474 lines
15 KiB
TeX
474 lines
15 KiB
TeX
\documentclass[10pt]{beamer}
|
|
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage{graphicx}
|
|
\usepackage{amssymb, amsthm}
|
|
\usepackage{setspace}
|
|
\usepackage{amsmath}
|
|
\usepackage{hyperref}
|
|
\usepackage{geometry}
|
|
\usepackage{enumerate}
|
|
\usepackage{physics}
|
|
\usepackage{listings}
|
|
%\usepackage{struktex}
|
|
\usepackage{qcircuit}
|
|
\usepackage{adjustbox}
|
|
|
|
|
|
\usetheme{metropolis}
|
|
|
|
\setbeamercolor{background canvas}{bg=white!20}
|
|
|
|
\title{An Efficient Quantum Computing Simulator using a Graphical Description for Many-Qbit Systems}
|
|
\subtitle{Simulation in the Stabilizer Formalism}
|
|
\author{Daniel Knüttel}
|
|
\date{21.02.2020}
|
|
\institute{Universität Regensburg}
|
|
|
|
\titlegraphic{\small\center Universität Regensburg\\
|
|
Faculty of the Institute of Theoretical Physics
|
|
\vspace{-11mm}\flushright\includegraphics[height=1.0cm]{logo.png}}
|
|
|
|
|
|
%\logo{\includegraphics[width=1cm]{logo.png}\hfill}
|
|
%\newcommand{\nologo}{\setbeamertemplate{logo}{}} % command to set the logo to nothing
|
|
%\newcommand{\congress}{Faculty of the Institute of Theoretical Physics}
|
|
|
|
%% footer
|
|
\makeatletter
|
|
\setbeamertemplate{footline}
|
|
{
|
|
%\leavevmode%
|
|
\hbox{%
|
|
|
|
\begin{beamercolorbox}[wd=.9\paperwidth,ht=2.25ex,dp=1ex,left]{Faculty of the Institute of Theoretical Physics}%
|
|
\end{beamercolorbox}%
|
|
|
|
\begin{beamercolorbox}[wd=.1\paperwidth,ht=2.25ex,dp=1ex,right]{Faculty of the Institute of Theoretical Physics}%
|
|
\insertframenumber{} / \inserttotalframenumber\hspace*{2ex}
|
|
\end{beamercolorbox}}%
|
|
|
|
}
|
|
\makeatother
|
|
|
|
\begin{document}
|
|
|
|
\maketitle
|
|
|
|
\section{Introduction}
|
|
|
|
{
|
|
\begin{frame}{Motivation}
|
|
|
|
\begin{itemize}
|
|
\item Some (physical) problems are classically hard to solve.
|
|
\pause
|
|
\item The quantum simulator: Mapping a hard problem to quantum hardware that can simulate this specific problem.
|
|
\pause
|
|
\item The (universal) quantum computer: able to simulate any unitary transformation on the system.
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Quantum Errors and Quantum Error Correction}
|
|
\begin{itemize}
|
|
\item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation).
|
|
\pause
|
|
\item Fault tolerant QC needs a way to correct for those errors.
|
|
\pause
|
|
\item Several strategies exist one important class of quantum error correction codes are \textbf{stabilizer codes}.
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
\section{Binary Quantum Computing}
|
|
|
|
{
|
|
\begin{frame}{Qbits}
|
|
|
|
\textbf{Definition}
|
|
{\itshape
|
|
A qbit is a two-level quantum mechanical system $\ket{0}, \ket{1}$ with $\braket{0}{1} = 0$.
|
|
|
|
In the following $Z = \sigma_Z, X = \sigma_X, Y = \sigma_Y$ will be used. $I$ is the identity matrix.
|
|
}
|
|
|
|
Where $Z\ket{0} = +1\ket{0}$ and $Z\ket{1} = -1\ket{1}$.
|
|
|
|
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Qbits and Gates}
|
|
\begin{itemize}
|
|
\item{
|
|
\textbf{Postulate}
|
|
{\itshape
|
|
A $n$-qbit system is the tensor product of the single-qbit systems.
|
|
}
|
|
}
|
|
\pause
|
|
%\item{
|
|
% For $n$ qbits define the integer state $\ket{j}$ as
|
|
|
|
% \begin{equation}
|
|
% \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
|
|
% \end{equation}
|
|
%}
|
|
\item{
|
|
A transformation $U \in SU(2^n)$ is called \textit{gate} acting on the system.
