1131 lines
40 KiB
TeX
1131 lines
40 KiB
TeX
\documentclass[10pt]{beamer}
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\usepackage{graphicx}
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\usepackage{amsmath}
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\usepackage{geometry}
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\usepackage{enumerate}
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\usepackage{physics}
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\usepackage{listings}
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%\usepackage{struktex}
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\usepackage{qcircuit}
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\usepackage{tikz}
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\usetheme{metropolis}
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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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%\logo{\includegraphics[width=1cm]{logo.png}\hfill}
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%\newcommand{\nologo}{\setbeamertemplate{logo}{}} % command to set the logo to nothing
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%\newcommand{\congress}{Faculty of the Institute of Theoretical Physics}
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%% footer
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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{Some physical systems are hard to observe or manipulate, relativistic fermions on a curved spacetime are
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a typical example.}
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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 Quantum Simulator}
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\begin{itemize}
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\item{Mapping a system which is hard to observe and/or hard to manipulate to an analogous system.}
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\item{A typical example is Graphene which has a band structure near the $K$ point similar to relativistic fermions.}
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\item{Original idea from Feynman.}
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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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\item{A quantum system to which any unitary transformation can be applied.}
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\item{Any quantum system with sufficiently small hilbert space can be simulated.}
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\item{Quantum algorithms such as the Phase Estimation Algorithm have physical applications.}
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\item{Applications in other fields: Quantum AI, breaking encryption (via prime factorization), Quantum Search, ...}
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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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\item{Preparing ground states of non-perturbative systems by adiabatic
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time evolution. Also works in QCD (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.}
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\item{Compute trace formuli (arXiv:2001.11145v1).}
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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}{Challenges}
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\begin{itemize}
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\item{Many Hilbert space have non-finite dimension.}
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\item{Choosing interesting subspaces can be hard.}
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\item{Treating the rest of the Hilbert space can be hard.}
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\item{Possible approach: Choose a subspace and treat the other subspace
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as a bath (Andreas Hackl).}
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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}
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\textbf{Definition}
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{\itshape
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A qbit is a two-level quantum mechanical system $\ket{0}, \ket{1}$ with $\braket{0}{1} = 0$.
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In the following $Z = \sigma_Z, X = \sigma_X, Y = \sigma_Y$ will be used. $I$ is the identity matrix.
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}
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$\ket{0}, \ket{1}$ are the eigenkets of the $Z$ operator:
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$Z\ket{0} = +1\ket{0}$ and $Z\ket{1} = -1\ket{1}$.
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\end{frame}
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}
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{
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\begin{frame}{Qbits and Gates}
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\begin{itemize}
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\item{
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\textbf{Postulate}
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{\itshape
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A $n$-qbit system is the tensor product of the single-qbit systems.
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}
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}
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%\pause
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%\item{
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% For $n$ qbits define the integer state $\ket{j}$ as
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% \begin{equation}
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% \ket{j} := \ket{\mbox{0b}i_0i_1...i_{n-1}} := \ket{i_0}_s \otimes \ket{i_1}_s \otimes ... \otimes \ket{i_{n-1}}_s
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% \end{equation}
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%}
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\item{
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A transformation $U \in U(2^n)$ is called \textit{gate} acting on the system.
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For $\bar{U} \in U(2)$ the gate $U_i$ acting on qbit $i$ is given by
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\begin{equation}
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U_i := \left(\bigotimes\limits_{k < i} I\right) \otimes \bar{U} \otimes \left(\bigotimes\limits_{k > i} I\right)
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\end{equation}
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}
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%\pause
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\item{
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For $\bar{U} \in U(2)$ and qbits $i \neq j$
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\begin{equation}
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CU_{i,j} := \ket{1}\bra{1}_j \otimes U_i + \ket{0}\bra{0}_j \otimes I
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\end{equation}
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is the controlled $\ket{U}$ gate acting on $i$ with control-qbit $j$.
