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%\usepackage{struktex}
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\usepackage{qcircuit}
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\usepackage{adjustbox}
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\setbeamercolor{background canvas}{bg=white!20}
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\title{An Efficient Quantum Computing Simulator using a Graphical Description for Many-Qbit Systems}
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\subtitle{Simulation in the Graph Formalism}
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\author{Daniel Knüttel}
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\date{21.02.2020}
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\institute{Universität Regensburg}
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\titlegraphic{\small\center Universität Regensburg\\
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Faculty of the Institute of Theoretical Physics
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\vspace{-11mm}\flushright\includegraphics[height=1.0cm]{logo.png}}
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\newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
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\makeatletter
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{
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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{beamercolorbox}[wd=.1\paperwidth,ht=2.25ex,dp=1ex,right]{Faculty of the Institute of Theoretical Physics}%
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\insertframenumber{} / \inserttotalframenumber\hspace*{2ex}
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\end{beamercolorbox}}%
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}
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\NewDocumentCommand{\DrawBox}{s O{}}{%
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}{%
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}%
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($(left)+(-0.2em,0.9em)$) rectangle
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($(RightPoint)+(0.2em,-0.3em)$);}
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\makeatother
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\begin{document}
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\maketitle
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\section{Introduction}
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{
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\begin{frame}{Motivation: Exponentially Hard (Physical) Problems}
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\begin{itemize}
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\item{Some mathematical problems cannot be solved in polynomial time, for instance prime factorization.}
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\item{There exist several physical systems which are interesting to study but hard so simulate such as
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QCD simulations at finite chemical potential or real time scattering amplitudes in QCD.}
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\item{The exponential behaviour in time (and space) complexity brings classical supercomputers to their limits.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{The Universal Quantum Computer}
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\begin{itemize}
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% 2^n complex vector, arbitrary unitary, measurements
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\item{A universal quantum computer is a $2^n$ dimensional complex vector
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to which any unitary can be applied with a quantum mechanical measurement process.}
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\item{Algorithms using the exponentially large Hilbert space can
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solve some classically exponentially hard problems in polynomial time.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Motivation: Using Quantum Computing for Physics}
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\begin{itemize}
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% Literaturübersicht.
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\item{
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Possible applications of quantum computing include:
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\begin{itemize}
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\item{Preparing ground states of non-perturbative systems by adiabatic
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time evolution (\textit{https://www.scottaaronson.com/qclec/26.pdf}).\\
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Also works in QCD (\textit{arXiv:2001.11145v1}).}
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%\item{Use algorithms such as Phase Estimation to analyze the spectrum
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% of a system.}
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\item{Study the time evolution by applying the transfer matrix (\textit{see later}).}
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\item{Compute trace formuli (\textit{arXiv:2001.11145v1}).}
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\end{itemize}
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}
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% Future use & current use.
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% (Tensor networks)
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\item{
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In the near future only small and noisy machines will be available.
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Some Hilbert spaces have infinite dimension (bosonic states).
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Two main strategies exist to deal with this problem:
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\begin{itemize}
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\item{Selecting a physically interesting subspace.
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% (tensor networks)
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}
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\item{Dividing the Hilbert space in system and bath, treating
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the bath statistically (\textit{Andreas Hackl}).}
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\end{itemize}
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}
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\item{
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Long Term: Fault tolerant computing.
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}
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\end{itemize}
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\end{frame}
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}
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\section{Binary Quantum Computing}
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{
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\begin{frame}{One Qbit}
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\begin{itemize}
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\item{One qbit has the Hilbert space $\mathcal{H}$ with basis vectors $\ket{0}$, $\ket{1}$.}
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\item{The Pauli matrices $X, Y, Z$ are gates commonly used.
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$X$ is also called the $NOT$ gate as it maps $\ket{0}$ to $\ket{1}$ and
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vice versa.
