[thesis] modify comments, phrasing, red outline

src/thesis/chapters/2_fundamentals.tex

@@ -914,7 +914,7 @@ Instead of employing only the individual qubit states, the
 information is stored in the correlations between the qubits
 \cite[Sec.~2]{preskill_quantum_2018}.
 
-% The size of the vector spaced
+% The size of the vector space
 
 As we can see in \autoref{eq:product_state}, the number of
 computational basis states needed to express the full composite state
@@ -927,7 +927,7 @@ It is also what motivated the research into performing computations
 using quantum hardware in the first place
 \cite[Sec.~3]{feynman_simulating_1982}.
 
-% Basic types of gates: The X,Y,Z operators, Bloch sphere
+% Basic types of gates
 
 After examining the modelling of single- and multi-qubit systems,
 we now shift our focus to describing the evolution of their states.
@@ -936,8 +936,8 @@ Unlike classical systems, where there are only two possible states and
 thus the only possible state change is a bit-flip, a gerenal qubit
 state as shown in \autoref{eq:gen_qubit_state} lives on a continuum of values.
 We thus technically also have an infinite number of possible state changes.
-Luckily, we can express any single-qubit coherent operator as a
+Luckily, we can express any operator as a linear combination of the
-linear combination of the \emph{Pauli operators}
+\emph{Pauli operators}
 \cite[Sec.~2.2]{roffe_quantum_2019}
 \begin{align*}
     \begin{array}{c}
@@ -1010,18 +1010,38 @@ Other important operators include the \emph{Hadamard} and
 \noindent Many more operators relevant to quantum computing exist, but they are
 not covered here as they are not central to this work.
 
+%%%%%%%%%%%%%%%%
+\subsection{Quantum Circuits}
+\label{Quantum Circuits}
+
+\red{[TODO] \cite[Sec.~1.3.4]{nielsen_quantum_2010}}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Quantum Error Correction}
 \label{sec:Quantum Error Correction}
 
+% Why we need quantum error correction
+
+% The unique challenges of QEC compared to classical FEC
+
+% The unique challenges of QEC compared to classical FEC
+
 \red{
-    \textbf{Content:}
+    \textbf{General Notes:}
     \begin{itemize}
+        \item Note that there are other codes than stabilizer codes
+            (and research and give some examples), but only
+            stabilizer codes are considered in this work
+        \item Degeneracy
+        \item The QEC decoding problem (considering degeneracy)
+            \cite[Sec.~2.3]{yao_belief_2024}
         \item Why we need commutativity of the stabilizers [Journal,
             p.~51], [Got97, p.~6]
+    \end{itemize}
+    \textbf{Content:}
+    \begin{itemize}
         \item  General context
             \begin{itemize}
-                \item Why we want QC
                 \item Why we need QEC (correcting errors due to noisy gates)
                 \item Main challenges of QEC compared to classical
                     error correction
@@ -1047,33 +1067,6 @@ not covered here as they are not central to this work.
         \item CSS codes
         \item Color codes?
         \item Surface codes?
-        \item Fault tolerant error correction (gates with which we do
-            error correction are also noisy)
-            \begin{itemize}
-                \item Transversal operations
-                \item \dots
-            \end{itemize}
-        \item Circuit level noise
-        \item Detector error model
-            \begin{itemize}
-                \item Columns of the check matrix represent different
-                    possible error patterns $\rightarrow$ Check matrix
-                    doesn't quite correspond to the codewords we used
-                    initially anymore, but some similar structure ist
-                    still there (compare with syndrome)
-            \end{itemize}
-    \end{itemize}
-    \textbf{General Notes:}
-    \begin{itemize}
-        \item Give a brief overview of the history of QEC
-        \item Note (and research if this is actually correct) that QC
-            was developed on an abstract level before thinking of
-            what hardware to use
-        \item Note that there are other codes than stabilizer codes
-            (and research and give some examples), but only
-            stabilizer codes are considered in this work
-        \item Degeneracy
-        \item The QEC decoding problem (considering degeneracy)
     \end{itemize}
 }