterfall region \cite{costello_spatially_2014}.
\subsection{Iterative Decoding}
\label{subsec:Iterative Decoding}
% Introduction
@@ -1373,12 +1374,10 @@ We can describe it using the check matrix
\right]
.%
\end{align}
We can understand each row as defining a stabilizer.
% TODO: Check X vs. Z
The first $n$ columns correspond to $X$ operators acting on the
corresponding physical qubit, the rest to the $Z$ operators.
% TODO: Write
\begin{figure}[t]
\centering
@@ -1400,26 +1399,6 @@ corresponding physical qubit, the rest to the $Z$ operators.
\label{fig:sec}
\end{figure}
% %%%%%%%%%%%%%%%%
% \subsection{Decoding Stabilizer Codes}
% \label{subsec:Decoding Stabilizer Codes}
%
%
%
% \noindent\indent\red{[The QEC decoding problem
% \cite[Sec.~2.3]{yao_belief_2024}]} \\
% \indent\red{[+ Degeneracy]} \\
% \indent\red{[``The task of decoding is therefore to infer, from a
% measured syndrome, the most likely error coset rather than the exact
% physical error.''
% % tex-fmt: off
% \cite[Sec.~II~B)]{koutsioumpas_colour_2025}%
% % tex-fmt: on
% ]} \\
% \indent\red{[Fixing the error after finding it
% \cite[Sec.~10.5.5]{nielsen_quantum_2010} -> This
% may require introducing the gates as unitary]}
%%%%%%%%%%%%%%%%
\subsection{Calderbank-Shor-Steane Codes}
\label{subsec:Calderbank-Shor-Steane Codes}
@@ -1477,14 +1456,17 @@ $\mathcal{C}_2 \subset \mathcal{C}_1$.
Various methods of constructing \ac{qec} codes exist
\cite{swierkowska_eccentric_2025}.
\red{[topological codes]}
\red{[(?) Mention color and surface codes]}.
A more recent development is that of quantum \ac{ldpc} (\acs{qldpc}) codes.
% Why QLDPC codes are interesting
\indent\red{[Constant overhead scaling]} \\
\indent\red{[Scaling of minimum distance with code length]} \\
Topological codes, for example, encode information in the features of
a lattice and are intrinsically robust against local errors.
Among these, the \emph{surface code} is the most widely studied.
Another example are concatenated codes, which nest one code within
another, allowing for especially simple and flexible constructions
\cite[Sec.~3.2]{swierkowska_eccentric_2025}.
An area of research that has recently seen more attention is that of
quantum \ac{ldpc} (\acs{qldpc}) codes.
They have much better encoding efficiency than, e.g., the surface
code, scaling up of which would be prohibitively expensive
\cite[Sec.~I]{bravyi_high-threshold_2024}.
% Bivariate Bicycle codes
@@ -1527,9 +1509,168 @@ Additionally, they posess short-depth syndrome measurement circuits,
leading to lower time requirements for the syndrome extraction
and thus lower error rates \cite[Sec.~1]{bravyi_high-threshold_2024}.
% Decoding QLDPC codes
% Syndrome-based BP
\indent\red{[Decoding QLDPC codes (syndrome-based BP)]} \\
\indent\red{[Short cycles]} \\
\indent\red{[Degeneracy + short cycles -> BP+OSD, BPGD]} \\
As we saw in \autoref{subsec:Stabilizer Measurements}, we work only
with the parity information contained in the syndrome, to avoid
disturbing the quantum states of indivudual qubits.
This necessitates a modification of the standard \ac{bp} algorithm
introduced in \autoref{subsec:Iterative Decoding}
\cite[Sec.~3.1]{yao_belief_2024}.
