Fix {ll,rr}bracket; Introduce Pauli group

src/thesis/chapters/2_fundamentals.tex

@@ -895,6 +895,12 @@ We have
 \end{align}
 We call $\ket{x_0, \ldots, x_n}~, x_i \in \{0,1\}$ the
 \emph{computational basis states} \cite[Sec.~4.6]{nielsen_quantum_2010}.
+To additionally simplify set notation, we define
+\begin{align*}
+    \mathcal{M}^{\otimes n} := \underbrace{\mathcal{M}\otimes \ldots
+    \otimes \mathcal{M}}_{n \text{ times}}
+    .%
+\end{align*}
 
 % Entanglement
 
@@ -970,11 +976,14 @@ Luckily, we can express any operator as a linear combination of the
         \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.
 In fact, if we allow for complex coefficients, the $X$ and $Z$
 operators are sufficient to express any other operator as a linear
 combination \cite[Sec.~2.2]{roffe_quantum_2019}.
+$I$ is the identity operator and $X$ and $Z$ are referred to as
+\emph{bit-flips} and \emph{phase-flips} respectively.
+We call the set $\mathcal{G}_n = \left\{ \pm I,\pm jI, \pm X,\pm jX,
+\pm Y,\pm jY, \pm Z, \pm jZ \right\}^{\otimes n}$ the \emph{Pauli
+group} over $n$ qubits.
 
 In the context of modifying qubit states, we also call operators \emph{gates}.
 When working with multi-qubit systems, we can also apply Pauli gates
@@ -1101,16 +1110,17 @@ three main restrictions apply \cite[Sec.~2.4]{roffe_quantum_2019}:
 % General idea (logical vs. physical gates) + notation
 
 Much like in classical error correction, in \ac{qec} information
-is protected by mapping it onto codewords in an expanded space,
+is protected by mapping it onto codewords in a higher-dimensional space,
 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}.
+$n \in \mathbb{N}$ \emph{physical qubits}, $n>k$.
-We circumvent the no-cloning restriction by not copying the state of
+We circumvent the no-cloning restriction by not copying the state of any of
-the $k$ logical qubits, instead spreading it out over all $n$
+the $k$ logical qubits, instead spreading the total state out over all $n$
-physical ones \cite[Intro.]{calderbank_good_1996}
+physical ones \cite[Intro.]{calderbank_good_1996}.
-To differentiate a quantum codes from classical ones, we denote a
+To differentiate quantum codes from classical ones, we denote a
 code with parameters $k,n$ and minimum distance $d_\text{min}$ using
-double brackets, as $[[ n,k,d_\text{min} ]]$ \cite[Sec.~4]{roffe_quantum_2019}.
+double brackets, as $\llbracket n,k,d_\text{min} \rrbracket$
+\cite[Sec.~4]{roffe_quantum_2019}.
 
 %%%%%%%%%%%%%%%%
 \subsection{Stabilizer Measurements}

src/thesis/main.tex

@@ -19,6 +19,7 @@
 % ]{biblatex}
 \usepackage{todonotes}
 \usepackage{quantikz}
+\usepackage{stmaryrd}
 
 \usetikzlibrary{calc, positioning, arrows, fit}