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}
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\item{The Hadamard gate
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$H = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)$
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transforms from Pauli $Z$ to $X$ basis.}
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\item{The $R_\phi = \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)$
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gate rotates a state vector around the $Z$ axis.}
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\item{The $R_{\frac{\pi}{2}}$ gate transforms from $X$ to $Y$ basis.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Universal Gates}
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\begin{itemize}
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\item{A quantum computer should be able to simulate any unitary on $\mathcal{H}^{\otimes n}$ the
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$n$ qbit Hilbert space.}
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\item{Similarly to classical computers a universal set of operations is required.}
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\item{\tikzmark{left}One can show that any unitary acting on $\mathcal{H}^{\otimes n}$
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can be generated using the $CX$ and universal gates
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acting on $\mathcal{H}$ with
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\begin{equation}
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CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i.
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\end{equation}
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}
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\item{The gates $\{H, R_\phi\}$ are universal on $\mathcal{H}$.\hfill\tikzmark{right}}
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\end{itemize}
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\DrawBox
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\end{frame}
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}
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{
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\begin{frame}{Measurements and Computational Hardness}
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\begin{itemize}
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\item{When measuring a qbit $i$ the observable $Z_i$ is measured.}
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\item{Operations on the $2^n$ dimensional state will have to update $2^n$ complex
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coefficients. This cannot be performed in $\mbox{poly}(n)$ time.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
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\begin{itemize}
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\item{
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For a set of $n$ spins in a magnetic field one can rescale the Hamiltonian
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of the system to
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\begin{equation}
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H = -\sum\limits_{i=1}^{n-1} Z_i Z_{i-1} + g\sum\limits_{i=0}^{n-1} X_i
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\end{equation}
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}
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%\pause
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\item{
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The time evolution of such a system is given by the transfer matrix
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\begin{equation}
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T := \exp(-itH) \in U(2^n)
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\end{equation}
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}
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%\pause
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\item{
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By associating every qbit with one spin (both are two-level systems)
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one should be able to simulate the behaviour of the spin chain using
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a quantum computer.
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
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\begin{itemize}
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\item{
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Trotterizing the matrix exponential
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\begin{equation}
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\begin{aligned}
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\exp(t(A + B)) &= \left(\exp(\frac{t}{2N}A)\exp(\frac{t}{N}B)\exp(\frac{t}{2N}A)\right)^N \\
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& + \mathcal{O}\left(\frac{t^3}{N^3}\right)\\
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\end{aligned}
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\end{equation}
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}
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\item{
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For $n=3$ spins one gets a circuit
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{\centering\adjustbox{max width=\textwidth}{
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\Qcircuit @C=1em @R=.7em {
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& \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 \\
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& \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 \\
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& \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 \\
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}
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}}
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}
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\item{Applying this circuit $N$ times gives an approximation for the time evolution of a state.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=\linewidth]{spin_chain/time_evo_6spin_g3.png}
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\end{center}
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\end{figure}
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\end{frame}
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}
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\section{Stabilizers}
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{
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\begin{frame}{The Gottesman-Knill Theorem}
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\begin{itemize}
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\item{When restriciting the $R_\phi$ gate to $\phi = \frac{\pi}{2}$
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many states can still be simulated.}
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\item{\textbf{Theorem}(\textit{Gottesman-Knill}):
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All states that can be reached using $H, R_\frac{\pi}{2}, CZ$ and measurements
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starting from the $\ket{0}^{\otimes n}$ state can be
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\textbf{simulated} and \textbf{sampled efficiently on a classical computer}, i.e. in $\mbox{poly}(n, m)$ time
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where $m$ is the amount of gates/measurements.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{The Gottesman-Knill Theorem}
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\begin{itemize}
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\item{Note that a general state has $2^n$ complex coefficients.
