713 lines
27 KiB
TeX
713 lines
27 KiB
TeX
\documentclass[10pt]{beamer}
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%\usepackage{struktex}
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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 Stabilizer 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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\makeatletter
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\setbeamertemplate{footline}
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{
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%\leavevmode%
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\hbox{%
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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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\end{beamercolorbox}%
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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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\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 are exponentially hard to solve, 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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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Quantum Errors and Quantum Error Correction}
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\begin{itemize}
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\item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation).
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%\pause
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\item Fault tolerant QC needs a way to correct for those errors.
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%\pause
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\item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}.
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\item{Parts of the theoretical description of quantum errors can be used for physical
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problems (see above).}
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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}{Qbits and Gates}
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\begin{itemize}
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\item{A qbit is a two level quantum mechanical system. The Hilbert space
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$\mathcal{H}$ is two dimensional and has the basis vectors $\ket{0}, \ket{1}$.}
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\item{$n$ qbits have the Hilbert space $\mathcal{H}^{\otimes n}$.}
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\item{Gates on a quantum computers are unitary operators acting on $\mathcal{H}^{\otimes n}$.}
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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}{Notable Gates on One Qbit}
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\begin{itemize}
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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}{Notable gates on $n$ Qbits}
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\begin{itemize}
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\item{For a unitary $U$ acting on $\mathcal{H}$
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\begin{equation}
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U_i := \left(\bigotimes\limits_{l < i} I\right) \otimes U \otimes \left(\bigotimes\limits_{l > i} I\right)
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\end{equation}
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is the $U$ gate acting on qbit $i$.}
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\item{For two qbits $i\neq j$ the controlled $X$ gate is
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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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\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 $\mathcal{H}^{\otimes n}$.}
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\item{Similarly to classical computers a universal set of operations is required.}
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\item{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 $\mathcal{H}$.}
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\item{The gates $\{H, R_\phi\}$ are universal on $\mathcal{H}$.}
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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}{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{The Hilbert space $\mathcal{H}^{\otimes n}$ has the integer basis
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\begin{equation}
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\ket{j} = \ket{\mbox{0b}j_{n-1}...j_1j_0} = \bigotimes\limits_{l=0}^{n-1} \ket{j_l}.
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\end{equation}
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}
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\item{A general state $\ket{\psi}$ has $2^n$ coefficients in this basis.}
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\item{In general an operation on the state $\ket{\psi}$ will have to update $2^n$ coefficients.
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Mapping a general state $\ket{\psi}$ to $\ket{\psi'}$ 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^2}{N^2}\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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simulated and sampled efficiently, 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.
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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}{The multilocal Pauli Group and the Clifford Group}
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\begin{itemize}
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\item{Why is simulating $H, R_\frac{\pi}{2}, CZ$ and measurements
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so much easier?}
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%\pause
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\item{
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The Pauli group is $P := \{\pm 1, \pm i\} \cdot \{X, Y, Z, I\}$.
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$P_n := P^{\otimes n}$ is called the multilocal Pauli group.
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}
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%\pause
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\item{
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$C_n$ is the normalizer of $P_n$, i.e. it maps $P_n$ to itself.\\
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One can show that $C_n$ is generated by $H, R_\frac{\pi}{2}, CZ_{i,j}$.
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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}{Stabilizers and Stabilizer Spaces}
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\begin{itemize}
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\item{
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Choose a finite Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
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One can show that all elements of $S$ are hermitian.
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}
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\item{
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One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$
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if every element of $S$ can be expressed as a product of the $S^{(i)}$ and the
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$S^{(i)}$ are the minimal amount of matrices required for this property.
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}
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\item{
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One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$ the stabilizer space
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$V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = \ket{\psi} \forall i\}$
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has dimension $1$. $\ket{\psi}$ is therefore up to a trivial phase unique.
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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}{Some Notable Stabilizer States}
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\begin{itemize}
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\item{The state $\ket{\mbox{0b}00}$ is stabilized by
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$\langle Z_0, Z_1\rangle$.}
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\item{Applying the Hadamard gate to the first qbit changes the state to
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$\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}01}\right)$.
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This state is stabilized by
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$\langle H_0 Z_0 H_0^\dagger, Z_1 \rangle = \langle X_0, Z_1 \rangle$.
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}
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\item{Applying a $CX_{1, 0}$ gate yields
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$\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}11}\right)$
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the famous EPR/Bell state which is stabilized by
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$\langle CX_{1, 0} X_0 CX_{1, 0}^\dagger, CX_{1, 0} Z_1 CX_{1, 0}^\dagger \rangle = \langle X_0 X_1, Z_0 Z_1 \rangle$.
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}
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\item{When measuring qbit $0$ the resulting state is either $\ket{\mbox{0b}00}$ or $\ket{\mbox{0b}11}$
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and the stabilizers are either $\langle Z_0, Z_1\rangle$ or $\langle -Z_0, -Z_1\rangle$.}
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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}{Dynamics and Measurements}
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\begin{itemize}
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\item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state.
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The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.}
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\item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.}
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\item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are
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possible.}
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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}{Graphical States}
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\begin{itemize}
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\item{The graphical representation is an optimized way to write
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the stabilizers.}
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\item{
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For a set of vertices $V = \{0, ..., n-1\}$, some edges
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$E \subset V \otimes V$, and local Clifford operators
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$O = \{o_0, ..., o_{n-1}\}$ the tuple $(V, E, O)$
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is the graphical representation of a stabilizer state.
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}
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\item{The state associated with $(V, E, O)$ is given by
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\begin{equation}
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\ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right) \prod\limits_{\{i,j\} \in E} CZ_{i,j} \ket{+}^{\otimes n}
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\end{equation}
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where $\ket{+} = H\ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$.
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}
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\item{One can show that all stabilizer states can be brought into this form.}
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\item{The stabilizers of this state are $K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j$.}
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%\item{The stabilizers associated with $(V, E, O)$ are
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% \begin{equation}
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% \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
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% \end{equation}
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% where
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% \begin{equation}
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% K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j.
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% \end{equation}
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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}{Graphical States}
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% \begin{itemize}
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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}{Dynamics of Graphical States}
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\begin{itemize}
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\item{Applying a local Clifford gate $U_i$ is trivial:
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Just the vertex operator is updated to $U o_i$.}
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\item{If $o_a, o_b \in \{I, Z, R_\frac{\pi}{2}, R_\frac{\pi}{2}^\dagger\}$ applying a $CZ_{a,b}$
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just toggles the edge $\{a,b\}$ in $E$.%: $E' = E \Delta \{\{a,b\}\}$.
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%With the symmetric set difference $\Delta$.
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}
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\item{If the vertices $a, b$ are isolated the resulting state after
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applying $CZ_{a,b}$ can be precomputed. }
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\item{When none of the above is possible one can clear at least one
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vertex operator (i.e. transforming it to $I$).}
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\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}
|