Instead of attempting to find the most likely codeword directly, the
algorithm will now try to find an error pattern $\hat{\bm{e}} \in
\mathbb{F}_2^n$ that satisfies
\begin{align*}
\bm{H} \hat{\bm{e}}^\text{T} = \bm{s}
.%
\end{align*}
To this end, we initialize the channel \acp{llr} as
\begin{align*}
\tilde{L}_i = \log{\frac{P(X_i = 0)}{P(X_i = 1)}} = \frac{1 - p_i}{p_i}
,%
\end{align*}
where $p_i$ is the prior probability of error of \ac{vn} $i$.
Additionally, we amend the \ac{cn} update to consider the parity
indicated by the syndrome, calculating
\begin{align*}
L_{i\leftarrow j} = 2\cdot (-1)^{s_j} \cdot \tanh^{-1} \left( \prod_{i'\in
\mathcal{N}(j)\setminus \{i\}} \tanh \frac{L_{i'\rightarrow j}}{2} \right)
.
\end{align*}
The resulting syndrome-based \ac{bp} algorithm is shown in
algorithm \ref{alg:syndome_bp}.
% tex-fmt: off
\tikzexternaldisable
\begin{algorithm}[t]
\caption{Binary syndrome-based belief propagation (BP) algorithm.}
\label{alg:syndome_bp}
\begin{algorithmic}[1]
\State \textbf{Initialize:} $\tilde{L}_i \leftarrow
\log \frac{1-p_i}{p_i}$ for all $i \in \mathcal{I}$
\State \textbf{Initialize:} $L_{i \rightarrow j} \leftarrow
\tilde{L}_i$ for all $i \in \mathcal{I},\, j \in \mathcal{N}_\text{V}(i)$
\State \textbf{Initialize:} $\hat{e} \leftarrow \bm{0}$
\For{$\ell = 1, \ldots, n_\text{iter}$}
\For{$j \in \mathcal{J}$}
\For{$i \in \mathcal{N}_\text{C}(j)$}
\State $\displaystyle L_{i \leftarrow j} \leftarrow
2\cdot(-1)^{s_j}\cdot\tanh^{-1}
\!\left(
\prod_{i' \in \mathcal{N}_\text{C}(j)\setminus\{i\}}
\tanh\frac{L_{i'\rightarrow j}}{2}
\right)$
\EndFor
\EndFor
\For{$i \in \mathcal{I}$}
\For{$j \in \mathcal{N}_\text{V}(i)$}
\State $\displaystyle L_{i \rightarrow j} \leftarrow
\tilde{L}_i +
\sum_{j' \in \mathcal{N}_\text{V}(i)\setminus\{j\}}
L_{i \leftarrow j'}$
\EndFor
\EndFor
\For{$i \in \mathcal{I}$}
\State $\displaystyle \hat{e}_i \leftarrow
\mathbbm{1}\left\{
\tilde{L}_i +
\sum_{j \in \mathcal{N}_\text{V}(i)} L_{i \leftarrow j} < 0
\right\}$
\EndFor
\If{$\bm{H}\hat{\bm{e}}^\text{T} = \bm{s}$}
\State \textbf{break}
\EndIf
\EndFor
\State \textbf{return} $\hat{\bm{e}}$
\end{algorithmic}
\end{algorithm}
\tikzexternalenable
% tex-fmt: on
% Degeneracy and short cycles
Decoding \ac{qldpc} codes brings with it some unique challenges.
One issue is that of \emph{quantum degeneracy}.
Because errors that differ by a stabilizer have the same impact on
all codewords, there can be multiple minimum-weight solutions to the
quantum decoding problem \cite[Sec.~II.C.]{babar_fifteen_2015}
\cite[Sec.~V]{roffe_decoding_2020}.
This leads to the decoding algorithm getting confused about the
direction to proceed in \cite[Sec.~5]{yao_belief_2024}.
Another problem is that due to the commutativity property of the stabilizers,
quantum codes inherently contain short cycles
\cite[Sec.~IV.C]{babar_fifteen_2015}.
