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Add TODOs to fault tolerance chapter

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Andreas Tsouchlos
2026-04-25 21:58:00 +02:00
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src/thesis/chapters/3_fault_tolerant_qec.tex
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% TODO: Make all [H] -> [t]
\chapter{Fault Tolerant QEC}
\section{Fault Tolerance}
% Intro
\content{Syndrome extraction circuitry itself introduces errors}
\content{High level explanation of fault tolerance (with figure)}
\content{Mathematical definition of fault tolerance}
% Practical considerations
\content{We generally need to perform multiple rounds of syndrome extraction}
\content{The number of rounds of syndrome extraction is usually
chosen equal to the $d_\text{min}$ of the code}
\content{One-shot decoding property}
\begin{figure}[H]
\centering
\begin{tikzpicture}
\node[rectangle, draw, fill=orange!20, minimum
height=2cm, minimum width=2.5cm, align=left] at (0,0)
(internal) {Internal\\ Errors};
\node[signal, draw, fill=orange!20, minimum height=2cm,
minimum width=2.5cm, align=left, signal pointer angle=140]
at (-2.45, 0) (input) {Input\\ Errors};
\node at (1.97,0) {\huge =};
\node[rectangle, draw, fill=orange!20, minimum height=2cm,
minimum width=2.5cm, align=left] at (4,0) (output)
{Output\\ Errors};
\node[above] at (input.north) {\small Input State};
\node[above] at (internal.north) {\small QEC};
\node[above] at (output.north) {\small Output State};
\end{tikzpicture}
\caption{Sources of error in a fault-tolerant \ac{qec} system.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Noise Models}
\subsection{Depolarizing Channel}
\subsection{Phenomenological Noise}
\subsection{Circuit-Level Noise}
\section{Detector Error Models}
\subsection{Measurement Syndrome Matrix}
\subsection{Detector Error Matrix}
\subsection{Detector Error Models}
\section{Practical Considerations}
\subsection{Practical Methodology}
\label{sec:Noise Models}
\content{Per-round-LER explanation}
% Intro
\subsection{Stim}
\content{Explanation of what a noise model is}
\content{Mention there are different types of noise models, each with
different possible error locations}
\begin{figure}[t]
% Figure intro
\content{\autoref{fig:pure_syndrome_extraction} shows the syndrome
extraction circuit of a three-qubit repetition code with stabilizers
$Z_1Z_2$ and $Z_2Z_3$. This code is only able to deal with X errors.
We will use it as a propotypical model to examine the different types
of noise models}
\content{This is now a concrete implementation of the syndrome
measurement circuit using CNOT gates, as opposed to the system-level
view in \autoref{subsec:Stabilizer Codes}}
\content{\autoref{fig:noise_model_types} shows a number of diffent
types of noise models}
% Bit-flip noise
\content{Bit-flip noise}
\content{Introduce \emph{data qubits}}
\content{Only X errors on data qubits}
\content{Most similar to classical channel coding}
\content{\textbf{TODO}: What is this useful for? Just as a first step?}
% Depolarizing channel
\content{Depolarizing channel}
\content{X/Y/Z errors on data qubits}
\content{\textbf{TODO}: What does this model? Memory experiment with
ideal syndrome extraction?}
\content{\textbf{TODO}: Why is it called depolarizing?}
% Phenomenological noise
\content{Phenomenological noise}
\content{First noise model that considers errors during syndrome extraction}
\content{X errors before each syndrome extraction round}
\content{X errors before measurements}
\content{\textbf{TODO}: Why is this useful? Derks et al. mentioned
something about it being useful to derive fault-tolerant circuits}
\content{\textbf{TODO}: Make sure phenomenological noise is only X errors}
% Circuit-level noise
\content{Circuit-level noise}
\content{This is generally what we strive to be able to decode under}
\content{X/Y/Z errors before each syndrome extraction round}
\content{$n$-qubit Pauli errors after each $n$-qubit Pauli gate}
\content{Define $n$-qubit Pauli errors}
\content{X errors right before the measurements}
\content{Note that the only errors right before the measurements that
have any effect on the measurement outcomes are X errors. That is why
it is enough to consider this type of error at this point in the circuit.}
% Different noise models for circuit-level noise
\content{Comparison from Gidney's paper}
\content{In this work we only consider standard circuit-based
depolarizing noise}
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\begin{align*}
\bm{H} =
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}
\end{align*}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
% tex-fmt: off
\begin{quantikz}%[row sep=4mm, column sep=4mm]
\lstick[3]{$\ket{\psi}_\text{L}$} & \ctrl{3} & & & & & \\
& & \ctrl{2} & \ctrl{3} & & & \\
& & & & \ctrl{2} & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
\end{quantikz}
% tex-fmt: on
\end{minipage}%
\caption{
Syndrome extraction circuit for the three-qubit repetition
code under bit-flip noise.