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}
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%\pause
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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}{Gates}
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Some notable gates are
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\begin{itemize}
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\item{
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the Hadamard gate
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\begin{equation}
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H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)
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\end{equation}
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which transforms from the $Z$ to the $X$ basis}
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%\pause
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\item{the rotation gate
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\begin{equation}
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R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)
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\end{equation}
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that performs a rotation around the $Z$ axis.}
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%\pause
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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}{Gates}
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\begin{itemize}
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\item{The $Z$ and $S$ gates are given by:
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\begin{equation}
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Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) = R_{\pi}
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\end{equation}
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\begin{equation}
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S := \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right) = R_{\frac{\pi}{2}}
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\end{equation}
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The $S$ gate transforms from $X$ to $Y$ basis.
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}
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%\pause
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\item{
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\textbf{Theorem}
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{\itshape
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Any gate $U \in U(2^n)$ can be generated using the $H$, $R_\phi$
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and $CZ_{i,j}$ gate.
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}
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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}{Integer States}
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\begin{itemize}
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\item{
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The eigenstates of the $Z_i$ are called integer states. They have the form
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\begin{equation}
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\ket{j} = \ket{\mbox{0b}l_1...l_n} = \ket{l_1}_s \otimes ... \otimes \ket{l_n}s
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\end{equation}
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}
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%\pause
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\item{
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For $n$ qbits there exist $2^n$ such states and they form a basis
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\begin{equation}
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\ket{\psi} = \sum\limits_{i=0}^{n-1} \ket{i}\braket{i}{\psi} = \sum\limits_{i=0}^{n-1} c_i\ket{i}
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\end{equation}
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with the condition $\sum\limits_{i=0}^{n-1} c_i^2 = 1$.
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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}{Measurement}
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\begin{itemize}
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\item{Measurements are performed in $Z$ basis, i.e. for a qbit $i$
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$Z_i$ is measured.
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}
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\item{
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The results of the measurements are associated with a classical result $s \in \{0, 1\}$
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using
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\begin{equation}
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Z_i\ket{\psi'} = (-1)^s \ket{\psi'}
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\end{equation}
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after the measurement.
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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 Circuits}
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\begin{itemize}
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\item{
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Writing a unitary transformation as a product of the generator gates is unreadable.
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To fix this problem quantum circuits have been introduced.
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}
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%\pause
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\item{
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Qbits are represented by horizontal lines.
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}
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%\pause
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\item{
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Gates acting on a qbit are boxes on the lines.
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}
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%\pause
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\item{
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Control-qbits are connected to the gate via a vertical line.
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}
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%\pause
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\item{
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Circuits are read left to right.
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}
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%\pause
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\item{
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Example:
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\Qcircuit @C=1em @R=.7em {
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& \gate{H} & \ctrl{1} & \qw &\qw \\
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& \gate{H} & \gate{Z} & \gate{H} &\qw \\
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}
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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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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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\exp(t(A + B)) = \left(\exp(\frac{t}{N}A)\exp(\frac{t}{N}B)\right)^N + \mathcal{O}\left(\frac{t^2}{N^2}\right)
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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 $H, S, CZ$ 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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\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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\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{\textbf{Definition}
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{\itshape
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$P := \{\pm1X, \pm1Y, \pm1Z, \pm1I, \pm iX, \pm iY, \pm iZ, \pm iI\}$ (with the matrix product)
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is called the Pauli group.\\
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$P_n := \left\{p_1 \otimes ... \otimes p_n \middle| p_i \in P \right\}$ is called the multilocal Pauli group.
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}}
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%\pause
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\item{
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\textbf{Definition}
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{\itshape
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$C_n := \left\{U \in U(2^n) \middle| \forall p \in P_n: U^\dagger p U \in P_n\right\}$
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is called the Clifford group
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}
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}
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%\pause
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\item{
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One can show that $C_n$ is generated by $H, S, CZ_{i,j}$.
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}
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\end{itemize}
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\end{frame}
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}
|
|
|
|
{
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\begin{frame}{Generators of a Group}
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\begin{itemize}
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\item{
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\textbf{Definition}
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{\itshape
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|
For a finite group $G$ one says $G$ is generated by $g_1, ..., g_N$ iff any $g \in G$ can be expressed
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as a product of $g_1, ..., g_N$. These generators are chosen to be the minimal set for which this
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condition holds.
