Move 3-qubit repetition code check matrix; Rewrite DEM intro

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -1,18 +1,42 @@
 % TODO: Make all [H] -> [t]
-\chapter{Fault Tolerant QEC}
+\chapter{Fault-Tolerant Quantum Error Correction}
 
 % Intro
 
-An important challenge of \ac{qec} that was recognized early on is
-the fact that the error correction machinery itself may introduce new
-errors \cite[Sec.~III]{shor_scheme_1995}.
-Specifically for stabilizer codes, errors may happen during the
-syndrome extraction process, since it is implemented in quantum hardware itself.
-We call the errors the \ac{qec} procedure is supposed to correct
-\emph{input errors} and the errors introduced by the procedure itself
-\emph{internal errors}.
-In order to be \emph{fault-tolerant}, the procedure must be able to
-address both types of errors.
+In the previous chapter, we introduced the fundamental concepts of
+classical and quantum error correction.
+In this chapter, we shift our focus to some practical aspects of the
+implementation of \ac{qec} systems.
+
+% Fault tolerance as a general notion
+
+The reason we turned to \ac{qec} in the first place is to allow us to
+perform quantum computing despite the errors inherent to operations
+using qubits.
+While the use of error correcting codes may facilitate this, it also
+introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}:
+\begin{itemize}
+    \item We must be able to perform operations on the encoded state
+        in such a way that we do not lose the protection against errors.
+    \item \ac{qec} systems are themselves partially implemented in
+        quantum hardware. In addition to the errors we have
+        originally introduced them for, these systems must
+        be able to acount for the fact they are implemented on noisy
+        hardware themselves.
+\end{itemize}
+In the literature, both of these points are viewed under the umbrella
+of \emph{fault tolerance}.
+We focus only on the second aspect in this work.
+
+It was recognized early on as a challenge of \ac{qec} that the correction
+machinery itself may introduce new faults \cite[Sec.~III]{shor_scheme_1995}.
+Specifically for stabilizer codes, this can happen during the
+syndrome extraction process.
+We distinguish between the errors the QEC procedure is supposed to
+correct, which we call \emph{input errors}, and
+those introduced by the procedure itself, which we call \emph{internal errors}.
+In order to be fault-tolerant, the procedure must be able to
+address both.
 
 % Definition of fault tolerance
 
@@ -33,7 +57,7 @@ indicating which errors occurred, with
 \end{align*}
 \autoref{fig:fault_tolerance_overview} illustrates the flow of errors.
 Specifically for \ac{css} codes, a \ac{qec} procedure is deemed
-\emph{fault-tolerant}, if \cite[Def.~4.2]{derks_designing_2025}
+fault-tolerant, if \cite[Def.~4.2]{derks_designing_2025}
 \begin{gather*}
     \lVert \bm{e}_{\text{output},X} \rVert \le t \hspace{5mm}
     \forall\,  \bm{e}_\text{input}, \bm{e}_\text{internal} \in \{0,1\}^N:
@@ -51,9 +75,9 @@ errors the code is able to correct.
 The vectors $\bm{e}_{\text{output},X}$ and $\bm{e}_{\text{output},Z}$
 denote only $X$ and $Z$ errors respectively.
 
-In order to deal with errors during the syndrome extraction that flip
-single syndrome bits, multiple rounds of syndrome measurements must
-be performed.
+% TODO: Properly introduce d_min for QEC, specifically for CSS codes
+In order to deal with internal errors that flip syndrome bits,
+multiple rounds of syndrome measurements must be performed.
 Typically, the number of syndrome extraction rounds is chosen as $d_\text{min}$.
 
 % % This is the definition of a fault-tolerant QEC gadget
@@ -136,9 +160,22 @@ different error locations in the circuit.
 
 We will illustrate the most widely used types of error models on the
 example of the three-qubit repetition code for $X$ errors.
-This code has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
-\autoref{fig:pure_syndrome_extraction} shows the respective
-check matrix and syndrome extraction circuit.
+This is a code with check matrix
+\begin{align*}
+    \bm{H} =
+    \left[
+        \begin{array}{ccc|ccc}
+            0 & 0 & 0 & 0 & 0 & 0 \\
+            0 & 0 & 0 & 0 & 0 & 0 \\
+            0 & 0 & 0 & 1 & 1 & 0 \\
+            0 & 0 & 0 & 0 & 1 & 1
+        \end{array}
+    \right]
+    .
+\end{align*}
+We can see that it has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
+\autoref{fig:pure_syndrome_extraction} shows the corresponding
+syndrome extraction circuit.
 We refer to the qubits carrying the logical state
 $\ket{\psi}_\text{L}$ as \emph{data qubits}.
 Note that this is a concrete implementation using CNOT gates, as
@@ -147,16 +184,21 @@ opposed to the system-level view introduced in
 We visualize the different types of noise models in
 \autoref{fig:noise_model_types}.
 