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Computing operations on this state is therefore exponentially hard in
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$n$.}
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\item{Note that using the $R_\phi$ is universal and $R_\frac{\pi}{4}$
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allows rational approximations of universal gates.}
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\item{Restricting $\phi$ to $\frac{\pi}{2}$ allows the simulation of
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large numbers of qbits on a classical computer.\\
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{\bf Goal of this project is to create a simulator that can
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simulate several thousand up to millions of qbits.
|
|
|
|
}}
|
|
|
|
|
2020-02-28 15:09:51 +00:00
|
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
|
|
|
|
|
2020-03-09 13:33:39 +00:00
|
|
|
{
|
2020-04-01 11:13:35 +00:00
|
|
|
\begin{frame}{Graphical States}
|
2020-03-09 13:33:39 +00:00
|
|
|
\begin{itemize}
|
|
|
|
\item{
|
2020-04-09 09:46:22 +00:00
|
|
|
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.
|
2020-03-09 13:33:39 +00:00
|
|
|
}
|
2020-04-01 11:13:35 +00:00
|
|
|
\item{The state associated with $(V, E, O)$ is given by
|
2020-03-05 13:23:00 +00:00
|
|
|
\begin{equation}
|
2020-04-01 11:13:35 +00:00
|
|
|
\ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right) \prod\limits_{\{i,j\} \in E} CZ_{i,j} \ket{+}^{\otimes n}
|
2020-03-05 13:23:00 +00:00
|
|
|
\end{equation}
|
2020-04-01 11:13:35 +00:00
|
|
|
where $\ket{+} = H\ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$.
|
2020-03-05 13:23:00 +00:00
|
|
|
}
|
2020-04-01 11:13:35 +00:00
|
|
|
\item{One can show that all stabilizer states can be brought into this form.}
|
2020-04-14 16:44:50 +00:00
|
|
|
\item{For the derivation of the stabilizer and graphical states see my bachelor's thesis.}
|
2020-03-05 13:23:00 +00:00
|
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
|
|
}
|
2020-03-09 13:33:39 +00:00
|
|
|
|
2020-04-14 16:44:50 +00:00
|
|
|
{
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{frame}{Example: The $5$ Qbit EPR State}
|
|
|
|
|
2020-04-14 16:44:50 +00:00
|
|
|
\begin{itemize}
|
2020-04-15 14:02:38 +00:00
|
|
|
\item{Start from $\ket{0}^{\otimes n}$.}
|
|
|
|
\item{Get to the state $\ket{\psi} = \frac{\ket{0}^{\otimes n} + \ket{1}^{\otimes n}}{\sqrt{2}}$.}
|
|
|
|
\item{Use the circuit \\
|
|
|
|
\[
|
|
|
|
\Qcircuit @C=1em @R=.7em {
|
|
|
|
& \gate{H} & \ctrl{1} & \qw & \ctrl{2} & \qw & \ctrl{3} & \qw & \ctrl{4} &\qw \\
|
|
|
|
& \qw & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
|
|
|
|
& \qw & \qw & \qw & \gate{X} & \qw & \qw & \qw & \qw &\qw \\
|
|
|
|
& \qw & \qw & \qw & \qw & \qw & \gate{X} & \qw & \qw &\qw \\
|
|
|
|
& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{X} &\qw \\
|
|
|
|
}
|
|
|
|
\]
|
2020-04-14 09:50:09 +00:00
|
|
|
}
|
2020-04-15 14:02:38 +00:00
|
|
|
\item{The state has the form
|
|
|
|
\begin{equation}
|
|
|
|
\ket{\psi} = \left(\prod\limits_{1 < i < 5} CX_{i,0}\right) H_0 \ket{0}.