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One also writes
|
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\begin{equation}
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G = \langle g_1,...,g_N \rangle \equiv \langle g_i\rangle_i
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\end{equation}
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}
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}
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\item{
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The generators are not unique. For instance $C_1$ can be generated using $H, S$ or $\sqrt{-iX}, \sqrt{iZ}$.
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}
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%\pause
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\item{
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The generators of a group have some kind of independence property.
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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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{
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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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\textbf{Definition}
|
|
{\itshape
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|
A finite abelian subgroup $S$ of $P_n$ is called a set of stabilizers iff $-I \notin S$ and
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all elements of $S$ commute.
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}
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}
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\item{
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From $-I \notin S$ follows that all elements of $S$ are hermitian.
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}
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\item{
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From the definition as tensor products of Pauli matrices follows that the
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elements of $S$ have eigenvalues $\pm1$.
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}
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\item{
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These properties together yield that all elements of $S$ can be diagonalized simultaneously.
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Further there exists a vector space $V_S$ with $s \ket{\psi} = +1 \ket{\psi}$ for all $s \in S$
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and all $\ket{\psi} \in V_S$. This space is called the stabilizer space of $S$
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and all $\ket{\psi}$ are called stabilizer states.
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}
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\end{itemize}
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|
\end{frame}
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}
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{
|
|
\begin{frame}{Stabilizer States}
|
|
\begin{itemize}
|
|
\item{
|
|
One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$ the stabilizer space $V_S$
|
|
has dimension $1$.
|
|
}
|
|
\item{
|
|
Therefore the state $\ket{\psi}$ that is the +1 eigenstate of all stabilizers is (up to a
|
|
global phase) unique.
|
|
}
|
|
\item{Notable stabilizer states are:
|
|
\begin{itemize}
|
|
\item{ $\ket{0b0..0} = \ket{0}\otimes ... \otimes \ket{0}$ and $\ket{0b1..1} = \ket{1}\otimes ... \otimes \ket{1}$
|
|
which are stabilized by $\langle Z_i \rangle_i$ and $\langle -Z_i \rangle_i$ respectively.
|
|
}
|
|
\item{
|
|
$\ket{+}$ stabilized by $\langle X_i \rangle_i$ (and $\ket{-}$ analogously).
|
|
}
|
|
\item{
|
|
$\frac{\ket{0b00} + \ket{0b11}}{\sqrt{2}}$ the Bell/EPR state which is stabilized by $\langle X_1X_2, Z_1Z_2\rangle$.
|
|
}
|
|
\end{itemize}
|
|
}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Dynamics of Stabilizer States}
|
|
\begin{itemize}
|
|
\item{
|
|
Under a transformation $U \in U(2^n)$ the state changes to
|
|
\begin{equation}
|
|
\ket{\psi'} = U \ket{\psi}
|
|
\end{equation}
|
|
}
|
|
\item{
|
|
The stabilizers of $\ket{\psi}$ change to
|
|
\begin{equation}
|
|
\ket{\psi'} = U\ket{\psi} = US^{(i)}\ket{\psi} = US^{(i)}U^\dagger U\ket{\psi} = US^{(i)}U^\dagger\ket{\psi'}
|
|
\end{equation}
|
|
using this it is clear that for $U \in C_n$ the state $\ket{\psi'}$ is a stabilizer state again with the stabilizers
|
|
$S' = \langle US^{(i)}U^\dagger\rangle_{i=1,...,n}$
|
|
}
|
|
\item{
|
|
Under the transformations $C_n$ one can describe the dynamics of the stabilizer states by their
|
|
stabilizers.
|
|
}
|
|
\item{Because the stabilizers are given by $n$ matrices which are the tensor product of $n$ Pauli matrices
|
|
this can be simulated in $\mathcal{O}\left(n^2\right)$ time instead of $\mathcal{O}\left(2^n\right)$.}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Measurements on Stabilizer States}
|
|
\begin{itemize}
|
|
\item{
|
|
Consider a Pauli observable $g_a \in \{(-1)^s X_a, (-1)^s Y_a, (-1)^s Z_a\}$ and the projector
|
|
onto its eigenspace $\frac{I + g_a}{2}$.
