some final work

presentation/main.tex

@@ -6,6 +6,7 @@
 \usepackage{setspace} 
 \usepackage{amsmath} 
 \usepackage{hyperref}
+\usepackage{url}
 \usepackage{geometry} 
 \usepackage{enumerate}
 \usepackage{physics}
@@ -13,7 +14,7 @@
 %\usepackage{struktex}
 \usepackage{qcircuit}
 \usepackage{adjustbox}
-\usepackage{tikz}
+%\usepackage{tikz}
 
 
 \usetheme{metropolis}
@@ -71,7 +72,7 @@
         \item{A universal quantum computer is a $2^n$ dimensional complex vector
             to which any unitary can be applied with a quantum mechanical measurement process.}
         \item{Algorithms using the exponentially large Hilbert space can 
-            solve classically exponentially hard problems in polynomial time.}
+            solve some classically exponentially hard problems in polynomial time.}
     \end{itemize}
 
 \end{frame}
@@ -275,9 +276,24 @@
             starting from the $\ket{0}^{\otimes n}$ state can be
             simulated and sampled efficiently, i.e. in $\mbox{poly}(n, m)$ time
             where $m$ is the amount of gates/measurements.}
+    \end{itemize}
+
+\end{frame}
+}
+{
+\begin{frame}{The Gottesman-Knill Theorem}
+    \begin{itemize}
         \item{Note that a general state has $2^n$ complex coefficients. 
             Computing operations on this state is therefore exponentially hard in
             $n$.}
+        \item{Note that using the $R_\phi$ is universal and $R_\frac{\pi}{4}$
+            allows rational approximations of universal gates.}
+        \item{Restricting $\phi$ to $\frac{\pi}{2}$ allows the simulation of
+            large numbers of qbits on a classical computer.\\
+            {\bf Goal of this project is to create a simulator that can
+                simulate several thousand up to millions of qbits.
+            }}
+
     \end{itemize}
 
 \end{frame}
@@ -349,13 +365,9 @@
     \begin{itemize}
         \item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state.
             The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.}
-        \item{A Pauli observable $g_a \in \{X_a, Y_a, Z_a\}$ will either commute with all stabilizers
-            (in this case $g_a$ is a stabilizer, the measurement is deterministic and the stabilizers are unchanged) or
-            or anticommute with at least one stabilizer.}
-        \item{If $g_a$ anticommutes with a subset $A := \{S^{(i)} | \{S^{(i)}, g_a\} = 0 \}$
-            the probability to measure $+1$ or $-1$ is $\frac{1}{2}$ and the stabilizers $A$
-            are changed.}
-        \item{When going from $+g_a$ to $-g_a$ the results are changed from $+1$ to $-1$ and vice versa.}
+        \item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.}
+        \item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are
+            possible.}
     \end{itemize}
 \end{frame}
 }
@@ -363,14 +375,13 @@
 {
 \begin{frame}{Graphical States}
     \begin{itemize}
-        \item{The graphical representation of stabilizer states is an optimized way to write 
+        \item{The graphical representation is an optimized way to write 
             the stabilizers.}
         \item{
-            $(V, E, O)$ is called the graphical representation of
-                a stabilizer state if $V = \{0, ..., n-1\}$,
-                $E \subset \{\{i,j\} | i,j \in V, i \neq j \}$ and
-                $O = \{o_0, ..., o_{n-1}\}$ where $o_i \in C_1$.
-                $G = (V, E)$ is a graph, $O$ are called vertex operators.
+            For a set of vertices $V = \{0, ..., n-1\}$, some edges
+            $E \subset V \otimes V$, and local Clifford operators 
+            $O = \{o_0, ..., o_{n-1}\}$ the tuple $(V, E, O)$
+            is the graphical representation of a stabilizer state.
         }
         \item{The state associated with $(V, E, O)$ is given by
             \begin{equation}
@@ -570,8 +581,20 @@
 
 {
 \begin{frame}{Implementation}
+    \noindent\begin{minipage}{0.5\textwidth}
     \center
     \includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio=true]{screenshot_github.png}
+    \end{minipage} \hfill
+    \begin{minipage}{0.4\textwidth}
+    {\small
+    \begin{itemize}
+        \item{The code is GPLv3.0 licensed.}
+        \item{GitHub repository: \url{https://github.com/daknuett/pyqcs}}
+        \item{Install it via PyPI: \lstinline{pip3 install pyqcs}}
+        \item{The code is tested automatically on TravisCI.}
+    \end{itemize}
+    }
+    \end{minipage}
 \end{frame}
 }
 
@@ -680,4 +703,10 @@
     \end{itemize}
 \end{frame}
 }
+
+{
+\begin{frame}{References}
+    See the bibliography of my bachelor thesis.
+\end{frame}
+}
 \end{document}