did a lot a work on the presentation

presentation/main.tex

@@ -330,7 +330,8 @@
     \begin{itemize}
         \item{\textbf{Definition}
             {\itshape
-            $P := \{\pm1, \pm i\} \cdot \{X, Y, Z, I\}$ is called the Pauli group.\\
+                $P := \{\pm1X, \pm1Y, \pm1Z, \pm1I, \pm iX, \pm iY, \pm iZ, \pm iI\}$ (with the matrix product)
+                is called the Pauli group.\\
             $P_n := \left\{p_1 \otimes ... \otimes p_n \middle| p_i \in P \right\}$ is called the multilocal Pauli group.
             }}
             \pause
@@ -426,7 +427,7 @@
                     $\ket{+}$ stabilized by $\langle X_i \rangle_i$ (and $\ket{-}$ analogously).
                 }
                 \item{
-                    $\ket{0b00} + \ket{0b11}$ the Bell/EPR state which is stabilized by $\langle X_1X_2, Z_1Z_2\rangle$.
+                    $\frac{\ket{0b00} + \ket{0b11}}{\sqrt{2}}$ the Bell/EPR state which is stabilized by $\langle X_1X_2, Z_1Z_2\rangle$.
                 }
             \end{itemize}
             }
@@ -501,6 +502,33 @@
 \end{frame}
 }
 
+{
+\begin{frame}{Measurements on Stabilizer States}
+    \begin{itemize}
+        \item{As the stabilizers and the observable are both multilocal Pauli operators
+            they either commute or anticommute.}
+        \item{Let $S^{(i)}, S^{(j)} \in A$ then 
+            \begin{equation}
+            \begin{aligned}
+                \ket{\psi'} &= \frac{I + g_a}{2} \ket{\psi}\\
+                            &= \frac{I + g_a}{2}S^{(i)}S^{(j)}\ket{\psi}\\
+                            &= S^{(i)}\frac{I - g_a}{2}S^{(j)}\ket{\psi}\\
+                            &= S^{(i)}S^{(j)}\frac{I + g_a}{2}\ket{\psi}.\\
+            \end{aligned}
+            \end{equation}
+            I.e. the new state is stabilized by the product $S^{(i)}S^{(j)}$.}
+        \item{
+            This yields that the new state is stabilized by
+            \begin{equation}
+                \langle \{g_a\} \cup \{S^{(j)}S^{(i)} | S^{(i)} \in A \setminus \{S^{(j)}\} \rangle.
+            \end{equation}
+        }
+
+    \end{itemize}
+\end{frame}
+}
+
+
 \section{Graphical Description of Stabilizer States}
 
 {
@@ -555,7 +583,7 @@
                 M_a := \sqrt{-iX_i} \prod\limits_{\{i,j\} \in E} \sqrt{iZ_j}
             \end{equation}
             toggles the neighbourhood $n_a := \left\{ j \middle| \{a,j\} \in E\right\}$
-            of a.
+            of $a$. I.e. it toggles the edges $n_a \otimes n_a$.
             }
         \item{
             Many Clifford operations cannot be described by the VOP-free graph states.\\
@@ -574,4 +602,265 @@
     \end{itemize}
 \end{frame}
 }
+
+{
+\begin{frame}{Graph States}
+    \begin{itemize}
+        \item{\textbf{Definition}
+            {\itshape
+                $(V, E, O)$ is called a graph with vertex operators (VOPs) iff $(V,E)$ is a 
+                graph as in the definition above and $O = (o_1, ..., o_n)$ where
+                the $o_i \in C_1$.
+
+                The Stabilizers are given by
+                
+                \begin{equation}
+                    \left(\bigotimes\limits_{i} o_i\right) K_G^{(i)}  \left(\bigotimes\limits_{i} o_i\right)^\dagger
+                \end{equation}
+
+                and the stabilizer state is
+
+                \begin{equation}
+                    \ket{G} = \left(\bigotimes\limits_{i} o_i\right) \ket{\bar{G}}.
+                \end{equation}
+        }}
+        \item{One can show that there exist $24$ local Clifford gates. Therefore $O$ 
+            can be represented by $n$ integers from $0$ to $23$.}
+        \item{It is clear that any single qbit Clifford gate changes the VOPs
+                to $o_i \rightarrow c_i o_i$.}
+    \end{itemize}
+\end{frame}
+}
+
+{
+\begin{frame}{The $CZ$ Gate on Graph States}
+    \begin{itemize}
+        \item{As it is clear how local Clifford gates act on a graph state it is 
+            enough to show how a $CZ$ gate acts on the graph states to proof that
+            any Clifford gate can be applied to a graph state.}
+        \item{Consider the gate $CZ_{a,b}$ for $a,b \in V$.}
+        \item{If $CZ_{a,b}$ commutes with $o_a$ and $o_b$ the gate just toggles the edge
+            $\{a,b\}$ in $E$.}
+    \end{itemize}
+    
+\end{frame}
+}
+
+{
+\begin{frame}{$CZ$ on Two Qbits}
+     \begin{itemize}
+        \item{If the vertices are a clique (i.e. are connected only to each other) or
+            are isolated they separate from the rest of the state. In this case
+            one can consider just the space of two qbits. 
