bachelor_thesis/presentation/main.tex

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\title{An Efficient Quantum Computing Simulator using a Graphical Description for Many-Qbit Systems}
\subtitle{Simulation in the Stabilizer Formalism}
\author{Daniel Knüttel} 
\date{21.02.2020}
\institute{Universität Regensburg}

\titlegraphic{\small\center Universität Regensburg\\
    Faculty of the Institute of Theoretical Physics
    \vspace{-11mm}\flushright\includegraphics[height=1.0cm]{logo.png}}

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  \begin{beamercolorbox}[wd=.9\paperwidth,ht=2.25ex,dp=1ex,left]{Faculty of the Institute of Theoretical Physics}%
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\begin{document}

\maketitle

\section{Introduction}

{
\begin{frame}{Motivation: Exponentially Hard (Physical) Problems}
    \begin{itemize}
        \item{Some mathematical problems are exponentially hard to solve, for instance prime factorization.}
        \item{There exist several physical systems which are interesting to study but hard so simulate such as
            QCD simulations at finite chemical potential or real time scattering amplitudes in QCD.}
        \item{The exponential behaviour in time (and space) complexity brings classical supercomputers to their limits.}
    \end{itemize}
\end{frame}
}


{
\begin{frame}{The Universal Quantum Computer}
    \begin{itemize}
        % 2^n complex vector, arbitrary unitary, measurements
        \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 some classically exponentially hard problems in polynomial time.}
    \end{itemize}

\end{frame}
}
{
\begin{frame}{Motivation: Using Quantum Computing for Physics}
    \begin{itemize}
        % Literaturübersicht.
        \item{
            Possible applications of quantum computing include:
            \begin{itemize}
                \item{Preparing ground states of non-perturbative systems by adiabatic
                        time evolution (\textit{https://www.scottaaronson.com/qclec/26.pdf}).\\
                    Also works in QCD (\textit{arXiv:2001.11145v1}).}
                %\item{Use algorithms such as Phase Estimation to analyze the spectrum
                %    of a system.}
                \item{Study the time evolution by applying the transfer matrix (\textit{see later}).}
                \item{Compute trace formuli (\textit{arXiv:2001.11145v1}).}
            \end{itemize}
        }
        % Future use & current use.
        % (Tensor networks)
        \item{
            In the near future only small and noisy machines will be available.
            Some Hilbert spaces have infinite dimension (bosonic states).
            Two main strategies exist to deal with this problem:
            \begin{itemize}
                \item{Selecting a physically interesting subspace.
                        % (tensor networks)
                    }
                \item{Dividing the Hilbert space in system and bath, treating
                    the bath statistically (\textit{Andreas Hackl}).}
            \end{itemize}
        }
    
    \end{itemize}
\end{frame}
}

{
\begin{frame}{Quantum Errors and Quantum Error Correction}
    \begin{itemize}
        \item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation).
        %\pause
        \item Fault tolerant QC needs a way to correct for those errors.
        %\pause
        \item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}.
        \item{Parts of the theoretical description of quantum errors can be used for physical
            problems (see above).}
    \end{itemize}
\end{frame}
}

\section{Binary Quantum Computing}

{
\begin{frame}{Qbits and Gates}
    \begin{itemize}
        \item{A qbit is a two level quantum mechanical system. The Hilbert space
            $\mathcal{H}$ is two dimensional and has the basis vectors $\ket{0}, \ket{1}$.}
        \item{$n$ qbits have the Hilbert space $\mathcal{H}^{\otimes n}$.}
        \item{Gates on a quantum computers are unitary operators acting on $\mathcal{H}^{\otimes n}$.}
    \end{itemize}
\end{frame}
}

{
\begin{frame}{Notable Gates on One Qbit}
    \begin{itemize}
        \item{The Pauli matrices $X, Y, Z$ are gates commonly used.
            $X$ is also called the $NOT$ gate as it maps $\ket{0}$ to $\ket{1}$ and
            vice versa.
            }
        \item{The Hadamard gate 
            $H = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)$
        transforms from Pauli $Z$ to $X$ basis.}
        \item{The $R_\phi = \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)$
            gate rotates a state vector around the $Z$ axis.}
        \item{The $R_{\frac{\pi}{2}}$ gate transforms from $X$ to $Y$ basis.}
    \end{itemize}

\end{frame}
}

{
\begin{frame}{Notable gates on $n$ Qbits}
    \begin{itemize}
        \item{For a unitary $U$ acting on $\mathcal{H}$ 
            \begin{equation}
                U_i := \left(\bigotimes\limits_{l < i} I\right) \otimes U \otimes \left(\bigotimes\limits_{l > i} I\right)
            \end{equation} 
            is the $U$ gate acting on qbit $i$.}