-% Bit-flip noise
+%%%%%%%%%%%%%%%%
+\subsection{Bit-Flip Noise}
+\label{subsec:Bit-Flip Noise}
 
 The simplest type of noise model is \emph{bit-flip} noise.
 This corresponds to the classical \ac{bsc}, i.e., only $X$ errors on the
 data qubits are possible \cite[Appendix~A]{gidney_new_2023}.
-Note that we cannot use bit-flip noise to develop fault-tolerant
-systems, as it doesnt't account for errors during the syndrome extraction.
 This type of noise model is shown in \autoref{subfig:bit_flip}.
 
-% Depolarizing channel
+Note that we cannot use bit-flip noise to develop fault-tolerant
+systems, as it doesnt't account for errors during the syndrome extraction.
+
+%%%%%%%%%%%%%%%%
+\subsection{Depolarizing Channel}
+\label{subsec:Depolarizing Channel}
 
 Extending bit-flip noise to consider $X,Z$ or $Y$ instead of just $X$
 errors, we obtain the \emph{depolarizing channel}
@@ -165,10 +207,16 @@ errors, we obtain the \emph{depolarizing channel}
 It is well-suited for modeling memory experiments, where data qubits
 are stored idly for some period of time and errors accumulate due to
 decoherence.
+
+While the depolarizing channel is still not suited for the design and
+simulation of fault-tolerant systems, it is already complex enough to
+be used to gauge the suitability of a code for the \ac{qec} problem.
 Bit-flip noise and the depolarizing channel are sometimes referred to
 as \emph{code capacity noise models}.
 
-% Phenomenological noise
+%%%%%%%%%%%%%%%%
+\subsection{Phenomenological Noise}
+\label{subsec:Phenomenological Noise}
 
 The \emph{phenomenological noise model} is the first type of noise model we
 examine that accounts for faults during the syndrome extraction.
@@ -176,15 +224,22 @@ Here, we consider multiple rounds of syndrome measurements with a
 depolarizing channel before each round.
 Additionally, we allow for measurement errors by having $X$ error
 locations right before each measurement \cite[Appendix~A]{gidney_new_2023}.
-Note that it is enough to only consider $X$ errors at this point,
+Note that it is enough to only consider $X$ errors at these points,
 since that is the only type of error directly affecting the
 measurement outcomes.
-While not fully capturing all possible error mechanisms,
-phenomenological noise is already expressive enough to be used for
-the design of fault-tolerant circuitry \cite[Sec.~4.2]{derks_designing_2025}.
 This model is depicted in \autoref{subfig:phenomenological}.
 
-% Circuit-level noise
+While not fully capturing all possible error mechanisms,
+phenomenological noise is already a significant step beyond the code
+capacity noise models.
+Additionally, there are applications where the
+consideration of phenomenological noise is enough.
+It can, for example, be used for guiding the design of fault-tolerant
+circuitry [DTTBE25, Sec. 4.2].
+
+%%%%%%%%%%%%%%%%
+\subsection{Circuit-Level Noise}
+\label{subsec:Circuit-Level Noise}
 
 The most general type of noise model is \emph{circuit-level noise}.
 Here we not only consider noise inbetween syndrome extraction rounds
@@ -193,58 +248,27 @@ Specifically, we allow arbitrary $n$-qubit Pauli errors after each
 $n$-qubit gate \cite[Def.~2.5]{derks_designing_2025}.
 An $n$-qubit Pauli error is simply a series of correlated Pauli
 errors on each related individual qubit.
+This type of noise model is shown in \autoref{subfig:circuit_level}.
+
 While phenomenological noise is useful for some design aspects of
 fault tolerant circuitry, for simulations, circuit-level noise should
 always be used \cite[Sec.~4.2]{derks_designing_2025}.
-This type of noise model is shown in \autoref{subfig:circuit_level}.
-
-% Practical simulation aspects
-
-While these types of noise models give us some constraints on the
-types and locations of errors, the question of how exactly to choose
-their probabilities remains.
-When performing simulations, we typically sweep a range of values of
-the \emph{physical error rate} $p$, and there are different ways of
-obtaining the noise model $\bm{p}$ from this physical error rate.
-For the depolarizing channel in particular, we usually choose the
-same error rate $p/3$ for $X$, $Z$, and $Y$ errors.
-For circuit-level noise, various options exist, such as the \emph{SI1000}
-(superconducting inspired) or the \emph{EM3} (entangling
-measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}.
-These differ in the way they compute individual error probabilities
-from the physical error rate.
-In this work we only consider \emph{standard circuit-based depolarizing
-noise}, as this is the standard approach in the literature.
-We thus set the error probabilities of all error locations in the
-circuit-level noise model to the same value, the physical error rate $p$.
+Note that this introduces new challenges during the decoding process,
+as the decoding complexity is increased considerably due to the many
+error locations.
 