As discussed in \autoref{subsec:Iterative Decoding}, these lead to
the violation of the independence assumption of the messages passed
during decoding, impeding performance.
% BPGD
The aforementioned issues both manifest themselves as convergence problems
of the \ac{bp} algorithm, and different ways of modifying the algorithm
to aide with convergence exist.
One approach is to use \ac{bp} with guided decimation (\acs{bpgd})
\cite[Alg.~1]{yao_belief_2024}.
Here, a number $T\in \mathbb{N}$ of \ac{bp} iterations are performed,
before \emph{decimating} the most reliable \ac{vn}, i.e., performing
a hard decision and excluding it from further decoding.
This constrains the solution space more and more as the decoding
progresses, encouraging the algorithm to converge to one of the
solutions \cite[Sec.~5]{yao_belief_2024}.
Algorithm \ref{alg:bpgd} shows this process.
Note that as the Tanner graph only has $n$ \acp{vn}, this is a
natural constraint on the maximum number of iterations.
% TODO: Explain that setting the channel LLR to infinity is the same
% as a hard decision and ignoring the VN in the further decoding
% tex-fmt: off
\tikzexternaldisable
\begin{algorithm}[t]
\caption{Belief propagation with guided decimation (BPGD) algorithm.}
\label{alg:bpgd}
\begin{algorithmic}[1]
\State \textbf{Initialize:} $\tilde{L}_i \leftarrow
\log \frac{1-p_i}{p_i}$ for all $i \in \mathcal{I}$
\State \textbf{Initialize:} $L_{i \rightarrow j} \leftarrow
\tilde{L}_i$ for all $i \in \mathcal{I},\, j \in \mathcal{N}_\text{V}(i)$
\State \textbf{Initialize:} $\hat{e} \leftarrow \bm{0}$
\State \textbf{Initialize:} $\mathcal{I}' \leftarrow \mathcal{I}$
\For{$r = 1, \ldots, n$}
\For{$\ell = 1, \ldots, T$}
\State Perform \ac{cn} update
\State Perform \ac{vn} update
\State $L^\text{total}_i \leftarrow \tilde{L}_i + \sum_{j \in \mathcal{N}_\text{V}(i)} L_{i \leftarrow j}$
\EndFor
\For{$i \in \mathcal{I}$}
\State $\displaystyle \hat{e}_i \leftarrow
\mathbbm{1}\left\{ L^\text{total}_i \right\}$
\EndFor
\If{$\bm{H}\hat{\bm{e}}^\text{T} = \bm{s}$}
\State \textbf{break}
\Else
\State $i_\text{max} \leftarrow \argmax_{i \in \mathcal{I}'} \lvert L^\text{total}_i \rvert $
\If{$L^\text{total}_{i_\text{max}} < 0$}
\State $\tilde{L}_{i_\text{max}} \leftarrow -\infty$
\Else
\State $\tilde{L}_{i_\text{max}} \leftarrow +\infty$
\EndIf
\State $\mathcal{I}' \leftarrow \mathcal{I}'\setminus\{i_\text{max}\}$
\EndIf
\EndFor
\State \textbf{return} $\hat{\bm{e}}$
\end{algorithmic}
\end{algorithm}
\tikzexternalenable
% tex-fmt: on

6
src/thesis/main.tex
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@@ -6,12 +6,15 @@
\usepackage{amsfonts}
\usepackage{mleftright}
\usepackage{bm}
\usepackage{bbm}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepackage{acro}
\usepackage{braket}
\usepackage{listings}
\usepackage{caption}
% \usepackage[
% backend=biber,
% style=ieee,
@@ -20,9 +23,10 @@
\usepackage{todonotes}
\usepackage{quantikz}
\usepackage{stmaryrd}
\usepackage{algorithm}
\usepackage[noEnd=false]{algpseudocodex}
\usetikzlibrary{calc, positioning, arrows, fit}
\usetikzlibrary{external}
\tikzexternalize