|
|
}
|
|
\item{If $g_a$ commutes with all $S^{(i)}$ the observable $g_a$ is diagonal in this basis and
|
|
the stabilizer state $\ket{\psi}$ is the $+1$ eigenstate of $g_a$. Therefore the measurement is
|
|
deterministic and the stabilizers remain unchanged.}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Measurements on Stabilizer States}
|
|
\begin{itemize}
|
|
\item{If $g_a$ does not commute with all stabilizers $A := \left\{S^{(i)} \middle| \left[g_a, S^{(i)}\right] \neq 0\right\}$
|
|
is the set of stabilizers that anticommute with $g_a$.}
|
|
\item{
|
|
To compute the probability to measure a result of $s=0$ one can use the trace formula
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
P(s=0) &= \left|\Tr(\frac{I + g_a}{2} \ket{\psi}\bra{\psi})\right| \\
|
|
&= \left|\Tr(\frac{I + g_a}{2} S^{(j)} \ket{\psi}\bra{\psi})\right| \\
|
|
&= \left|\Tr(S^{(j)}\frac{I - g_a}{2} \ket{\psi}\bra{\psi})\right| \\
|
|
&= \left|\Tr(\frac{I - g_a}{2} \ket{\psi}\bra{\psi})\right| \\
|
|
&= P(s=1)\\
|
|
\end{aligned}
|
|
\end{equation}
|
|
|
|
where $S^{(j)} \in A$. The stabilizer $S^{(j)}$ pulled to the right using the cyclic property of the
|
|
trace and absorbed into the $\bra{\psi}$.
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Measurements on Stabilizer States}
|
|
\begin{itemize}
|
|
\item{As the stabilizers and the observable are both multilocal Pauli operators
|
|
they either commute or anticommute.}
|
|
\item{Let $S^{(i)}, S^{(j)} \in A$ then
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
\ket{\psi'} &= \frac{I + g_a}{2} \ket{\psi}\\
|
|
&= \frac{I + g_a}{2}S^{(i)}S^{(j)}\ket{\psi}\\
|
|
&= S^{(i)}\frac{I - g_a}{2}S^{(j)}\ket{\psi}\\
|
|
&= S^{(i)}S^{(j)}\frac{I + g_a}{2}\ket{\psi}.\\
|
|
\end{aligned}
|
|
\end{equation}
|
|
I.e. the new state is stabilized by the product $S^{(i)}S^{(j)}$.}
|
|
\item{
|
|
This yields that the new state is stabilized by
|
|
\begin{equation}
|
|
\langle \{g_a\} \cup \{S^{(j)}S^{(i)} | S^{(i)} \in A \setminus \{S^{(j)}\} \rangle.
|
|
\end{equation}
|
|
}
|
|
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
|
|
\section{Graphical Description of Stabilizer States}
|
|
|
|
{
|
|
\begin{frame}{Graphs}
|
|
\begin{itemize}
|
|
\item{\textbf{Definition}
|
|
{\itshape
|
|
The tuple $(V, E)$ is called a graph iff $V$ is a set of vertices with $|V| = n \in \mathbb{N}$ elements.
|
|
In the following $V = \{0, ..., n-1\}$ will be used.
|
|
$E$ is the set of edges $E \subset \left\{\{i, j\} \middle| i,j \in V, i \neq j\right\}$.
|
|
}}
|
|
\item{
|
|
Example for a valid graph:\\
|
|
\includegraphics[width=\linewidth,height=0.5\textheight,keepaspectratio]{graphs/valid_graph.png}
|
|
}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{ VOP-free Graph States}
|
|
\begin{itemize}
|
|
\item{\textbf{Definition}
|
|
{\itshape
|
|
For $G = (V,E)$, $i \in V$ define
|
|
\begin{equation}
|
|
K_G^{(i)} := X_i \prod\limits_{\{i,j\} \in E} Z_j
|
|
\end{equation}
|
|
the stabilizers associated with the graph $G$.
|
|
}}
|
|
\item{
|
|
The state stabilized by all $K_G^{(i)}$ is
|
|
\begin{equation}
|
|
\ket{\bar{G}} = \prod\limits_{\{i,j\} \in E} CZ_{i,j} \ket{+}.
|
|
\end{equation}
|
|
This state is called vertex operator-free (VOP-free) graph state.
|
|
}
|
|
\item{
|
|
Applying a $CZ_{i,j}$ gate toggles the edge $\{i,j\}$ in $E$.
