some work

thesis/chapters/quantum_computing.tex

@@ -21,19 +21,13 @@ common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ wit
 \label{ref:singleqbitgates}
 
 \begin{equation}
-    X := \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right) 
+    \begin{aligned}
-\end{equation}
+    & X := \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right) \\
-\begin{equation}
+    & Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) \\
-    Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) 
+    & H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) \\
-\end{equation}
+    & R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)\\
-\begin{equation}
+    & I := \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right) \\
-    H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) 
+    \end{aligned}
-\end{equation}
-\begin{equation}
-    R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)
-\end{equation}
-\begin{equation}
-    I := \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right) 
 \end{equation}
 
 Note that $X = HZH$ and $Z = R_{\pi}$, so the set of $H, R_\phi$ is sufficient. 
@@ -47,6 +41,20 @@ transforming to the other Pauli eigenstates is done using $H$ and $SH$:
     S H Z H^\dagger S^\dagger = S X S^\dagger = Y
 \end{equation}
 
+The following states are the $\pm 1$ eigenstates of the $X$, $Y$, $Z$ operators
+and will be used in some calculations later.
+
+\begin{equation}
+    \begin{aligned}
+        &\ket{+_X} \equiv \ket{+} := \frac{1}{\sqrt{2}} (\ket{0} + \ket{1}) \\
+        &\ket{-_X} \equiv \ket{-} := \frac{1}{\sqrt{2}} (\ket{0} - \ket{1}) \\
+        &\ket{+_Y} := \frac{1}{\sqrt{2}} (\ket{0} + i\ket{1}) \\
+        &\ket{-_Y} := \frac{1}{\sqrt{2}} (\ket{0} - i\ket{1}) \\
+        &\ket{+_Z} := \ket{0} \\ 
+        &\ket{-_Z} := \ket{1} \\
+    \end{aligned}
+\end{equation}
+
 \subsubsection{Many Qbits}
 
 \begin{postulate}
@@ -130,8 +138,6 @@ matrices to the basis states)
 
 \subsubsection{Measurements}
 
-\textbf{FIXME} I don't like this at all.
-
 \begin{postulate}
     Let 
     $$\ket{\psi} = \alpha\ket{\phi_1} \otimes \ket{1}_j + \beta\ket{\phi_0} \otimes \ket{0}_j$$ 
@@ -145,6 +151,9 @@ matrices to the basis states)
 \end{postulate}
 
 Measuring a qbit will also yield a classical result $0$ or $1$ with the respective probabilities.
+Measurements are always performed in the computational basis, i.e. for a qbit
+$i$ $Z_i$ is measured. With the eigenstate of the $+1$ eigenvalue being the $\ket{0}$
+and $\ket{1}$ for the $-1$ eigenvalue of $Z$.
 
 \begin{corrolary}
     In general the measurement of a qbit is not invertible, in particular it cannot be represented as a
@@ -162,27 +171,31 @@ As a measurement is not unitary it is not a gate as in the definition above.
 In the following discussion the term \textit{measurement gate} will be used from time
 to time as a measurement can be treated similarely while doing numerics. 
 
-Measurements are always performed in the computational basis, i.e. for a qbit
-$i$ $Z_i$ is measured. Let the state to be measured be 
-
-\begin{equation}
-    \ket{\psi} = \alpha\ket{0}\otimes\ket{\psi_0} + \beta\ket{1}\otimes\ket{\psi_1}
-\end{equation}
-
-then a result $0$ is measured with probability $|\alpha|^2$ and $1$ with probability 
-$|\beta|^2 = 1 - |\alpha|^2$. The wave function is then collapsed to 
-
-\begin{equation}
-    \begin{aligned}
-    \ket{\psi'} = \left\{\begin{array}{c}\ket{0}\otimes\ket{\psi_0}, \mbox{ for a result 0 } \\
-        \ket{1}\otimes\ket{\psi_1}, \mbox{ for a result 1 }
-    \end{array}\right\}
-    \end{aligned}
-\end{equation}
 
 \subsection{Quantum Circuits}
 
-Quantum circuits are a simple and well-readable way to express the application 
+\textbf{FIXME: citation needed}
-of several gates on a state. 
 