+        }
+         \item{
+             The amount of stabilizer states on these two qbits is finite. An upper bound
+             is given by $2\cdot24^2$, i.e. $24$ Clifford operators on each vertex and
+             the graph with or without and edge.
+            }
+        \item{All those states and the result after applying a $CZ$ gate can be
+            computed.}
+        \item{If one vertex has the VOP $I$ the result can be chosen
+            such that the VOP remains $I$.}
+        \item{This is used to implement the $CZ$ on isolated (or 2-clique) vertices.}
+     \end{itemize}
+\end{frame}
+}
+
+{
+\begin{frame}{Clearing Vertex Operators}
+    \begin{itemize}
+    \item{Recalling the transformation $M_a$ it is clear that the
+        graph state $\ket{G}$ is invariant under this transformation:
+        \begin{enumerate}[1]
+            \item Toggle the neighbourhood of $a$.
+            \item Right-multiply $M_a^\dagger$ to the VOPs.
+        \end{enumerate}
+        This transformation is called $L_a$.
+    }
+    \item{Using $\sqrt{-iX}$ and $\sqrt{iZ}$ as generators of $C_1$ one can
+        express any VOP as a product of $\sqrt{-iX},\sqrt{iZ}$.}
+    \item{Let the vertex $a$ have a neighbour $j \neq b$, then the VOP
+        on $a$ can be reduced to the identity by moving from right to left through the
+        product and applying $L_a$ for a $\sqrt{-iX}$ and $L_i$ for a $\sqrt{iZ}$.}
+    \item{Any vertex $j \in n_a \setminus \{i\}$ picks up powers of $\sqrt{iZ}$ during this
+        operation. As $\sqrt{iZ}$ commutes with $CZ$ this is no problem.}
+    \end{itemize}
+\end{frame}
+}
+
+{
+\begin{frame}{Clearing Vertex Operators to Apply a $CZ$}
+    \begin{itemize}
+        \item{The VOP on $a$ ($b$) can be cleared if $a$ ($b$) has at least one
+            neighbour that is not $b$ ($a$).}
+
+        \item{To apply a $CZ_{a,b}$ use the following algorithm to clear the 
+            VOPs as much as possible:
+            \begin{enumerate}[1]
+                \item Try to clear the VOP of $a$.
+                \item{ Try to clear the VOP on $b$.}
+                \item{If the VOP of $a$ wasn't clear yet, try to clear it again.}
+            \end{enumerate}
+        }
+        \item{After this procedure it is certain that at least one VOP is $I$
+            and possibly both VOPs commute with $CZ$. If they both commute 
+            applying the $CZ$ is done by toggling the edge $\{a,b\}$.
+            }
+        \item{If one VOP does not commute with $CZ$ this vertex is either isolated
+            or connected to the other operand vertex only. One can show that the 
+            two qbit $CZ$ method can be applied here if the identity on the vertex
+            connected to the rest of the graph is preserved.}
+    \end{itemize}
+\end{frame}
+}
+
+{
+\begin{frame}{Measurements on Graph States}
+    \begin{itemize}
+        \item{When measuring $Z_a$ the projector $P_a := \frac{I \pm Z_a}{a}$ can be 
+            pulled behind the vertex operator $o_a$ by transforming the observable:
+        
+            \begin{equation}
+            \begin{aligned}
+                P_a \ket{G} &= \left(\prod\limits{i \neq a} o_i\right) P_a o_a \ket{\bar{G}} \\
+                            &= \left(\prod\limits{i \neq a} o_i\right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
+                            &=  \left(\prod\limits{i} o_i \right) \tilde{P}_a \ket{\bar{G}}\\
+            \end{aligned}
+            \end{equation}
+
+            With a new projector $\tilde{P}_a = \frac{I + g_a}{2}$.