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Dynamics of VOP-free Graph States}
|
|
\begin{itemize}
|
|
\item{
|
|
For $a \in V$ the transformation
|
|
\begin{equation}
|
|
M_a := \sqrt{-iX_i} \prod\limits_{\{i,j\} \in E} \sqrt{iZ_j}
|
|
\end{equation}
|
|
toggles the neighbourhood $n_a := \left\{ j \middle| \{a,j\} \in E\right\}$
|
|
of $a$. I.e. it toggles the edges $n_a \otimes n_a$.
|
|
}
|
|
\item{
|
|
Many Clifford operations cannot be described by the VOP-free graph states.\\
|
|
Example:
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
&G = \left(\{0, 1\}, \{\}\right)\\
|
|
&\ket{\bar{G}} = \ket{+}\\
|
|
&U = H_0H_1 \\
|
|
&U \ket{\bar{G}} = \ket{\mbox{0b}00}\\
|
|
\end{aligned}
|
|
\end{equation}
|
|
|
|
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Graph States}
|
|
\begin{itemize}
|
|
\item{\textbf{Definition}
|
|
{\itshape
|
|
$(V, E, O)$ is called a graph with vertex operators (VOPs) iff $(V,E)$ is a
|
|
graph as in the definition above and $O = (o_1, ..., o_n)$ where
|
|
the $o_i \in C_1$.
|
|
|
|
The Stabilizers are given by
|
|
|
|
\begin{equation}
|
|
\left(\bigotimes\limits_{i} o_i\right) K_G^{(i)} \left(\bigotimes\limits_{i} o_i\right)^\dagger
|
|
\end{equation}
|
|
|
|
and the stabilizer state is
|
|
|
|
\begin{equation}
|
|
\ket{G} = \left(\bigotimes\limits_{i} o_i\right) \ket{\bar{G}}.
|
|
\end{equation}
|
|
}}
|
|
\item{One can show that there exist $24$ local Clifford gates. Therefore $O$
|
|
can be represented by $n$ integers from $0$ to $23$.}
|
|
\item{It is clear that any single qbit Clifford gate changes the VOPs
|
|
to $o_i \rightarrow c_i o_i$.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{The $CZ$ Gate on Graph States}
|
|
\begin{itemize}
|
|
\item{As it is clear how local Clifford gates act on a graph state it is
|
|
enough to show how a $CZ$ gate acts on the graph states to proof that
|
|
any Clifford gate can be applied to a graph state.}
|
|
\item{Consider the gate $CZ_{a,b}$ for $a,b \in V$.}
|
|
\item{If $CZ_{a,b}$ commutes with $o_a$ and $o_b$ the gate just toggles the edge
|
|
$\{a,b\}$ in $E$.}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{$CZ$ on Two Qbits}
|
|
\begin{itemize}
|
|
\item{If the vertices are a clique (i.e. are connected only to each other) or
|
|
are isolated they separate from the rest of the state. In this case
|
|
one can consider just the space of two qbits.
|
|
}
|
|
\item{
|
|
The amount of stabilizer states on these two qbits is finite. An upper bound
|
|
is given by $2\cdot24^2$, i.e. $24$ Clifford operators on each vertex and
|
|
the graph with or without and edge.
|
|
}
|
|
\item{All those states and the result after applying a $CZ$ gate can be
|
|
computed.}
|
|
\item{If one vertex has the VOP $I$ the result can be chosen
|
|
such that the VOP remains $I$.}
|
|
\item{This is used to implement the $CZ$ on isolated (or 2-clique) vertices.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Clearing Vertex Operators}
|
|
\begin{itemize}
|
|
\item{Recalling the transformation $M_a$ it is clear that the
|
|
graph state $\ket{G}$ is invariant under this transformation:
|
|
\begin{enumerate}[1]
|
|
\item Toggle the neighbourhood of $a$.
