some work

thesis/chapters/quantum_computing.tex

@@ -56,6 +56,7 @@ and will be used in some calculations later.
 \end{equation}
 
 \subsubsection{Many Qbits}
+\label{ref:many_qbits}
 
 \begin{postulate}
     A $N$ qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $N$ single qbit 
@@ -174,12 +175,14 @@ to time as a measurement can be treated similarely while doing numerics.
 
 \subsection{Quantum Circuits}
 
-\textbf{FIXME: citation needed}
-
-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:
+As mentioned in \ref{ref:many_qbits} one can approximate an arbitrary $n$
+qbit gate $U$ as a product of some single qbit gates and either $CX$ or $CZ$.
+Writing (possibly huge) products of matrices is quite unpractical and very much
+unreadable. To address this problem quantum circuits have been introduced.
+These represent the qbits as a horizontal line, a gate acting on a qbit is
+a box with a name on the respective line. Quantum circuits are read from
+left to right. This means that a gate $U_i = Z_i X_i H_i$ has the
+circuit representation
 
 \[
 \Qcircuit @C=1em @R=.7em {
@@ -198,4 +201,6 @@ the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is
 }
 \]
 
-\textbf{TODO: more info about quantum circuits}
+\subsection{Quantum Algorithms}
+
+

thesis/chapters/stabilizer.tex

@@ -92,7 +92,7 @@ properties to understand the properties of the group.
     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
+In the following discussions $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as
 the properties of a set of stabilizers that are used in the discussions can be studied using only its
 generators.
 
@@ -117,4 +117,91 @@ can be diagonalized simultaneously. This motivates and justifies the following d
 
 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 
+The dimension of $V_S$ is not immediately clear. One can however show that
+for a set of stabilizers $\langle S^{(i)} \rangle_{i=1, ..., n-m}$ the dimension
+$dim V_S = 2^m$ \cite[Chapter 10.5]{nielsen_chuang_2010}. This yields the following important
+result:
+
+\begin{theorem}
+    For a $n$ qbit system and a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
+    space $V_S$ has $dim V_S = 1$, in particular there exists an up to a trivial phase unique
+    state $\ket{\psi}$ that is stabilized by $S$.
+    
+    Without proof.
+\end{theorem}
+
+In the following discussions for $n$ qbits a set $S = \langle S^{(i)} \rangle_{i=1,...,n}$ 
+of $n$ independent stabilizers will be assumed.
+
+
+\subsubsection{Dynamics of Stabilizer States}
+
+Consider a $n$ qbit state $\ket{\psi}$ that is the $+1$ eigenstate of $S = \langle S^{(i)} \rangle_{i=1,...,n}$
+and a unitary transformation $U$ that describes the dynamics of the system, i.e. 
+
+\begin{equation}
+    \ket{\psi'} = U \ket{\psi}
+\end{equation}
+
+It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$ anymore. There are
+however some statements that can still be made:
+
+\begin{equation}
+    \begin{aligned}
+    \ket{\psi'} &= U \ket{\psi} \\
+                &= U S^{(i)} \ket{\psi} \\
+                &= U S^{(i)} U^\dagger U\ket{\psi} \\
+                &= U S^{(i)} U^\dagger \ket{\psi'} \\
+                &= S^{\prime(i)} \ket{\psi'} \\
+    \end{aligned}
+\end{equation}
+
+Note that in \ref{def:stabilizer} it has been demanded that stabilizers are a 
+subgroup of the multilocal Pauli operators. This does not hold true for an arbitrary
+$U$ but there exists a group for which $S'$ will be a set of stabilizers.
+
+\begin{definition}
+    For $n$ qbits
+    \begin{equation}
+        C_n := \left\{U \in SU(n) | UpU^\dagger \in P_n \forall p \in P_n\right\}
+    \end{equation}
+
+    is called the Clifford group. $C_1 =: C_L$ is called the local Clifford group.
+\end{definition}
+
+\begin{theorem}
+    \begin{enumerate}
+        \item{$C_L$ can be generated using only $H$ and $S$.}
+        \item{$C_L$ can be generated from $\sqrt{iZ} = \exp(\frac{i\pi}{4}) S^\dagger$ 
+            and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i \\ -i & 1 \end{array}\right)$. 
+
+                Also $C_L$ is generated by a product of at most $5$ matrices $\sqrt{iZ}$, $\sqrt{-iX}$.
+                }
+            \item{$C_n$ can be generated using $C_L$ and $CZ$ or $CX$.} 
+    \end{enumerate} 
+\end{theorem}
+
+\begin{proof}
+    \begin{enumerate}
+        \item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
+        \item{
+            One can easily verify that $\sqrt{iZ} \in C_L$ and $\sqrt{-iX} \in C_L$.
+            Further one can show easily that (up to a global phase)
+            $H = \sqrt{iZ} \sqrt{-iX}^3 \sqrt{iZ}$ and $S = \sqrt{iZ}^3$.
+
+            The length of the product can be seen when explicitly calculating 
+            $C_L$.
+            }
+        \item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
+    \end{enumerate} 
+\end{proof}
+
+This is quite an important result: As under a transformation $U \in C_n$ $S'$ is a set of 
+$n$ independent stabilizers and $\ket{\psi'}$ is stabilized by $S'$ one can consider
+the dynamics of the stabilizers instead of the actual state. This is considerably more
+efficient as only $n$ stabilizers have to be modified, each being just the tensor
+product of $n$ Pauli matrices. This has led to the simulation using stabilizer tableaux
+\cite{gottesman_aaronson2008}.
+
+Interestingly also measurements are dynamics covered by the stabilizers.
+When an observable $g_i \in \{\pm X_i, \pm Y_i \pm Z_i\}$ is measured