        \item{For two qbits $i\neq j$ the controlled $X$ gate is 
            \begin{equation}
                CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i.
            \end{equation}}
    \end{itemize}
\end{frame}
}
{
\begin{frame}{Universal Gates}
    \begin{itemize}
        \item{A quantum computer should be able to simulate any unitary on $\mathcal{H}^{\otimes n}$.}
        \item{Similarly to classical computers a universal set of operations is required.}
        \item{One can show that any unitary acting on $\mathcal{H}^{\otimes n}$
            can be generated using the $CX$ and universal gates
            acting on $\mathcal{H}$.}
        \item{The gates $\{H, R_\phi\}$ are universal on $\mathcal{H}$.}
    \end{itemize}
\end{frame}
}

{
\begin{frame}{Measurements and Computational Hardness}
    \begin{itemize}
        \item{When measuring a qbit $i$ the observable $Z_i$ is measured.}
        \item{The Hilbert space $\mathcal{H}^{\otimes n}$ has the integer basis
            \begin{equation}
                \ket{j} = \ket{\mbox{0b}j_{n-1}...j_1j_0} = \bigotimes\limits_{l=0}^{n-1} \ket{j_l}.
            \end{equation}
        }
        \item{A general state $\ket{\psi}$ has $2^n$ coefficients in this basis.}
        \item{In general an operation on the state $\ket{\psi}$ will have to update $2^n$ coefficients.
            Mapping a general state $\ket{\psi}$ to $\ket{\psi'}$ cannot be performed in $\mbox{poly}(n)$ time.}
    \end{itemize}

\end{frame}
}
{
\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
    \begin{itemize}
        \item{
            For a set of $n$ spins in a magnetic field one can rescale the Hamiltonian
            of the system to
            \begin{equation}
                H = -\sum\limits_{i=1}^{n-1} Z_i Z_{i-1} + g\sum\limits_{i=0}^{n-1} X_i
            \end{equation}
            }
        %\pause
        \item{
            The time evolution of such a system is given by the transfer matrix
            \begin{equation}
                T := \exp(-itH) \in U(2^n)
            \end{equation}
        }
    %\pause
    \item{
            By associating every qbit with one spin (both are two-level systems)
            one should be able to simulate the behaviour of the spin chain using 
            a quantum computer.
        }
    \end{itemize}
\end{frame}
}
{
\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
    \begin{itemize}
        \item{
            Trotterizing the matrix exponential
            \begin{equation}
            \begin{aligned}
                \exp(t(A + B)) &= \left(\exp(\frac{t}{2N}A)\exp(\frac{t}{N}B)\exp(\frac{t}{2N}A)\right)^N \\
                                & + \mathcal{O}\left(\frac{t^2}{N^2}\right)\\
            \end{aligned}
            \end{equation}
            }
        \item{
            For $n=3$ spins one gets a circuit
            {\centering\adjustbox{max width=\textwidth}{
\Qcircuit @C=1em @R=.7em {
& \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
& \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} &\qw \\
& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} &  \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} &\qw \\
}

            }}
            }
        \item{Applying this circuit $N$ times gives an approximation for the time evolution of a state.}
    \end{itemize}
\end{frame}
}
{
\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{spin_chain/time_evo_6spin_g3.png}
\end{center}
\end{figure}
\end{frame}
}
\section{Stabilizers}

{
\begin{frame}{The Gottesman-Knill Theorem}
    \begin{itemize}
        \item{When restriciting the $R_\phi$ gate to $\phi = \frac{\pi}{2}$
            many states can still be simulated.}
        \item{\textbf{Theorem}(\textit{Gottesman-Knill}):
            All states that can be reached using $H, R_\frac{\pi}{2}, CZ$ and measurements
            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}
}