|
|
\item Right-multiply $M_a^\dagger$ to the VOPs.
|
|
\end{enumerate}
|
|
This transformation is called $L_a$.
|
|
}
|
|
\item{Using $\sqrt{-iX}$ and $\sqrt{iZ}$ as generators of $C_1$ one can
|
|
express any VOP as a product of $\sqrt{-iX},\sqrt{iZ}$.}
|
|
\item{Let the vertex $a$ have a neighbour $j \neq b$, then the VOP
|
|
on $a$ can be reduced to the identity by moving from right to left through the
|
|
product and applying $L_a$ for a $\sqrt{-iX}$ and $L_i$ for a $\sqrt{iZ}$.}
|
|
\item{Any vertex $j \in n_a \setminus \{i\}$ picks up powers of $\sqrt{iZ}$ during this
|
|
operation. As $\sqrt{iZ}$ commutes with $CZ$ this is no problem.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Clearing Vertex Operators to Apply a $CZ$}
|
|
\begin{itemize}
|
|
\item{The VOP on $a$ ($b$) can be cleared if $a$ ($b$) has at least one
|
|
neighbour that is not $b$ ($a$).}
|
|
|
|
\item{To apply a $CZ_{a,b}$ use the following algorithm to clear the
|
|
VOPs as much as possible:
|
|
\begin{enumerate}[1]
|
|
\item Try to clear the VOP of $a$.
|
|
\item{ Try to clear the VOP on $b$.}
|
|
\item{If the VOP of $a$ wasn't clear yet, try to clear it again.}
|
|
\end{enumerate}
|
|
}
|
|
\item{After this procedure it is certain that at least one VOP is $I$
|
|
and possibly both VOPs commute with $CZ$. If they both commute
|
|
applying the $CZ$ is done by toggling the edge $\{a,b\}$.
|
|
}
|
|
\item{If one VOP does not commute with $CZ$ this vertex is either isolated
|
|
or connected to the other operand vertex only. One can show that the
|
|
two qbit $CZ$ method can be applied here if the identity on the vertex
|
|
connected to the rest of the graph is preserved.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Clearing VOPs: Example}
|
|
\begin{itemize}
|
|
\item{Vertex operators are represented by an integer; most of the VOPs are not important.}
|
|
\item{Some notable VOPs are $0 = H$, $1 = S$, $2 = I$, $19 = \sqrt{iZ}^2\sqrt{-iX}^3$.}
|
|
\item{
|
|
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_01.png}
|
|
Clearing the VOP on vertex $1$ uses the transformation $La(1) La(1) La(1) La(2) La(2)$.
|
|
}
|
|
\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}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Measurements on Graph States}
|
|
\begin{itemize}
|
|
\item{When measuring $Z_a$ the projector $P_a := \frac{I \pm Z_a}{a}$ can be
|
|
pulled behind the vertex operator $o_a$ by transforming the observable:
|
|
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
P_a \ket{G} &= \left(\prod\limits_{i \neq a} o_i\right) P_a o_a \ket{\bar{G}} \\
|
|
&= \left(\prod\limits_{i \neq a} o_i\right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
|
|
&= \left(\prod\limits_{i} o_i \right) \tilde{P}_a \ket{\bar{G}}\\
|
|
\end{aligned}
|
|
\end{equation}
|
|
|
|
With a new projector $\tilde{P}_a = \frac{I + g_a}{2}$.