|
|
|
|
\end{equation}}
|
|
|
|
\end{itemize}
|
2020-03-09 13:33:39 +00:00
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
|
|
|
|
{
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{frame}{Example: The $5$ Qbit EPR State}
|
2020-03-09 13:33:39 +00:00
|
|
|
\begin{itemize}
|
2020-04-15 14:02:38 +00:00
|
|
|
\item{$CX_{i,j} = H_i CZ_{i,j} H_i$.}
|
|
|
|
\item{The new circuit is
|
|
|
|
\[
|
|
|
|
\Qcircuit @C=1em @R=.7em {
|
|
|
|
& \gate{H} & \ctrl{1} & \qw & \ctrl{2} & \qw & \ctrl{3} & \qw & \ctrl{4} & \qw &\qw \\
|
|
|
|
& \gate{H} & \gate{Z} & \qw & \qw & \qw & \qw & \qw & \qw & \gate{H} &\qw \\
|
|
|
|
& \gate{H} & \qw & \qw & \gate{Z} & \qw & \qw & \qw & \qw & \gate{H} &\qw \\
|
|
|
|
& \gate{H} & \qw & \qw & \qw & \qw & \gate{Z} & \qw & \qw & \gate{H} &\qw \\
|
|
|
|
& \gate{H} & \qw & \qw & \qw & \qw & \qw & \qw & \gate{Z} & \gate{H} &\qw \\
|
2020-04-01 11:13:35 +00:00
|
|
|
}
|
2020-04-15 14:02:38 +00:00
|
|
|
\]
|
|
|
|
}
|
|
|
|
\item{Switching from starting state $\ket{0}^{\otimes n}$ to $\ket{+}^{\otimes n}$ gives the
|
|
|
|
graphical representation.}
|
2020-03-09 14:29:32 +00:00
|
|
|
\end{itemize}
|
|
|
|
|
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
{
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{frame}{Example: The $5$ Qbit EPR State}
|
|
|
|
\noindent\begin{minipage}{0.4\textwidth}
|
|
|
|
\center
|
|
|
|
\adjustbox{max width=\textwidth}{
|
|
|
|
\Qcircuit @C=1em @R=.7em {
|
|
|
|
& \gate{H} & \ctrl{1} & \qw & \ctrl{2} & \qw & \ctrl{3} & \qw & \ctrl{4} &\qw \\
|
|
|
|
& \qw & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
|
|
|
|
& \qw & \qw & \qw & \gate{X} & \qw & \qw & \qw & \qw &\qw \\
|
|
|
|
& \qw & \qw & \qw & \qw & \qw & \gate{X} & \qw & \qw &\qw \\
|
|
|
|
& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{X} &\qw \\
|
|
|
|
}
|
|
|
|
}\\
|
|
|
|
\begin{equation}
|
|
|
|
\begin{aligned}
|
|
|
|
\ket{\psi} &= \left(\prod\limits_{1 < i < 5} CX_{i,0}\right) H_0 \ket{0}^{\otimes n}\\
|
|
|
|
\end{aligned}
|
|
|
|
\end{equation}
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage} \hfill
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{minipage}{0.5\textwidth}
|
|
|
|
\center
|
|
|
|
\includegraphics[width=\textwidth, height=\textheight, keepaspectratio=true]{example_graphs/graph_EPR_state.png}
|
|
|
|
|
|
|
|
The graphical state with $E =\{\{0,1\}, \{0,2\}, \{0,3\}, \{0,4\}\}$ and $O = (I, H, H, H, H)$.
|
|
|
|
|
|
|
|
\begin{equation}
|
|
|
|
\begin{aligned}
|
|
|
|
\ket{G} = \left(\prod\limits_{1 < i < 5} H_i\right) \left(\prod\limits_{1 < i < 5} CZ_{i,0}\right) \ket{+}^{\otimes n}
|
|
|
|
\end{aligned}
|
|
|
|
\end{equation}
|
|
|
|
|
|
|
|
\end{minipage}
|
2020-03-09 14:29:32 +00:00
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
{
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{frame}{Dynamics of Graphical States}
|
|
|
|
\begin{itemize}
|
|
|
|
\item{
|
|
|
|
\noindent\begin{minipage}{0.4\textwidth}
|
|
|
|
Applying a local Clifford gate $U_i$ is trivial:
|
|
|
|
Just the vertex operator is updated to $U o_i$.
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage} \hfill
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{minipage}{0.5\textwidth}
|
|
|
|
\tikzmark{left}Apply $SH = R_{\frac{\pi}{2}}H$ to qbit $2$:\\
|
|
|
|
\includegraphics[width=\textwidth, height=\textheight, keepaspectratio=true]{example_graphs/graph_update_VOP.png}\tikzmark{right}
|
|
|
|
\DrawBox
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage}
|
2020-04-15 14:02:38 +00:00
|
|
|
}
|
|
|
|
\item{
|
|
|
|
\noindent\begin{minipage}{0.4\textwidth}
|
|
|
|
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 $\Delta$.
