some more work

thesis/chapters/quantum_computing.tex

@@ -142,8 +142,7 @@ $CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1}
 
 One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough
 to generate an arbitrary $N$ qbit gate\cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}.
-The matrix representation of $CX$ and $CZ$ for two qbits is given by (this is quickly verified by applying the
+The matrix representation of $CX$ and $CZ$ for two qbits is given by
-matrices to the basis states)
 
 \begin{equation}
     CX_{1, 0} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{array}\right)
@@ -216,6 +215,37 @@ the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is
 }
 \]
 
+Several qbits can be abbreviated by writing a slash on the qbit line:
+
+\[
+\Qcircuit @C=1em @R=.7em {
+    & {/} \qw & \qw & \gate{Z^{\otimes n}} & \qw \\
+}
+\]
+
+
 \subsection{Quantum Algorithms}
 
+The great hope behind quantum computing is that it will speed up some computations
+exponentially using algorithms that utilize the laws of quantum mechanics. Current 
+algorithms are based upon quantum search and quantum fourier transform \cite{nielsen_chuang_2010}.
+The latter is particular interesting for physical problems as it is used in the phase estimation
+algorithm that can be used to analyze the spectrum of the transfer matrix
+
+\begin{equation}
+    T = \exp(itH)
+\end{equation}
+
+The eigenvalue of $T$ can now be estimated by using the phase estimation circuit:
+
+\[
+\Qcircuit @C=1em @R=.7em {
+    & \lstick{\ket{0}}      & {/^N} \qw & \gate{H^{\otimes n}} & \ctrl{1} & \gate{FT^\dagger} & \qw & \meter & \rstick{x}\\
+    & \lstick{\ket{\varphi}} & {/} \qw & \qw & \gate{T} & \qw & {/} \qw & \qw& \rstick{\ket{\varphi}} \\
+}
+\]
+
+Where $T\ket{\varphi} = \exp(2\pi i\varphi) \ket{\varphi}$ and the measurement result $\tilde{\varphi} = \frac{x}{2^n}$ is an
+estimate for $\varphi$. If a success rate of $1-\epsilon$ and an accuracy of $| \varphi - \tilde{\varphi} | < 2^{-m}$
+is wanted $N = m + \log_2(2 + \frac{1}{2\epsilon})$ qbits are required\cite{nielsen_chuang_2010}\cite{lehner2019}.
 

thesis/chapters/stabilizer.tex

@@ -31,7 +31,7 @@ the elements of $P$ either commute or anticommute.
     For $n$ qbits
 
     \begin{equation}
-        P_n := \left\{\bigotimes\limits_{i=0}^{n-1} p_i | p_i \in P\right\}
+        P_n := \left\{\bigotimes\limits_{i=0}^{n-1} p_i \middle| p_i \in P\right\}
     \end{equation}
 
     is called the multilocal Pauli group on $n$ qbits. 
@@ -68,12 +68,12 @@ via the tensor product as do the (anti-)commutator relationships.
     \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
+            $S^{(i)} \in S$ has the form $\pm i^l \left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$ where
-            $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$
+            $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $\left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$
             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 $(S^{(i)})^2 = -I \Leftrightarrow l=1$ 
+        \item{Following the argumentation above $\left(S^{(i)}\right)^2 = -I \Leftrightarrow l=1$ 
-            therefore $(S^{(i)})^2 = -I \Leftrightarrow (S^{(i)})^\dagger \neq (S^{(i)})$.}
+            therefore $\left(S^{(i)}\right)^2 = -I \Leftrightarrow \left(S^{(i)}\right)^\dagger \neq S^{(i)}$.}
     \end{enumerate}
 
 \end{proof}
@@ -108,7 +108,7 @@ can be diagonalized simultaneously. This motivates and justifies the following d
     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\}
+        V_S := \left\{\ket{\psi} \middle| S^{(i)}\ket{\psi} = +1\ket{\psi} \forall S^{(i)} \in S\right\}
     \end{equation}
 
     is called the space of stabilizer states associated with $S$ and one says
@@ -124,7 +124,7 @@ result:
 
 \begin{theorem}
     \label{thm:unique_s_state}
-    For a $n$ qbit system and a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
+    For a $n$ qbit system and stabilizers $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ 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 $S$.
     
