Update week6.do.txt

Author: mhjensen

doc/src/week6/week6.do.txt

@@ -56,68 +56,92 @@ The Pauli matrices (and gate operations following therefrom) are defined as
 
 !split
 ===== Pauli-$X$ gate =====
+
 The Pauli-$X$ gate is also known as the _NOT_ gate, which flips the state of the qubit.
 !bt
 \begin{align}
 	X\ket{0} &= \ket{1}, \\
 	X\ket{1} &= \ket{0}.	
 \end{align}
 !et
-The Pauli-Y gate flips the bit and multiplies the phase by $ i $. 
+The Pauli-$Y$ gate flips the bit and multiplies the phase by $ i $. 
+!bt
 \begin{align}
 	Y\ket{0} &= i\ket{1}, \\
 	Y\ket{1} &= -i\ket{0}.
 \end{align}
-The Pauli-Z gate multiplies only the phase of $\ket{1}$ by $ -1 $.
+!et
+The Pauli-$Z$ gate multiplies only the phase of $\ket{1}$ by $ -1 $.
+!bt
 \begin{align}
 	Z\ket{0} &= \ket{0}, \\
 	Z\ket{1} &= -\ket{1}.
 \end{align}
+!et
+!split
+===== Hadamard gate =====
 
-\subsubsection{Hadamard Gate}
 The Hadamard gate is defined as
-\begin{equation}
+!bt
+\[
 	H = \frac{1}{\sqrt{2}} \begin{pmatrix}
 		1 & 1 \\
 		1 & -1
 	\end{pmatrix}.
-\end{equation}
+\]
+!et
+
 It creates a superposition of the $ \ket{0} $ and $ \ket{1} $ states.
+!bt
 \begin{align}
 	H\ket{0} &= \frac{1}{\sqrt{2}} \left( \ket{0} + \ket{1} \right), \\
 	H\ket{1} &= \frac{1}{\sqrt{2}} \left( \ket{0} - \ket{1} \right).
 \end{align}
+!et
 
-\subsubsection{Phase Gates}
-The S-phase gate is usually denoted as $ S $ and is defined as
+!split
+===== Phase Gates =====
+The phase gate is usually denoted as $S$ and is defined as
+!bt
 \begin{equation}
 	S = \begin{pmatrix}
 		1 & 0 \\
 		0 & i
 	\end{pmatrix}.
 \end{equation}
+!et
 It multiplies only the phase of the $ \ket{1} $ state by $ i $.
+!bt
 \begin{align}
 	S\ket{0} &= \ket{0}, \\
 	S\ket{1} &= i\ket{1}.
 \end{align}
-Its inverse
+!et
+
+!split
+===== The  inverse of the $S$-gate =====
+
+The inverse
+!bt
 \begin{equation}
 	S^\dagger = \begin{pmatrix}
 		1 & 0 \\
 		0 & -i
 	\end{pmatrix}
 \end{equation}
-is known as the $ S^\dagger $ gate which applies a $ -i $ phase shift to \ $ \ket{1}$.
+!et
+is known as the $ S^\dagger $ gate which applies an $\imath$ phase shift to $\ket{1}$.
+!bt
 \begin{align}
 	S^\dagger\ket{0} &= \ket{0}, \\
 	S^\dagger\ket{1} &= -i\ket{1}.
 \end{align}
+!et
+!split
+===== Two-qubit gates =====
 
-\subsection{Two Qubit Gates}
-\subsubsection{CNOT Gate}
-\label{ssub:cnot_gate}
 The CNOT gate is a two-qubit gate which acts on two qubits, a control qubit and a target qubit. The CNOT gate is defined as
+!bt
 \begin{equation}
 	\text{CNOT} = \begin{pmatrix}
 		1 & 0 & 0 & 0 \\
@@ -126,17 +150,22 @@ The CNOT gate is a two-qubit gate which acts on two qubits, a control qubit and
 		0 & 0 & 1 & 0
 	\end{pmatrix}.
 \end{equation}
+!et
 It is often used to perform linear entanglement on qubits.
+!bt
 \begin{align*}
 	\label{eq:cnot-behaviour}
 	\cnot \ket{00} &= \ket{00}, \\
 	\cnot \ket{01} &= \ket{01}, \\
 	\cnot \ket{10} &= \ket{11}, \\
 	\cnot \ket{11} &= \ket{10}.
 \end{align*}
-\subsubsection{SWAP gate}%
-\label{ssub:swap_gate}
+!et
+
+!split
+===== The SWAP gate =====
 The SWAP gate is a two-qubit gate which swaps the state of two qubits. It is defined as
+!bt
 \begin{equation}
 	\text{SWAP} = \begin{pmatrix}
 		1 & 0 & 0 & 0 \\
@@ -145,13 +174,14 @@ The SWAP gate is a two-qubit gate which swaps the state of two qubits. It is def
 		0 & 0 & 0 & 1
 	\end{pmatrix}.
 \end{equation}
+!et
 \begin{align*}
-	\swp \ket{00} &= \ket{00}, \\
-	\swp \ket{01} &= \ket{10}, \\
-	\swp \ket{10} &= \ket{01}, \\
-	\swp \ket{11} &= \ket{11}.
+	\text{SWAP}\ket{00} &= \ket{00}, \\
+	\text{SWAP} \ket{01} &= \ket{10}, \\
+	\text{SWAP} \ket{10} &= \ket{01}, \\
+	\text{SWAP} \ket{11} &= \ket{11}.
 \end{align*}
-
+!et
 
 !split
 ===== Pauli Strings =====