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage} \hfill
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{minipage}{0.5\textwidth}
|
|
|
|
\tikzmark{left}Apply $CZ_{2,0}$:\\
|
|
|
|
\includegraphics[width=\textwidth, height=\textheight, keepaspectratio=true]{example_graphs/graph_apply_CZ.png}\hfill\tikzmark{right}
|
|
|
|
\DrawBox
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage}
|
2020-04-15 14:02:38 +00:00
|
|
|
}
|
|
|
|
|
|
|
|
\end{itemize}
|
2020-03-09 14:29:32 +00:00
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
{
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{frame}{Dynamics of Graphical States}
|
|
|
|
\begin{itemize}
|
|
|
|
\item{
|
|
|
|
\noindent\begin{minipage}{0.4\textwidth}
|
|
|
|
If the vertices $a, b$ are isolated the resulting state after
|
|
|
|
applying $CZ_{a,b}$ can be precomputed.
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage} \hfill
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{minipage}{0.5\textwidth}
|
|
|
|
\tikzmark{left}Before:\\
|
|
|
|
\includegraphics[width=\textwidth, height=0.3\textheight, keepaspectratio=true]{example_graphs/graph_two_qbit_CZ_before.png}\\
|
|
|
|
After:\\
|
|
|
|
\includegraphics[width=0.5\textwidth, height=0.3\textheight, keepaspectratio=true]{example_graphs/graph_two_qbit_CZ_after.png}\hfill\tikzmark{right}
|
|
|
|
\DrawBox
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage}
|
2020-04-15 14:02:38 +00:00
|
|
|
}
|
|
|
|
\item{
|
|
|
|
|
|
|
|
\noindent\begin{minipage}{0.4\textwidth}
|
|
|
|
When the VOPs do not commute with $CZ$ and the vertices are not isolated one can clear at least one
|
|
|
|
vertex operator (i.e. transforming it to $I$). This changes the neighbourhood of the vertices.
|
|
|
|
\end{minipage} \hfill
|
|
|
|
\begin{minipage}{0.5\textwidth}
|
|
|
|
\tikzmark{left}The starting state:\\
|
|
|
|
\includegraphics[width=0.7\textwidth, height=0.5\textheight, keepaspectratio=true]{example_graphs/graph_clear_VOPs_CZ_before.png}\hfill\tikzmark{right}
|
|
|
|
\DrawBox
|
|
|
|
\end{minipage}
|
|
|
|
}
|
|
|
|
\end{itemize}
|
2020-03-09 14:29:32 +00:00
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
{
|
2020-04-15 14:02:38 +00:00
|
|
|
\begin{frame}{Dynamics of Graphical States}
|
|
|
|
\begin{itemize}
|
|
|
|
\item{
|
|
|
|
If both vertex operators can be cleared $\{a,b\}$ is toggled in $E$.
|
|
|
|
\noindent\begin{minipage}{0.4\textwidth}
|
|
|
|
\tikzmark{left}After clearing vertices $1,2$:\\
|
|
|
|
\includegraphics[width=0.7\textwidth, height=0.5\textheight, keepaspectratio=true]{example_graphs/graph_clear_VOPs_CZ_cleared.png}\hfill\tikzmark{right}
|
|
|
|
\DrawBox
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage} \hfill
|
|
|
|
\begin{minipage}{0.4\textwidth}
|
2020-04-15 14:02:38 +00:00
|
|
|
\tikzmark{left}Applying $CZ_{2, 1}$:\\
|
|
|
|
\includegraphics[width=0.7\textwidth, height=0.5\textheight, keepaspectratio=true]{example_graphs/graph_clear_VOPs_CZ_after.png}\tikzmark{right}
|
|
|
|
\DrawBox
|
2020-04-01 11:13:35 +00:00
|
|
|
\end{minipage}
|
2020-04-15 14:02:38 +00:00
|
|
|
}
|
|
|
|
\item{
|
|
|
|
If just one vertex operator has been cleared the other vertex is isolated
|
|
|
|
and one can precompute all resulting states.}
|
|
|
|
\end{itemize}
|
2020-03-11 09:48:51 +00:00
|
|
|
\end{frame}
|
|
|
|
}
|
2020-04-15 14:02:38 +00:00
|
|
|
|
|
|
|
|
2020-03-09 13:33:39 +00:00
|
|
|
\section{Implementation and Performance}
|
|
|
|
|
2020-04-01 15:03:55 +00:00
|
|
|
{
|
|
|
|
\begin{frame}{Implementation}
|
2020-04-09 09:46:22 +00:00
|
|
|
\noindent\begin{minipage}{0.5\textwidth}
|
2020-04-01 15:03:55 +00:00
|
|
|
\center
|
|
|
|
\includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio=true]{screenshot_github.png}
|
2020-04-09 09:46:22 +00:00
|
|
|
\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}
|
2020-04-01 15:03:55 +00:00
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
|
2020-03-09 13:33:39 +00:00
|
|
|
{
|
|
|
|
\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}
|
|
|
|
}
|
2020-04-01 15:03:55 +00:00
|
|
|
%{
|
|
|
|
%\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
|
|
|
|
% \includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png}
|
|
|
|
%\end{frame}
|
|
|
|
%}
|
2020-03-09 13:33:39 +00:00
|
|
|
|
|
|
|
{
|
|
|
|
\begin{frame}{Performance: Circuit Length on Graphical Representation}
|
2020-03-23 15:02:32 +00:00
|
|
|
\includegraphics[width=\textwidth]{../performance/regimes/scaling_circuits_linear.png}
|
2020-03-09 13:33:39 +00:00
|
|
|
\end{frame}
|
|
|
|
}
|
|
|
|
|
2020-04-01 15:03:55 +00:00
|
|
|
%{
|
|
|
|
%\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}
|
|
|
|
%}
|
|
|
|
%{
|
|
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|
%\begin{frame}{Window in a Graph in the High-Linear Regime}
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% \includegraphics[width=\textwidth]{../thesis/graphics/graph_high_linear_regime_cut.png}
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%\end{frame}