}
\label{fig:pure_syndrome_extraction}
\end{figure}
\begin{figure}[H]
\centering
\newcommand{\xerr}{\gate[style={fill=KITblue!50}]{\phantom{1}}}
@@ -125,67 +248,60 @@
\end{minipage}
\caption{Types of noise models.}
\label{fig:Types of noise models.}
\label{fig:noise_model_types}
\end{figure}
\begin{figure}[t]
\centering
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Detector Error Models}
\label{sec:Detector Error Models}
\begin{tikzpicture}
\node[rectangle, draw, fill=orange!20, minimum
height=2cm, minimum width=2.5cm, align=left] at (0,0)
(internal) {Internal\\ Errors};
\content{\textbf{TODO}: Look up how Derks et al. introduce DEMs}
\node[signal, draw, fill=orange!20, minimum height=2cm,
minimum width=2.5cm, align=left, signal pointer angle=140]
at (-2.45, 0) (input) {Input\\ Errors};
% Different ways of implementing fault tolerance
\node at (1.97,0) {\huge =};
\content{Ways of implementing fault tolerance different from DEMs}
\node[rectangle, draw, fill=orange!20, minimum height=2cm,
minimum width=2.5cm, align=left] at (4,0) (output)
{Output\\ Errors};
% Core idea
\node[above] at (input.north) {\small Input State};
\node[above] at (internal.north) {\small QEC};
\node[above] at (output.north) {\small Output State};
\end{tikzpicture}
\content{Model additional error locations in the code}
\content{Construct ``circuit code'' from original code}
\caption{Sources of error in a fault-tolerant \ac{qec} system.}
\end{figure}
% Benefits
\begin{figure}[t]
\centering
\content{Benefits of this approach}
\begin{minipage}{0.5\textwidth}
\begin{align*}
\bm{H} =
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}
\end{align*}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
% tex-fmt: off
\begin{quantikz}%[row sep=4mm, column sep=4mm]
\lstick[3]{$\ket{\psi}_\text{L}$} & \ctrl{3} & & & & & \\
& & \ctrl{2} & \ctrl{3} & & & \\
& & & & \ctrl{2} & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
\end{quantikz}
% tex-fmt: on
\end{minipage}%
%%%%%%%%%%%%%%%%
\subsection{Measurement Syndrome Matrix}
\label{subsec:Measurement Syndrome Matrix}
\caption{
Syndrome extraction circuit for the three-qubit repetition
code under bit-flip noise.
}
\label{fig:}
\end{figure}
% Core idea
\begin{figure}[t]
\content{Core idea: Matrix describes parity checks \\
$\rightarrow$ A column shows which parity checks the
corresponding VN contributes to \\
$\rightarrow$ View columns as syndromes corresponding to error
locations in the circuit
}
% Multiple rounds of syndrome extraction
% TODO: First introduce syndrome measurement matrix, mathematically
% (consult Derks et al.'s paper). Then use the three-qubit repetition
% code as an example only
\autoref{fig:rep_code_multiple_rounds_bit_flip} shows a circuit
performing three rounds of syndrome extraction for the three-qubit
repetition code introduced earlier.
We are only considering bit-flip noise at this point.
For each syndrome extraction round, we get an additional set of
syndrome measurements.
We combine these measurements by stacking them in a new vector $\bm{s}
\in \mathbb{F}_2^{n_\text{rounds}\cdot(n-k)}$.
To model this behavior mathematically, we append additional rows to
the check matrix.
We call this matrix the \emph{measurement syndrome matrix}
$\bm{\Omega}$.
\begin{figure}[H]
\centering
\begin{minipage}{0.3\textwidth}
@@ -293,9 +409,10 @@
Repeated syndrome extraction circuit for the three-qubit
repetition code under bit flip noise.