 \begin{figure}[t]
     \centering
 
-    \begin{minipage}{0.5\textwidth}
-        \begin{align*}
-            \bm{H} =
-            \left[
-                \begin{array}{ccc|ccc}
-                    0 & 0 & 0 & 0 & 0 & 0 \\
-                    0 & 0 & 0 & 0 & 0 & 0 \\
-                    0 & 0 & 0 & 1 & 1 & 0 \\
-                    0 & 0 & 0 & 0 & 1 & 1
-                \end{array}
-            \right]
-        \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}%
+    % 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
 
     \caption{
         Syndrome extraction circuit for the three-qubit repetition
@@ -374,19 +398,35 @@ circuit-level noise model to the same value, the physical error rate $p$.
 \section{Detector Error Models}
 \label{sec:Detector Error Models}
 
-\emph{Detector error models} constitue a standardized framework for
-passing information about the circuit used for \ac{qec} to a decoder.
-They are also useful in the design of fault-tolerant \ldots such as
-fault-tolerant quantum computing schemes \cite[Sec.~1]{derks_designing_2025}.
-% While alternate ways of considering fault tolerance exist, detector
-% error models
-% benefit from the fact that
-\content{Benefits of this approach \cite[Sec.~4.2]{derks_designing_2025}}
+\emph{Detector error models} (\acsp{dem}) constitue a standardized framework for
+passing information about a circuit used for \ac{qec} to a decoder.
+They are also useful as a theoretical tool to aid in the design of
+fault-tolerant \ac{qec} schemes.
+E.g., they can be used to easily determine whether a measurement
+schedule is fault-tolerant \cite[Example~12]{derks_designing_2025}.
+
+Other approaches of implementing fault tolerance exist, such as
+flag error correction, which uses additional ancilla qubits to detect
+potentially damaging high-weight errors \cite[Sec.~1]{chamberland_flag_2018}.
+However, \acp{dem} offer some unique advantages
+\cite[Sec.~4.2]{derks_designing_2025}:
+\begin{itemize}
+    \item They distinguish between errors based on their effect on
+        the measurements, not based on their location in the circuit.
+        This allows for merging equivalent errors, which decreases
+        decoding complexity.
+    \item Errors on the data qubits and on the measurements are
+        treated in a unified manner. This leads to a more powerful
+        description of the overall circuit.
+\end{itemize}
+In this work, we only consider the process of decoding under the
+\ac{dem} framework.
 
 % Core idea
 
-The goal we strive to achieve is to consider the possible error
-locations in the syndrome extraction circuitry during decoding.
+To achieve fault tolerance, the goal we strive towards is to
+consider the internal errors in addition to the input errors during
+the decoding process.
 The core idea behind detector error models is to do this by defining
 a new \emph{circuit code} that describes the circuit.
 Each \ac{vn} of this new code corresponds to an error location in the
@@ -665,7 +705,7 @@ 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}.
-Chaning the perspective in this way does not change the theoretical
+Changing the perspective in this way does not alter the theoretical
 error correcting capabilities of the circuit, but it may change the
 decoding performance when using a practical decoder.
 \red{[Possibly a few more words on this (maybe a mathematical
@@ -829,6 +869,9 @@ identical to obtain this structure.
 \subsection{Detector Error Models}
 \label{subsec:Detector Error Models}
 
+A \emph{detector error model} is the combination of the detector
+error matric $\bm{H}$ and the noise model $\bm{p}$.
+
 \content{Combination of detector error matrix and noise model}
 \content{Contains all information necessary for decoding
 \cite[Intro.]{derks_designing_2025}}
@@ -838,7 +881,25 @@ identical to obtain this structure.
 \section{Practical Considerations}
 \label{sec:Practical Considerations}
 
-% Intro
+% Practical simulation aspects
+
+While these types of noise models give us some constraints on the
+types and locations of errors, the question of how exactly to choose
+their probabilities remains.
+When performing simulations, we typically sweep a range of values of
+the \emph{physical error rate} $p$, and there are different ways of
+obtaining the noise model $\bm{p}$ from this physical error rate.
+For the depolarizing channel in particular, we usually choose the
+same error rate $p/3$ for $X$, $Z$, and $Y$ errors.
+For circuit-level noise, various options exist, such as the \emph{SI1000}
+(superconducting inspired) or the \emph{EM3} (entangling
+measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}.
+These differ in the way they compute individual error probabilities
+from the physical error rate.
+In this work we only consider \emph{standard circuit-based depolarizing
+noise}, as this is the standard approach in the literature.
+We thus set the error probabilities of all error locations in the
+circuit-level noise model to the same value, the physical error rate $p$.
 
 \content{Intro}