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%}
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%{
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%\begin{frame}{Performance: Circuit Length on Graphical Representation}
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% \begin{itemize}
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% \item{Pauli measurements reduce entanglement entropy.}
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% \item{Pauli measurements reduce the amount of edges in the graph.}
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% \item{When adding measurement to the random circuits the regimes
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% do not show up.}
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% \end{itemize}
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%\end{frame}
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%}
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2020-03-09 13:33:39 +00:00
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{
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\begin{frame}{Performance: Circuit Length on Graphical Representation}
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2020-04-01 15:03:55 +00:00
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\includegraphics[width=\textwidth]{../performance/regimes/scaling_circuits_measurements_linear.png}
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2020-03-09 13:33:39 +00:00
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\end{frame}
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}
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2020-04-01 15:03:55 +00:00
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\section{Conclusion and Outlook}
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2020-04-01 11:13:35 +00:00
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{
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2020-04-01 15:03:55 +00:00
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\begin{frame}{Performance}
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2020-03-09 13:33:39 +00:00
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\begin{itemize}
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2020-04-01 15:03:55 +00:00
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\item{Simulation using stabilizers (in particular using the graphical representation)
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is polynomial hard.}
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\item{Simulation using dense state vectors is exponentially hard.}
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\item{The performance of the graphical simulator depends on the state/circuit.}
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2020-03-09 13:33:39 +00:00
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\end{itemize}
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\end{frame}
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}
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2020-03-09 15:41:26 +00:00
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2020-04-01 11:13:35 +00:00
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{
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2020-04-01 15:03:55 +00:00
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\begin{frame}{Physical Applications}
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\begin{itemize}
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\item{The graphical simulator cannot be used for physical applications because
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interesting problems require real (or at least rational) probability amplitudes.}
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\item{Stabilizer states do not span a vector space.}
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\item{The idea of finding a more efficient representation for a subset of
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states can be used in physics.}
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\end{itemize}
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2020-03-09 13:33:39 +00:00
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\end{frame}
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}
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2020-04-09 09:46:22 +00:00
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{
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\begin{frame}{References}
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See the bibliography of my bachelor thesis.
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\end{frame}
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}
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2020-02-25 13:21:31 +00:00
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\end{document}
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