Clean up qubits and multi-qubit systems section

src/thesis/chapters/2_fundamentals.tex

@@ -851,11 +851,12 @@ operations we use \emph{quantum gates} \cite[Sec.~1.3]{nielsen_quantum_2010}.
 We define a qubit to be a component with determinate
 states $\ket{0}$ and $\ket{1}$.
 The general description of the state $\ket{\psi}$ of a qubit is thus
-\begin{align*}
+\begin{align}
+    \label{eq:gen_qubit_state}
     \ket{\psi} = \alpha\ket{0} + \beta\ket{1}, \hspace{5mm} \alpha,
     \beta \in \mathbb{C}
     .%
-\end{align*}
+\end{align}
 
 % The tensor product and multi-qubit states
 
@@ -891,7 +892,7 @@ We have
     \end{split}
 \end{align}
 We call $\ket{x_0, \ldots, x_n}~, x_i \in \{0,1\}$ the
-\emph{computational basis states}.
+\emph{computational basis states} \cite[Sec.~4.6]{wohlin_guidelines_2014}.
 
 % Entanglement
 
@@ -917,7 +918,7 @@ information is stored in the correlations between the qubits
 
 As we can see in \autoref{eq:product_state}, the number of
 computational basis states needed to express the full composite state
-is $2^n$ \cite[Sec.~4.6]{nielsen_quantum_2010}.
+is $2^n$.
 This is in contrast to classical systems, where the dimensionality of
 the state space only grows linearly with $n$.
 This exponential growth of the state space is what makes it difficult
@@ -930,65 +931,84 @@ using quantum hardware in the first place
 
 After examining the modelling of single- and multi-qubit systems,
 we now shift our focus to describing the evolution of their states.
-\red{[Bloch sphere]}
-We do this using operators, also called \emph{gates}.
-The \emph{Pauli operators} $I$,
-$X$, $Z$ and $Y$ \cite[Appendix~2]{roffe_quantum_2019} are the most
-fundamental ones:
+We model state changes as operators.
+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
+linear combination of the \emph{Pauli operators}
+\cite[Sec.~2.2]{roffe_quantum_2019}
 \begin{align*}
     \begin{array}{c}
         I\text{ Operator} \\
         \hline\\
         \ket{0} \mapsto \ket{0} \\
-        \ket{1} \mapsto \ket{1} \\
+        \ket{1} \mapsto \ket{1}
     \end{array}%
     \hspace{10mm}%
     \begin{array}{c}
         X\text{ Operator} \\
         \hline\\
         \ket{0} \mapsto \ket{1} \\
-        \ket{1} \mapsto \ket{0} \\
+        \ket{1} \mapsto \ket{0}
     \end{array}%
     \hspace{10mm}%
     \begin{array}{c}
         Z\text{ Operator} \\
         \hline\\
         \ket{0} \mapsto -\ket{0} \\
-        \ket{1} \mapsto -\ket{1} \\
+        \ket{1} \mapsto -\ket{1}
     \end{array}%
     \hspace{10mm}%
     \begin{array}{c}
         Y\text{ Operator} \\
         \hline\\
         \ket{0} \mapsto -j\ket{1} \\
-        \ket{1} \mapsto -j\ket{0} \\
+        \hspace{2.75mm}\ket{1} \mapsto -j\ket{0} \hspace*{1mm}.
     \end{array}
 \end{align*}
 $I$ is the identity operator and $X$ and $Z$ are referred to as
 \emph{bit-flips} and \emph{phase-flips} respectively.
-
-% % TODO: Move this further down to the digitization of errors?
-% $Y$ can be represented as a combination of $X$ and $Z$ as $Y = jXZ$.
-
-% Operators over multiple qubits
-
-We can also perform operations on multi-qubit states.
-
-% TODO: Maybe the Hadamard operator and X <-> Z?
-
-% CNOT gates
-
-\red{[CNOT gates]}
-
-\red{
-    \begin{itemize}
-        \item Qubits and multi-qubit states
-            \begin{itemize}
-                \item The X,Z and Y operators
-                \item (?) Notation of operators on multi-qubit states
-            \end{itemize}
-    \end{itemize}
-}
+We also call these operators \emph{gates}.
+When working with multi-qubit systems, we can also apply Pauli gates
+to individual qubits independently, e.g., $I_1 X_2 I_3 Z_4 Y_5$.
+We often omit the identity operators, instead writing $X_2 Z_4 Y_5$.
+Other important operators include the \emph{Hadamard} and
+\emph{controlled-NOT (CNOT)} gates \cite[Sec.~1.3]{nielsen_quantum_2010}
+\vspace*{-7mm}
+\begin{figure}[H]
+    \centering
+    \begin{minipage}[t]{0.4\textwidth}
+        \centering
+        \begin{align*}
+            \begin{array}{c}
+                H\text{ Operator} \\
+                \hline\\
+                \ket{0} \mapsto \frac{1}{\sqrt{2}} \left( \ket{0} +
+                \ket{1} \right) \\[2mm]
+                \ket{1} \mapsto \frac{1}{\sqrt{2}} \left( \ket{0} -
+                \ket{1} \right)
+            \end{array}
+        \end{align*}
+    \end{minipage}%
+    \begin{minipage}[t]{0.4\textwidth}
+        \centering
+        \begin{align*}
+            \begin{array}{c}
+                CNOT\text{ Operator} \\
+                \hline\\
+                \ket{00} \mapsto \ket{00} \\
+                \ket{01} \mapsto \ket{01} \\
+                \ket{10} \mapsto \ket{11} \\
+                \hspace{2.75mm}\ket{11} \mapsto \ket{10} \hspace*{1mm}.
+            \end{array}
+        \end{align*}
+    \end{minipage}%
+\end{figure}
+\vspace{-4mm}
+\noindent Many more operators relevant to quantum computing exist, but they are
+not covered here as they are not central to this work.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Quantum Error Correction}
@@ -1005,6 +1025,7 @@ We can also perform operations on multi-qubit states.
                 \item Why we need QEC (correcting errors due to noisy gates)
                 \item Main challenges of QEC compared to classical
                     error correction
+                \item Logical vs physical states, logical vs physical operators
             \end{itemize}
         \item Stabilizer codes
             \begin{itemize}