-\textbf{TODO}
+Quantum circuits are a simple and well-readable way to express the application 
+of several gates on a state. Qbits are represented by horizontal line on which
+time goes from left to right. On these lines single qbit gates are 
+represented by a box:
+
+\[
+\Qcircuit @C=1em @R=.7em {
+& \gate{H} & \gate{X} & \gate{Z} &\qw \\
+}
+\]
+
+The controlled gates (such as $CX$ and $CZ$) have a vertical line from the control qbit to
+the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is
+
+\[
+\Qcircuit @C=1em @R=.7em {
+& \qw & \ctrl{2} & \qw & \qw &\qw \\
+& \qw & \qw & \qw & \ctrl{1} &\qw \\
+& \qw & \gate{X} & \qw & \gate{Z} &\qw \\
+}
+\]
+
+\textbf{TODO: more info about quantum circuits}

thesis/chapters/stabilizer.tex

@@ -14,3 +14,107 @@ performance has since been improved to $n\log(n)$ time on average \cite{andersbr
 
 \subsection{Stabilizers and Stabilizer States}
 
+\subsubsection{Local Pauli Group and Multilocal Pauli Group}
+
+\begin{definition}
+    \begin{equation}
+        P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
+    \end{equation}
+
+    Is called the Pauli group.
+\end{definition}
+
+The group property of $P$ can be verified easily. Note that 
+the elements of $P$ either commute or anticommute.
+
+\begin{definition}
+    For $n$ qbits
+
+    \begin{equation}
+        P_n := \{\bigotimes\limits_{i=0}^{n-1} p_i | p_i \in P\}
+    \end{equation}
+
+    is called the multilocal Pauli group on $n$ qbits. 
+\end{definition}
+
+The group property of $P_n$ follows directly from its definition
+via the tensor product as do the (anti-)commutator relationships.
+Further are $p \in P_n$ hermitian and have the eigenvalues $\pm 1$ for
+$p \neq \pm I$, $+1$ for $p = I$  and $-1$ for $p = -I$.
+
+\subsubsection{Stabilizers}
+
+\begin{definition}
+    \label{def:stabilizer}
+    An abelian subgroup $S = \{S^{(0)}, ..., S^{(N)}\}$ of $P_n$ is called a set of stabilizers iff
+    \begin{enumerate}
+        \item{$\forall i,j = 1, ..., N$ $[S^{(i)}, S^{(j)}] = 0$ $S^{(i)}$ and $S^{(j)}$ commute}
+        \item{$-I \notin S$}
+    \end{enumerate}
+\end{definition}
+
+\begin{lemma}
+    If $S$ is a set of stabilizers, the following statements are follow
+    directly
+
+    \begin{enumerate}
+        \item{$\pm iI \notin S$}
+        \item{$(S^{(i)})^2 = I$ for all $i$}
+        \item{$S^{(i)}$ are hermitian for all $i$}
+    \end{enumerate}
+\end{lemma}
+\begin{proof}
+
+    \begin{enumerate}
+        \item{$(iI)^2 = (-iI)^2 = -I$. Which contradicts the definition of $S$.}
+        \item{From the definition of $S$ ($G_n$ respectively) follows that any
+            $S^{(i)} \in S$ has the form $\pm i^l (\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$ where
+            $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$
+            is hermitian $(S^{(i)})^2$ is either $+I$ or $-I$. As $-I \notin S$ $(S^{(i)})^2 = I$ follows directly. 
+            }
+        \item{Following the argumentation above $(S^{(i)})^2 = -I \Leftrightarrow l=1$ 
+            therefore $(S^{(i)})^2 = -I \Leftrightarrow (S^{(i)})^\dagger \neq (S^{(i)})$.}
+    \end{enumerate}
+
+\end{proof}
+
+As considering all elements of a group can be unpractical for some calculations
+the generators of a group are introduced. It is usually enough to discuss the generator's
+properties to understand the properties of the group.
+
+\begin{definition}
+    For a finite group $G$ and some $m \in \mathbb{N}$ one denotes the generators
+    of G
+
+    $$ \langle g_1, ..., g_m \rangle \equiv \langle g_i \rangle_{i=1,...,m}$$
+
+    where $g_i \in G$, every element in $G$ can be written as a product of the $g_i$
+    and $m$ is the smallest integer for which these statements hold.
+\end{definition}
+
+In the following discussions $\rangle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as
+the properties of a set of stabilizers that are used in the discussions can be studied using only its
+generators.
+
+\subsubsection{Stabilizer States}
+
+One important basic property of quantum mechanics is that hermitian operators have real eigenvalues
+and eigenspaces associated with these eigenvalues. Finding these eigenvalues and eigenvectors
+is what one calls solving a quantum mechanical system. One of the most fundamental insights of 
+quantum mechanics is that operators that commute have a common set of eigenvectors, i.e. they
+can be diagonalized simultaneously. This motivates and justifies the following definition
+
+\begin{definition}
+    For a set of stabilizers $S$ the vector space
+
+    \begin{equation}
+        V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = +1\ket{\psi} \forall S^{(i)} \in S\}
+    \end{equation}
+
+    is called the space of stabilizer states associated with $S$ and one says
+    $\ket{\psi}$ is stabilized by $S$.
+\end{definition}
+
+It is clear that it is sufficient to show the stabilization property for the generators of
+$S$, as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$.
+The dimension of $V_S$ is not immediately