}
\label{fig:rep_code_multiple_rounds_bit_flip}
\end{figure}
\begin{figure}[t]
\begin{figure}[H]
\begin{gather*}
\hspace*{-33.3mm}%
\begin{array}{c}
@@ -388,7 +505,52 @@
}
\end{figure}
\begin{figure}[t]
%%%%%%%%%%%%%%%%
\subsection{Detector Error Matrix}
\label{subsec:Detector Error Matrix}
% Core idea
% TODO: Make this a proper definition?
Instead of using the measurements as parity indicators directly, we
may wish to combine them in some way.
We call such combinations \emph{detectors}.
Formally, a detector is a parity constraint on a set of measurement
outcomes \cite[Def.~2.1]{derks_designing_2025}.
\content{Detector matrix}
\content{Detector error matrix}
\content{One way of defining the detectors is ...}
\begin{figure}[H]
\centering
\tikzset{
gate/.style={
draw, %line width=1pt,
minimum height=2cm,
}
}
% tex-fmt: off
\begin{quantikz}[row sep=2mm, column sep=4mm, wire types={q,q,q,n,n,n}]
\lstick[3]{$\ket{\psi}_\text{L}$} & \gate[3]{\text{SE}_1} & & \gate[3]{\text{SE}_2} & & \gate[3]{\text{SE}_3} & & \gate[3]{\text{SE}_4} & \\
& & & & & & & & & \setwiretype{n} & \ldots \\
& \wire[d][3]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & \\
& \ctrl[wire=c]{0}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \\
& & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \\
& \gate[1]{\bm{D}_1} & & \gate[1]{\bm{D}_2} & & \gate[1]{\bm{D}_3} & & \gate[1]{\bm{D}_4} & \\
\end{quantikz}
% tex-fmt: on
\caption{Construction of detectors from measurements in the general case.}
\end{figure}
\content{The three-qubit repetition code as an exmaple}
\begin{figure}[H]
\centering
\hspace*{-5mm}
@@ -444,27 +606,38 @@
\label{fig:Construction of the detectors from the measurements}
\end{figure}
\begin{figure}[t]
\centering
%%%%%%%%%%%%%%%%
\subsection{Detector Error Models}
\label{subsec:Detector Error Models}
\tikzset{
gate/.style={
draw, %line width=1pt,
minimum height=2cm,
}
}
\content{Combination of detector error matrix and noise model}
\content{Contains all information necessary for decoding
\cite[Intro.]{derks_designing_2025}}
\content{Not only useful for decoding, but also for ... (Derks et al.)}
% tex-fmt: off
\begin{quantikz}[row sep=2mm, column sep=4mm, wire types={q,q,q,n,n,n}]
\lstick[3]{$\ket{\psi}_\text{L}$} & \gate[3]{\text{SE}_1} & & \gate[3]{\text{SE}_2} & & \gate[3]{\text{SE}_3} & & \gate[3]{\text{SE}_4} & \\
& & & & & & & & & \setwiretype{n} & \ldots \\
& \wire[d][3]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & \\
& \ctrl[wire=c]{0}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \\
& & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \\
& \gate[1]{\bm{D}_1} & & \gate[1]{\bm{D}_2} & & \gate[1]{\bm{D}_3} & & \gate[1]{\bm{D}_4} & \\
\end{quantikz}
% tex-fmt: on
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Practical Considerations}
\label{sec:Practical Considerations}
\caption{Construction of detectors from measurements in the general case.}
\end{figure}
% Intro
\content{Intro}
%%%%%%%%%%%%%%%%
\subsection{Practical Methodology}
\label{subsec:Practical Methodology}
\content{Per-round-LER explanation}
%%%%%%%%%%%%%%%%
\subsection{Stim}
\label{subsec:Stim}
\content{Circuit code heavily depends on the exact circuit construction}
\content{Not easy to predict how errors at different locations
propagate through the circuit an what detectors they affect}
\content{Stim is a software package that generates DEMs from circuits}
\content{The user still has to define the circuit themselves, and
especially the detectors \cite[Sec~2.5]{derks_designing_2025}}

1
src/thesis/chapters/4_decoding_under_dems.tex
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@@ -1,3 +1,4 @@
% TODO: Make all [H] -> [t]
\chapter{Decoding under Detector Error Models}
% Intro