Finish first draft of stabilizer measurement section

src/thesis/chapters/2_fundamentals.tex

@@ -871,7 +871,7 @@ Take for example the two qubits
 \end{align*}
 % TODO: Fix the fact that \psi is used above for the single-qubit
 % case and below for the multi-qubit case
-We denote the state of the composite system as $\ket{\psi}$.
+We examine the state $\ket{\psi}$ of the composite system as.
 Assuming the qubits are independent, this is a \emph{product state}
 $\ket{\psi} = \ket{\psi_1}\otimes\ket{\psi_2}$.
 When not ambiguous, we may omit the tensor product symbol or even write
@@ -896,7 +896,7 @@ We call $\ket{x_0, \ldots, x_n}~, x_i \in \{0,1\}$ the
 
 % Entanglement
 
-States that are not able to be decomposed into such a product
+States that are not able to be decomposed into such products
 are called \emph{entangled} \cite[Sec.~2.2.8]{nielsen_quantum_2010}.
 An example of such states are the \emph{Bell states}
 \begin{align*}
@@ -1068,7 +1068,7 @@ thereby introducing redundancy.
 To this end, $k \in \mathbb{N}$ \emph{logical qubits} are mapped onto
 $n \in \mathbb{N},~n>k$ \emph{physical qubits}.
 We circumvent the no-cloning restriction by not copying the state of
-the $k$ logical qubits, but rather spreading it out over all $n$
+the $k$ logical qubits, instead spreading it out over all $n$
 physical ones \cite[Intro.]{calderbank_good_1996}
 To differentiate a quantum codes from classical ones, we denote a
 code with parameters $k,n$ and minimum distance $d_\text{min}$ using
@@ -1237,24 +1237,51 @@ E.g., $P_\mathcal{C}$ will eliminate all components of $E
 \ket{\psi}_\text{L}$ that lie in $\mathcal{F}$.
 This process, together with the fact that any coherent error can be
 decomposed into a linear combination of $X$ and $Z$ errors, means
-that it is enough for a \ac{qec} scheme only needs to be able to
-correct $X$ and $Z$ errors.
+that it is enough for a \ac{qec} to be able to correct only $X$ and $Z$ errors.
 This effect is referred to as error \emph{digitization}
 \cite[Sec.~2.2]{roffe_quantum_2019}.
 
 % The stabilizer group
 
-\indent\red{[Conditions for the stabilizer group
-\cite[Sec.~4.1]{roffe_quantum_2019}]} \\
-\indent\red{[Why we care about (anti-)commutativity of the
-        stabilizers with errors + Z-type operators for X type errors and vice
-versa ]} \\
-\indent\red{[(?) Stabilizer generators]}
-
-% A general stabilizer measurement circuit
-
-\indent\red{[General stabilizer measurement circuit
-\cite[Figure~4]{roffe_quantum_2019}]}
+Operators such as $Z_1Z_2$ above are called \emph{stabilizers}.
+An operator $P_i \in \mathcal{G}_n$ is called a stabilizer of an
+$[[n, k, d_\text{min}]]$ code $\mathcal{C}$, if
+\begin{itemize}
+    \item It stabilizes all logical states, i.e.,
+        $P_i\ket{\psi}_\text{L} = (+1)\ket{\psi}_\text{L} ~\forall~
+        \ket{\psi}_\text{L} \in \mathcal{C}$.
+    \item It commutes with all other stabilizers of the code. This
+        property is important to be able to measure the eigenvalue of
+        a stabilizer without disturbing the eigenvectors of the
+        others \cite[Sec.~1.2]{gottesman_stabilizer_1997}.
+\end{itemize}
+Formally, we define the \emph{stabilizer group} $\mathcal{S}$ as
+\cite[Sec.~4.1]{roffe_quantum_2019}
+\begin{align*}
+    \mathcal{S} = \left\{P_i \in \mathcal{G}_n ~:~ P_i \ket{\psi}_\text{L} =
+        (+1)\ket{\psi}_\text{L} \forall \ket{\psi}_\text{L} ~\cap~
+    [P_i,P_j] = 0 \forall i,j\right\}
+    .%
+\end{align*}
+We care in particular about the commuting properties of stabilizers
+with respect to possible errors.
+The measurement circuit for an arbitrary stabilizer $P_i$ modifies
+the state as \cite[Eq.~29]{roffe_quantum_2019}
+\begin{align*}
+    E\ket{\psi}_\text{L}\ket{0}_\text{A}
+    \hspace{3mm}\mapsto\hspace{3mm}
+    \frac{1}{2} \left( I + P_i
+    \right)E\ket{\psi}_\text{L}\ket{0}_\text{A} + \frac{1}{2}
+    \left( I - P_i \right)E\ket{\psi}_\text{A} \ket{1}_\text{A}
+    .%
+\end{align*}
+If a given error $E$ anticommutes with $P_i$, we have
+\begin{align*}
+    EP_i \ket{\psi}_{L} &= -P_i E \ket{\psi}_\text{L} \\
+    \Rightarrow E \ket{\psi}_{L} &= -P_i E \ket{\psi}_\text{L} \\
+    \Rightarrow \left( I + P_i \right)E\ket{\psi}_\text{L} &= 0
+\end{align*}
+and the stabilizer measurement returns 1.
 
 %%%%%%%%%%%%%%%%
 \subsection{Stabilizer Codes}
@@ -1290,6 +1317,7 @@ A full \emph{syndrome extraction circuit} is depicted in \autoref{fig:sec}.
 the encoded computations \cite[Sec.~2.6]{derks_designing_2025}]} \\
 \indent\red{[X and Z measurements can be performed with only CNOT and
 Hadamard gates \cite[Sec.~10.5.8]{nielsen_quantum_2010}]} \\
+\indent\red{[(?) Stabilizer generators]} \\
 \indent\red{[Parity-check matrix \cite[Sec.~10.5.1]{nielsen_quantum_2010}]}
 
 \begin{figure}[t]
@@ -1329,6 +1357,7 @@ handle $X$- and $Z$-type errors independently.
 We can then separate the stabilizer generators into some with only
 $Z$ operators and some with only $X$ operators.
 
+\indent\red{[Z-type operators for X type errors and vice versa ]} \\
 \indent\red{[Construction from two binary linear codes
 \cite[p.~452,469]{nielsen_quantum_2010}]}