+        }
+        \item{The anticommuting stabilizers are given by
+            \begin{itemize}
+                \item{$A_X = \{K_G^{(i)} | i \in n_a\}$ for $g_a = \pm X_a$,}
+                \item{$A_Y = \{K_G^{(i)} | i \in n_a \cup a\}$ for $g_a = \pm Y_a$ and}
+                \item{$A_Z = \{K_G^{(a)}$ for $g_a = Z_a$.}
+            \end{itemize}}
+        \item{This can be used to compute the probability amplitudes and update $(G,V,E)$ after
+            the measurement.}
+    \end{itemize}
+    
+\end{frame}
+}
+
+\section{Implementation and Performance}
+
+{
+\begin{frame}{Implementation}
+    \begin{itemize}
+        \item{Both a dense vector simulator and a simulator using the graphical
+            representation have been implemented in the \lstinline{python3} package
+            \lstinline{pyqcs}.}
+        \item{To increase simulation efficiency the core of both simulators has been
+            implemented in \lstinline{C}.}
+        \item{The dense vector states are stored in \lstinline{numpy} arrays.}
+        \item{The graph is stored in an length $n$ array of linked lists. The vertex operators
+            are stored in a \lstinline{uint8_t} array.}
+    \end{itemize}
+\end{frame}
+}
+
+{
+\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
+    \includegraphics[width=\textwidth]{../performance/scaling_qbits_linear.png}
+\end{frame}
+}
+{
+\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
+    \includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png}
+\end{frame}
+}
+
+{
+\begin{frame}{Performance: Circuit Length on Graphical Representation}
+    \includegraphics[width=\textwidth]{../performance/scaling_circuits_linear.png}
+\end{frame}
+}
+
+{
+\begin{frame}{Performance: Circuit Length on Graphical Representation}
+    \begin{itemize}
+        \item{There seem to be three regimes: Low-linear regime, intermediate regime and high-linear regime.}
+        \item{In the low-linear regime only few VOPs have to be cleared.
+                The length of this regime increases with the number of qbits.
+            }
+        \item{In the high-linear regime the neighbourhoods are big; the probability that VOPs
+            must be cleared is high. Clearing VOPs involves many vertices.}
+        \item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$ 
+            no linear behaviour.}
+    \end{itemize}
+    
+\end{frame}
+}
+
+\section{Limitations and Future Work}
+
+{
+\begin{frame}{Non-Universality}
+    \begin{itemize}
+        \item{Recalling the discussion around measurement in the stabilizer formalism
+            only probability amplitudes $0, \frac{1}{2}, 1$ are possible for any Pauli
+        observable.}
+        \item{As seen in the example simulation of a spin chain other probability amplitudes
+            are important and particularely interesting.}
+        \item{Only few unitaries and states can be simulated.}
+        \item{The stabilizers have lost the vector space property: 
+            only special superpositions are allowed.}
+    \end{itemize}
+\end{frame}
+}
+
+{
+{
+\begin{frame}{Possible Ways of Extending the Formalism I}
+    \begin{itemize}
+        \item{One possible way would be to use more qbits to increase the 
+            amount of possible measurement outcomes.}
+        \item{One could describe the evolution of a general hermitian unitary
+            by expressing it as a sum of multilocal Pauli operators. Applying
+            non-Clifford gates would reintroduce exponential growth.
+        }
+    \end{itemize}
+\end{frame}
+}
+\begin{frame}{Possible Ways of Extending the Formalism II}
+    \begin{itemize}
+        \item{The reason why the multilocal Pauli operators can be stored
+            so efficiently is that they are the tensor product of $SU(2)$ matrices.}
+        \item{
+            A key property of the Clifford group is that it preserves this feature:
+            multilocal Pauli operators are mapped to multilocal Pauli operator, the
+            result is a tensor product again.
+            }
+        \item{One interesting way to search for new/other states and circuits that can
+            be simulated efficiently would be to check what hermitian matrices
+            are tensor products of single qbit matrices and what operations would
+            preserve this property.
+            }
+    \end{itemize}
+\end{frame}
+}
+{
+\begin{frame}{Possible Ways of Extending the Formalism II}
+    \begin{itemize}
+        \item{Single qbit gates will always preserve the tensor product property:
+                \begin{equation}
+                g_i' = U_k g_i U_k^\dagger = \left(\bigotimes\limits_{j<k} g_{i,j}\right) \otimes Ug_{i,k}U^\dagger \otimes \left(\bigotimes\limits_{j>k} g_{i,j}\right)
+                \end{equation}
+            }
+        \item{It would probably be enough to search for matrices $g_1, g_2$
+            for which
+                \begin{equation}
+                    CX_{1,2} (g_1 \otimes g_2) CX_{1,2} = g_1' \otimes g_2'
+                \end{equation}
+            holds. The Pauli matrices are one group that fulfills this property.
+            }
+
+    \end{itemize}
+\end{frame}
+}
+
+
+
 \end{document}