|
|
}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Measurements on Graph States}
|
|
\begin{itemize}
|
|
\item{The anticommuting stabilizers are given by
|
|
\begin{itemize}
|
|
\item{$A_X = \{K_G^{(i)} | i \in n_a\}$ for $g_a = \pm X_a$,}
|
|
\item{$A_Y = \{K_G^{(i)} | i \in n_a \cup \{a\}\}$ for $g_a = \pm Y_a$ and}
|
|
\item{$A_Z = \{K_G^{(a)}\}$ for $g_a = \pm Z_a$.}
|
|
\end{itemize}}
|
|
\item{This can be used to compute the probability amplitudes and update $(G,V,E)$ after
|
|
the measurement.}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
\section{Implementation and Performance}
|
|
{
|
|
\begin{frame}{Implementation}
|
|
\center
|
|
\includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio=true]{screenshot_github.png}
|
|
\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}{Properties of Stabilizer States}
|
|
\begin{itemize}
|
|
\item{Stabilizer states support entanglement; One example is the Bell state $\frac{\ket{\mbox{0b}00} + \ket{\mbox{0b}11}}{\sqrt{2}}$.}
|
|
\item{Some forms of superposition exist in the stabilizer formalism.}
|
|
\item{Stabilizer states and their dynamics (including measurement) can be simulated exponentiallly faster than dense state vectors.}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Non-Universality}
|
|
\begin{itemize}
|
|
\item{Recalling the discussion around measurement in the stabilizer formalism
|
|
only probability amplitudes $0, \frac{1}{2}, 1$ are possible for any Pauli
|
|
observable.}
|
|
\item{As seen in the example simulation of a spin chain other probability amplitudes
|
|
are important and particularely interesting.}
|
|
\item{Only few unitaries and states can be simulated.}
|
|
\item{The stabilizers have lost the vector space property:
|
|
only special superpositions are allowed.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
{
|
|
\begin{frame}{Possible Ways of Extending the Formalism I}
|
|
\begin{itemize}
|
|
\item{One possible way would be to use more qbits to increase the
|
|
amount of possible measurement outcomes.}
|
|
\item{One could describe the evolution of a general hermitian unitary
|
|
by expressing it as a sum of multilocal Pauli operators. Applying
|
|
non-Clifford gates would reintroduce exponential growth.
|
|
}
|
|
\item{Expressing a non-stabilizer state as a sum of stabilizer states
|
|
and computing their time evolution under Clifford gates might
|
|
be possible. Handling relative phases to ensure interference is
|
|
correct may require special care.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
\begin{frame}{Possible Ways of Extending the Formalism II}
|
|
\begin{itemize}
|
|
\item{The reason why the multilocal Pauli operators can be stored
|
|
so efficiently is that they are the tensor product of $U(2)$ matrices.}
|
|
\item{
|
|
A key property of the Clifford group is that it preserves this feature:
|
|
multilocal Pauli operators are mapped to multilocal Pauli operator, the
|
|
result is a tensor product again. The Clifford group is the Normalizer of
|
|
the multilocal Pauli group.
|
|
}
|
|
%\item{One interesting way to search for new/other states and circuits that can
|
|
% be simulated efficiently would be to check what hermitian matrices
|
|
% are tensor products of single qbit matrices and what operations would
|
|
% preserve this property.
|
|
% }
|
|
\item{The Stabilizer Formalism has since been generalized to finite
|
|
Abelian groups over the Hilbert space. The Clifford gates are generalized
|
|
to normalizers.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
%{
|
|
%\begin{frame}{Possible Ways of Extending the Formalism II}
|
|
% \begin{itemize}
|
|
% \item{Single qbit gates will always preserve the tensor product property:
|
|
% \begin{equation}
|
|
% g_i' = U_k g_i U_k^\dagger = \left(\bigotimes\limits_{j<k} g_{i,j}\right) \otimes Ug_{i,k}U^\dagger \otimes \left(\bigotimes\limits_{j>k} g_{i,j}\right)
|
|
% \end{equation}
|
|
% }
|
|
% \item{It would probably be enough to search for matrices $g_1, g_2$
|
|
% for which
|
|
% \begin{equation}
|
|
% \langle CX_{1,2} (g_1 \otimes g_2) CX_{1,2}\rangle = \langle g_1' \otimes g_2'\rangle
|
|
% \end{equation}
|
|
% holds. The Pauli matrices are one group that fulfills this property.
|
|
% }
|
|
%
|
|
% \item{It is not immideately clear how measurement would work.}
|
|
%
|
|
% \end{itemize}
|
|
%\end{frame}
|
|
%}
|
|
|
|
|
|
|
|
\end{document}
|