504 lines
17 KiB
TeX
504 lines
17 KiB
TeX
\documentclass[10pt]{beamer}
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\usepackage[utf8]{inputenc}
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\usepackage{graphicx}
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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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\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}
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\begin{itemize}
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\item Some (physical) problems are classically hard to solve.
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\pause
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\item The quantum simulator: Mapping a hard problem to quantum hardware that can simulate this specific problem.
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\pause
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\item The (universal) quantum computer: able to simulate any unitary transformation on the system.
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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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\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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Where $Z\ket{0} = +1\ket{0}$ 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 $U \in SU(2^n)$ is called \textit{gate} acting on the system.
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For $\bar{U} \in SU(2)$ 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 $\bar{U} \in SU(2)$ 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 $U \in SU(2^n)$ can be approximated arbitrarely good 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 SU(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 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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$P := \{\pm1, \pm i\} \cdot \{X, Y, Z, I\}$ 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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$C_n := \left\{U \in SU(2^n) \middle| \forall p \in P_n: U^\dagger p U \in P_n\right\}$
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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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{
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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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\begin{frame}{Stabilizers and Stabilizer Spaces}
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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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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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{
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\begin{frame}{Stabilizer States}
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\begin{itemize}
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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 $V_S$
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has dimension $1$.
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}
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\item{
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Therefore the state $\ket{\psi}$ that is the +1 eigenstate of all stabilizers is (up to a
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global phase) unique.
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}
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\item{Notable stabilizer states are:
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\begin{itemize}
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\item{ $\ket{0b0..0} = \ket{0}\otimes ... \otimes \ket{0}$ and $\ket{0b1..1} = \ket{1}\otimes ... \otimes \ket{1}$
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which are stabilized by $\langle Z_i \rangle_i$ and $\langle -Z_i \rangle_i$ respectively.
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}
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\item{
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$\ket{+}$ stabilized by $\langle X_i \rangle_i$ (and $\ket{-}$ analogously).
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}
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\item{
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$\ket{0b00} + \ket{0b11}$ the Bell/EPR state which is stabilized by $\langle X_1X_2, Z_1Z_2\rangle$.
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}
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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}{Dynamics of Stabilizer States}
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\begin{itemize}
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\item{
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Under a transformation $U \in SU(2^n)$ the state changes to
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\begin{equation}
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\ket{\psi'} = U \ket{\psi}
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\end{equation}
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}
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\item{
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The stabilizers of $\ket{\psi}$ change to
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\begin{equation}
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\ket{\psi'} = U\ket{\psi} = US^{(i)}\ket{\psi} = US^{(i)}U^\dagger U\ket{\psi} = US^{(i)}U^\dagger\ket{\psi'}
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\end{equation}
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using this it is clear that for $U \in C_n$ the state $\ket{\psi'}$ is a stabilizer state again with the stabilizers
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$S' = \langle US^{(i)}U^\dagger\rangle_{i=1,...,n}$
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}
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\item{
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Under the transformations $C_n$ one can describe the dynamics of the stabilizer states by their
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stabilizers.
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}
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\item{Because the stabilizers are given by $n$ matrices which are the tensor product of $n$ Pauli matrices
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this can be simulated in $\mathcal{O}\left(n^2\right)$ time instead of $\mathcal{O}\left(2^n\right)$.}
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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 on Stabilizer States}
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\begin{itemize}
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\item{
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Consider a Pauli observable $g_a \in \{(-1)^s X_a, (-1)^s Y_a, (-1)^s Z_a\}$ and the projector
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onto its eigenspace $\frac{I + g_a}{2}$.
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}
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\item{If $g_a$ commutes with all $S^{(i)}$ the observable $g_a$ is diagonal in this basis and
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the stabilizer state $\ket{\psi}$ is the $+1$ eigenstate of $g_a$. Therefore the measurement is
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deterministic and the stabilizers remain unchanged.}
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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 on Stabilizer States}
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\begin{itemize}
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\item{If $g_a$ does not commute with all stabilizers $A := \left\{S^{(i)} \middle| \left[g_a, S^{(i)}\right] \neq 0\right\}$
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is the set of stabilizers that anticommute with $g_a$.}
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\item{
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To compute the probability to measure a result of $s=0$ one can use the trace formula
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\begin{equation}
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\begin{aligned}
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P(s=0) &= \Tr(\frac{I + g_a}{2} \ket{\psi}\bra{\psi}) \\
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&= \Tr(\frac{I + g_a}{2} S^{(j)} \ket{\psi}\bra{\psi}) \\
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&= \Tr(S^{(j)}\frac{I - g_a}{2} \ket{\psi}\bra{\psi}) \\
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&= \Tr(\frac{I - g_a}{2} \ket{\psi}\bra{\psi}) \\
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&= P(s=1)\\
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\end{aligned}
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\end{equation}
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where $S^{(j)} \in A$. The stabilizer $S^{(j)}$ pulled to the right using the cyclic property of the
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trace and absorbed into the $\bra{\psi}$.
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}
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\end{itemize}
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\end{frame}
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}
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\end{document}
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