{
\begin{frame}{The multilocal Pauli Group and the Clifford Group}
    \begin{itemize}
        \item{Why is simulating $H, R_\frac{\pi}{2}, CZ$ and measurements
            so much easier?}
            %\pause
        \item{
            The Pauli group is $P := \{\pm 1, \pm i\} \cdot \{X, Y, Z, I\}$.
            $P_n := P^{\otimes n}$ is called the multilocal Pauli group.
             }
            %\pause
        \item{
                $C_n$ is the normalizer of $P_n$, i.e. it maps $P_n$ to itself.\\
                One can show that $C_n$ is generated by $H, R_\frac{\pi}{2}, CZ_{i,j}$.
            }
    \end{itemize}
\end{frame}
}
{
\begin{frame}{Stabilizers and Stabilizer Spaces}
    \begin{itemize}
        \item{
            Choose a finite Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
            One can show that all elements of $S$ are hermitian.
            }
        \item{
            One says $S =  \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$
            if every element of $S$ can be expressed as a product of the $S^{(i)}$ and the 
            $S^{(i)}$ are the minimal amount of matrices required for this property.
            }
        \item{
            One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$  the stabilizer space 
            $V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = \ket{\psi} \forall i\}$
            has dimension $1$. $\ket{\psi}$ is therefore up to a trivial phase unique.
            }
    \end{itemize}

\end{frame}
}

{
\begin{frame}{Some Notable Stabilizer States}
    \begin{itemize}
        \item{The state $\ket{\mbox{0b}00}$ is stabilized by 
            $\langle Z_0, Z_1\rangle$.}
        \item{Applying the Hadamard gate to the first qbit changes the state to
                $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}01}\right)$.
                This state is stabilized by 
                $\langle H_0 Z_0 H_0^\dagger, Z_1 \rangle = \langle X_0, Z_1 \rangle$.
                }
        \item{Applying a $CX_{1, 0}$ gate yields 
                $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}11}\right)$
                the famous EPR/Bell state which is stabilized by
                $\langle CX_{1, 0} X_0 CX_{1, 0}^\dagger, CX_{1, 0} Z_1 CX_{1, 0}^\dagger \rangle = \langle X_0 X_1, Z_0 Z_1 \rangle$.
                }
        \item{When measuring qbit $0$ the resulting state is either $\ket{\mbox{0b}00}$ or $\ket{\mbox{0b}11}$
            and the stabilizers are either $\langle Z_0, Z_1\rangle$ or $\langle -Z_0, -Z_1\rangle$.}
    \end{itemize}
\end{frame}
}

{
\begin{frame}{Dynamics and Measurements}
    \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{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}
}

{
\begin{frame}{Graphical States}
    \begin{itemize}
        \item{The graphical representation is an optimized way to write 
            the stabilizers.}
        \item{
            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}
            \ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right) \prod\limits_{\{i,j\} \in E} CZ_{i,j} \ket{+}^{\otimes n}
            \end{equation}
            where $\ket{+} = H\ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$.
            }
        \item{One can show that all stabilizer states can be brought into this form.}
        \item{The stabilizers of this state are $K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j$.}
        %\item{The stabilizers associated with $(V, E, O)$ are
        %    \begin{equation}
        %        \left\langle\left(\bigotimes\limits_{j=0}^{n-1} o_j \right) K_G^{(i)} \left(\bigotimes\limits_{j=0}^{n-1} o_j \right)^\dagger\right\rangle_i
        %    \end{equation}
        %    where
        %    \begin{equation}
        %        K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j.
        %    \end{equation}
        %}
    \end{itemize}

\end{frame}
}

%{
%\begin{frame}{Graphical States}
%    \begin{itemize}
%    \end{itemize}
%\end{frame}
%}

{
\begin{frame}{Dynamics of Graphical States}
\begin{itemize}
    \item{Applying a local Clifford gate $U_i$ is trivial: 
        Just the vertex operator is updated to $U o_i$.}
    \item{If $o_a, o_b \in \{I, Z, R_\frac{\pi}{2}, R_\frac{\pi}{2}^\dagger\}$ applying a $CZ_{a,b}$
    just toggles the edge $\{a,b\}$ in $E$.%: $E' = E \Delta \{\{a,b\}\}$.
    %With the symmetric set difference $\Delta$.
    }
    \item{If the vertices $a, b$ are isolated the resulting state after
        applying $CZ_{a,b}$ can be precomputed. }
    \item{When none of the above is possible one can clear at least one
        vertex operator (i.e. transforming it to $I$).}
    \item{If both vertex operators can be cleared $\{a,b\}$ is toggled in $E$.
        If just one vertex operator has been cleared the other vertex is isolated
        and one can precompute all resulting states.}
\end{itemize}
\end{frame}
}

