Fix {ll,rr}bracket; Introduce Pauli group

src/thesis/chapters/2_fundamentals.tex

@@ -643,6 +643,8 @@ output \cite[Sec.~3.2.2]{griffiths_introduction_1995}.
 Operators are useful to describe the relations between different
 quantities relating to a particle.
 An example of this is the differential operator $\partial x$.
+Two operators $P_1$ and $P_2$ are said to \emph{commute}, if $P_1P_2
+= P_2P_1$ and \emph{anti-commute} if $P_1P_2 = -P_2P_1$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Observables}
@@ -871,7 +873,7 @@ 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 examine the state $\ket{\psi}$ of the composite system as.
+We examine the state $\ket{\psi}$ of the composite system.
 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
@@ -893,6 +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
 
@@ -933,7 +941,7 @@ After examining the modelling of single- and multi-qubit systems,
 we now shift our focus to describing the evolution of their states.
 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
+thus the only possible state change is a bit-flip, a general 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 operator as a linear combination of the
@@ -968,12 +976,20 @@ 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*}
+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 also call these operators \emph{gates}.
+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
-to individual qubits independently, e.g., $I_1 X_2 I_3 Z_4 Y_5$.
-We often omit the identity operators, instead writing $X_2 Z_4 Y_5$.
+to individual qubits independently, which we write ask e.g., $I_1 X_2
+I_3 Z_4 Y_5$.
+We often omit the identity operators, instead writing, e.g., $X_2 Z_4 Y_5$.
 Other important operators include the \emph{Hadamard} and
 \emph{controlled-NOT (CNOT)} gates \cite[Sec.~1.3]{nielsen_quantum_2010}
 \vspace*{-7mm}
@@ -1010,20 +1026,51 @@ 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.
 
-\indent\red{[We only need to consider X and Z errors]
-\cite[Equation~8]{roffe_quantum_2019}} \\
-\indent\red{[Explain commuting/anticommuting property of operators]
-[Journal~p.~46]}
-
 %%%%%%%%%%%%%%%%
 \subsection{Quantum Circuits}
 \label{Quantum Circuits}
 
-\noindent\indent\red{[Controlled operations]
-\cite[Sec.~4.3]{nielsen_quantum_2010}} \\
-\indent\red{[In case this reference is needed: Measurements
-\cite[Sec.~4.4]{nielsen_quantum_2010}]} \\
-\indent\red{[General circuit stuff] \cite[Sec.~1.3.4]{nielsen_quantum_2010}}
+% Intro
+
+Using these quantum gates, we can construct \emph{circuits} to manipulate
+the states of qubits \cite[Sec.~1.3.4]{nielsen_quantum_2010}.
+Circuits are read from left to right and each horizontal wire
+represents a qubit whose state evolves as it passes through
+successive gates.
+
+% General notation
+
+A single line carries a quantum state, while a double line
+denotes a classical bit, typically used to carry the result of a measurement.
+A measurement is represented by a meter symbol.
+In general, gates are represented as labeled boxes placed on one or more wires.
+An exception is the CNOT gate, where the operation is represented as
+the symbol $\oplus$.
+
+% Controlled gates & example
+
+We can additionally add a control input to a gate.
+This conditions its application on the state of another qubit
+\cite[Sec.~4.3]{nielsen_quantum_2010}.
+The control connection is represented by a vertical line connecting
+the gate to the corresponding qubit, where a filled dot is placed.
+A controlled gate applies the respective operation only if the
+control qubit is in state $\ket{1}$.
+An example of this is the CNOT gate introduced in
+\autoref{subsec:Qubits and Multi-Qubit States}, which is depicted in
+\autoref{fig:cnot_circuit}.
+
+\begin{figure}[t]
+    \centering
+
+    \begin{quantikz}
+        \lstick{$\ket{\psi}_1$} & \ctrl{1} & \\
+        \lstick{$\ket{\psi}_2$} & \targ{}  & \\
+    \end{quantikz}
+
+    \caption{CNOT gate circuit.}
+    \label{fig:cnot_circuit}
+\end{figure}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Quantum Error Correction}
@@ -1063,16 +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}.
-We circumvent the no-cloning restriction by not copying the state of
-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
+$n \in \mathbb{N}$ \emph{physical qubits}, $n>k$.
+We circumvent the no-cloning restriction by not copying the state of any of
+the $k$ logical qubits, instead spreading the total state out over all $n$
+physical ones \cite[Intro.]{calderbank_good_1996}.
+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}
@@ -1261,8 +1309,10 @@ Formally, we define the \emph{stabilizer group} $\mathcal{S}$ as
     \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*}
+where $[P_i,P_j] := P_iP_j - P_j P_i$ is called the \emph{commutator}
+of $P_i$ and $P_j$.
 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
@@ -1388,25 +1438,8 @@ $Z$ operators and some with only $X$ operators.
             \end{itemize}
             \textbf{Content:}
             \begin{itemize}
-                \item  General context
-                    \begin{itemize}
-                        \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}
-                        \item Definition of a stabilizer code
-                        \item The stabilizer its generators (note somewhere
-                                that the generators have to commute
-                                to be able to
-                            be measured without disturbing each other)
-                            (Why we need commutativity of the
-                                stabilizers [Journal,
-                            p.~51], [Got97, p.~6])
                         \item syndrome extraction circuit
                         \item Stabilizer codes are effectively the QM
                             % TODO: Actually binary linear codes or
@@ -1418,8 +1451,6 @@ $Z$ operators and some with only $X$ operators.
                             rather than working with the states directly
                             \cite[Sec.~10.5.1]{nielsen_quantum_2010}
                     \end{itemize}
-                \item Digitization of errors
-                \item CSS codes
                 \item Color codes?
                 \item Surface codes?
             \end{itemize}

src/thesis/main.tex

@@ -18,6 +18,8 @@
 %     sorting=nty,
 % ]{biblatex}
 \usepackage{todonotes}
+\usepackage{quantikz}
+\usepackage{stmaryrd}
 
 \usetikzlibrary{calc, positioning, arrows, fit}