@@ -157,14 +157,14 @@ however some statements that can still be made:
     \end{aligned}
 \end{equation}
 
-Note that in \ref{def:stabilizer} it has been demanded that stabilizers are a 
+Note that in Definition \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\}
+        C_n := \left\{U \in SU(n) \middle| 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.
@@ -235,12 +235,12 @@ the result of the measurement.
 
 \begin{lemma}
     \label{lemma:stab_measurement}
-    Let $J := \{ S^{(i)} | [g_a, S^{(i)}] \neq 0\} \neq \{\}$. When measuring 
+    Let $J := \left\{ S^{(i)} \middle| [g_a, S^{(i)}] \neq 0\right\} \neq \{\}$. When measuring 
     $\frac{I + (-1)^s g_a}{2} $
     $1$ and $0$ are obtained with probability $\frac{1}{2}$ and after choosing
     a $j \in J$ the new state $\ket{\psi'}$ is stabilized by 
     \begin{equation}
-        \langle \{(-1)^s g_a\} \cup \{S^{(i)} S^{(j)} | S^{(i)} \in J \setminus \{S^{(j)}\} \} \cup J^c\rangle
+        \langle \{(-1)^s g_a\} \cup \left\{S^{(i)} S^{(j)} \middle| S^{(i)} \in J \setminus \{S^{(j)}\} \right\} \cup J^c\rangle
     \end{equation}
 \end{lemma}
 
@@ -287,9 +287,9 @@ vertex operator-free will be clear in the following section about graph states.
     \label{def:graph}
     The tuple $(V, E)$ is called a graph iff $V$ is a set of vertices with $|V| = n \in \mathbb{N}$.
     In the following $V = \{0, ..., n-1\}$ will be used.
-    $E$ is the set of edges $E = \left\{\{i, j\} | i,i \in V, i \neq j\right\}$.
+    $E$ is the set of edges $E = \left\{\{i, j\} \middle| i,i \in V, i \neq j\right\}$.
 
-    For a vertex $i$ $n_i := \left\{j \in V | \{i, j\} \in E\right\}$ is called the neighbourhood
+    For a vertex $i$ $n_i := \left\{j \in V \middle| \{i, j\} \in E\right\}$ is called the neighbourhood
     of $i$.
 \end{definition}
 
@@ -436,11 +436,11 @@ that will be used later.
                         = K_{G'}^{(j)}K_{G}^{(a)}\ket{\bar{G}'} = K_{G'}^{(j)}\ket{\bar{G}'}
     \end{equation}
 
-    Because $\{K_G^{(i)} | i \notin n_a\} \cup \{S^{(i)} | i\in n_a\}$ and 
+    Because $\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{S^{(i)} \middle| i\in n_a\right\}$ and 
-    $\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}$ are both $n$ commuting
+    $\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}$ are both $n$ commuting
     multi-local Pauli operators where the $S^{(i)}$ can be generated from the $K_{G'}^{(i)}$
     and $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$ 
-    $\langle\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}\rangle$
+    $\langle\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}\rangle$
     are the stabilizers of $\ket{\bar{G}'}$ and the associated graph is changed as given
     in the third equation.
 \end{proof}
@@ -665,8 +665,8 @@ so it is easier to list the operators that anticommute:
 
 \begin{equation}
     \begin{aligned}
-        A_{\pm X_a} &= \left\{j | \{j, a\} \in E\right\}\\
+        A_{\pm X_a} &= \left\{j \middle| \{j, a\} \in E\right\}\\
-        A_{\pm Y_a} &= \left\{j | \{j, a\} \in E\right\} \cup \{a\} \\
+        A_{\pm Y_a} &= \left\{j \middle| \{j, a\} \in E\right\} \cup \{a\} \\
         A_{\pm Z_a} &= \{a\}\\
     \end{aligned}
 \end{equation}
@@ -695,8 +695,8 @@ $K_G^{(a)}$ is chosen. The graph is changed according to
 
 \begin{equation}
     \begin{aligned}
-        E'_{Z} &= E \setminus \left\{\{i,a\} | i \in V\right\}\\
+        E'_{Z} &= E \setminus \left\{\{i,a\} \middle| i \in V\right\}\\
-        E'_{Y} &= E \Delta (n_a \otimes n_a) \setminus \left\{\{i,a\} | i \in V\right\}\\
+        E'_{Y} &= E \Delta (n_a \otimes n_a) \setminus \left\{\{i,a\} \middle| i \in V\right\}\\
     \end{aligned}
 \end{equation}
 
@@ -714,7 +714,7 @@ For $g_a = X_a$ one has to choose a $b \in n_a$ and the transformations are
         E'_{X} = E &\Delta (n_b \otimes n_a) \\
                 & \Delta ((n_b \cap n_a) \otimes (n_b \cap n_a)) \\
                 & \Delta (\{b\} \otimes (n_a \setminus \{b\})) \\
-                & \setminus \left\{\{i,a\} | i \in V\right\}\\
+                & \setminus \left\{\{i,a\} \middle| i \in V\right\}\\
     \end{aligned}
 \end{equation}
 

thesis/main.bib

@@ -147,3 +147,11 @@
     author={M. Hein, J. Eisert, H.J. Briegel},
     note={https://arxiv.org/abs/quant-ph/0307130v7}
 }
+@article{
+    lehner2019,
+    title={Lecture Notes on Quantum Computing SS 2019},
+    year=2019,
+    author={Christoph Lehner},
+    note={Unpublished Work}
+}
+