{
\begin{frame}{Clearing Vertex Operators}
    \begin{itemize}
        \item{One can show that for a non-isolated vertex $j$
            the unitary $M_j = \sqrt{-iX_j} \prod\limits_{\{l,j\} \in E} \sqrt{iZ_l}$
            when applied to a graphical state with $O = \{I, ..., I\}$ toggles the neighbourhood
            of $j$.}
        \item{The operation $L_j$ which simultaneously toggles the neighbourhood of $j$ and right-multiplies
            $M_j^\dagger$ to the vertex operators keeps the state $\ket{G}$ invariant.}
        \item{The group $C_1$ is generated by $\sqrt{-iX}, \sqrt{iZ}$. 
            Any $U \in C_1$ is the product of at most $5$ of those matrices.
            }
        \item{
            Let $a$ be the vertex to be cleared. Then there is one neighbour $j \neq b$ of $a$.
            Clearing the vertex operator $o_a$ is done by moving from right to left through the product 
            and applying

            \begin{itemize}
                \item{$L_a$ if the current matrix is $\sqrt{-iX}$ or}
                \item{$L_j$ if the current matrix is $\sqrt{iZ}$.}
            \end{itemize}

            }
    \end{itemize}
\end{frame}
}

{
\begin{frame}{Clearing VOPs: Example}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_01.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
    $19 = \sqrt{iZ}^2\sqrt{-iX}^3$
    \end{minipage}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_02.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
    $ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
    \end{minipage}
\end{frame}
}
{
\begin{frame}{Clearing VOPs: Example}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_02.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
    $ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
    \end{minipage}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
    $11 = \sqrt{iZ}^2\sqrt{-iX}$
    \end{minipage}
\end{frame}
}
{
\begin{frame}{Clearing VOPs: Example}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
    $11 = \sqrt{iZ}^2\sqrt{-iX}$
    \end{minipage}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
    $5 = \sqrt{iZ}^2$
    \end{minipage}
\end{frame}
}
{
\begin{frame}{Clearing VOPs: Example}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
    $5 = \sqrt{iZ}^2$
    \end{minipage}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
    $7 = \sqrt{iZ}$
    \end{minipage}
\end{frame}
}
{
\begin{frame}{Clearing VOPs: Example}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
    $7 = \sqrt{iZ}$
    \end{minipage}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$\\
    \end{minipage}
\end{frame}
}
{
\begin{frame}{Clearing VOPs: Example}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_01.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
    $19 = \sqrt{iZ}^2\sqrt{-iX}^3$
    \end{minipage}
    \noindent\begin{minipage}{0.5\textwidth}
    \includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
    \end{minipage} \hfill
    \begin{minipage}{0.4\textwidth}
    Vertex operator \\
    $2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$
    \end{minipage}
\end{frame}
}
\section{Implementation and Performance}

{
\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}
}

{
\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/regimes/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}
%}

%{
%\begin{frame}{Graph in the Low-Linear Regime}
%    \includegraphics[width=\textwidth]{../thesis/graphics/graph_low_linear_regime.png}
%\end{frame}
%}
%{
%\begin{frame}{Window in a Graph in the Intermediate Regime}
%    \includegraphics[width=\textwidth]{../thesis/graphics/graph_intermediate_regime_cut.png}
%\end{frame}
%}
%{
%\begin{frame}{Window in a Graph in the High-Linear Regime}
%    \includegraphics[width=\textwidth]{../thesis/graphics/graph_high_linear_regime_cut.png}
%\end{frame}
%}

%{
%\begin{frame}{Performance: Circuit Length on Graphical Representation}
%    \begin{itemize}
%        \item{Pauli measurements reduce entanglement entropy.}
%        \item{Pauli measurements reduce the amount of edges in the graph.}
%        \item{When adding measurement to the random circuits the regimes 
%            do not show up.}
%    \end{itemize}
%\end{frame}
%}

{
\begin{frame}{Performance: Circuit Length on Graphical Representation}
    \includegraphics[width=\textwidth]{../performance/regimes/scaling_circuits_measurements_linear.png}
\end{frame}
}

\section{Conclusion and Outlook}

{
\begin{frame}{Performance}
    \begin{itemize}
        \item{Simulation using stabilizers (in particular using the graphical representation)
            is polynomial hard.}
        \item{Simulation using dense state vectors is exponentially hard.}
        \item{The performance of the graphical simulator depends on the state/circuit.}
    \end{itemize}
\end{frame}
}

{
\begin{frame}{Physical Applications}
    \begin{itemize}
        \item{The graphical simulator cannot be used for physical applications because 
            interesting problems require real (or at least rational) probability amplitudes.}
        \item{Stabilizer states do not span a vector space.}
        \item{The idea of finding a more efficient representation for a subset of
            states can be used in physics.}
    \end{itemize}
\end{frame}
}

{
\begin{frame}{References}
    See the bibliography of my bachelor thesis.
\end{frame}
}
\end{document}