|
|
For $\bar{U} \in SU(2)$ the gate $U_i$ acting on qbit $i$ is given by
|
|
\begin{equation}
|
|
U_i := \left(\bigotimes\limits_{k < i} I\right) \otimes \bar{U} \otimes \left(\bigotimes\limits_{k > i} I\right)
|
|
\end{equation}
|
|
}
|
|
\pause
|
|
\item{
|
|
For $\bar{U} \in SU(2)$ and qbits $i \neq j$
|
|
\begin{equation}
|
|
CU_{i,j} := \ket{1}\bra{1}_j \otimes U_i + \ket{0}\bra{0}_j \otimes I
|
|
\end{equation}
|
|
is the controlled $\ket{U}$ gate acting on $i$ with control-qbit $j$.
|
|
}
|
|
\pause
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
|
|
{
|
|
\begin{frame}{Gates}
|
|
Some notable gates are
|
|
\begin{itemize}
|
|
\item{
|
|
the Hadamard gate
|
|
\begin{equation}
|
|
H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)
|
|
\end{equation}
|
|
which transforms from the $Z$ to the $X$ basis}
|
|
\pause
|
|
\item{the rotation gate
|
|
\begin{equation}
|
|
R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)
|
|
\end{equation}
|
|
that performs a rotation around the $Z$ axis.}
|
|
\pause
|
|
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Gates}
|
|
\begin{itemize}
|
|
\item{The $Z$ and $S$ gates are given by:
|
|
\begin{equation}
|
|
Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) = R_{\pi}
|
|
\end{equation}
|
|
\begin{equation}
|
|
S := \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right) = R_{\frac{\pi}{2}}
|
|
\end{equation}
|
|
|
|
The $S$ gate transforms from $X$ to $Y$ basis.
|
|
|
|
}
|
|
\pause
|
|
\item{
|
|
\textbf{Theorem}
|
|
{\itshape
|
|
Any gate $U \in SU(2^n)$ can be approximated arbitrarely good using the $H$, $R_\phi$
|
|
and $CZ_{i,j}$ gate.
|
|
}
|
|
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Integer States}
|
|
\begin{itemize}
|
|
\item{
|
|
The eigenstates of the $Z_i$ are called integer states. They have the form
|
|
\begin{equation}
|
|
\ket{j} = \ket{\mbox{0b}l_1...l_n} = \ket{l_1}_s \otimes ... \otimes \ket{l_n}s
|
|
\end{equation}
|
|
}
|
|
\pause
|
|
\item{
|
|
For $n$ qbits there exist $2^n$ such states and they form a basis
|
|
|
|
\begin{equation}
|
|
\ket{\psi} = \sum\limits_{i=0}^{n-1} \ket{i}\braket{i}{\psi} = \sum\limits_{i=0}^{n-1} c_i\ket{i}
|
|
\end{equation}
|
|
|
|
with the condition $\sum\limits_{i=0}^{n-1} c_i^2 = 1$.
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Measurement}
|
|
\begin{itemize}
|
|
\item{Measurements are performed in $Z$ basis, i.e. for a qbit $i$
|
|
$Z_i$ is measured.
|
|
}
|
|
\item{
|
|
The results of the measurements are associated with a classical result $s \in \{0, 1\}$
|
|
using
|
|
|
|
\begin{equation}
|
|
Z_i\ket{\psi'} = (-1)^s \ket{\psi'}
|
|
\end{equation}
|
|
|
|
after the measurement.
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Quantum Circuits}
|
|
\begin{itemize}
|
|
\item{
|
|
Writing a unitary transformation as a product of the generator gates is unreadable.
|
|
To fix this problem quantum circuits have been introduced.
|
|
}
|
|
\pause
|
|
\item{
|
|
Qbits are represented by horizontal lines.
|
|
}
|
|
\pause
|
|
\item{
|
|
Gates acting on a qbit are boxes on the lines.
|
|
}
|
|
\pause
|
|
\item{
|
|
Control-qbits are connected to the gate via a vertical line.
|
|
}
|
|
\pause
|
|
\item{
|
|
Circuits are read left to right.
|
|
}
|
|
\pause
|
|
\item{
|
|
Example:
|
|
\Qcircuit @C=1em @R=.7em {
|
|
& \gate{H} & \ctrl{1} & \qw &\qw \\
|
|
& \gate{H} & \gate{Z} & \gate{H} &\qw \\
|
|
}
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
|
|
\begin{itemize}
|
|
\item{
|
|
For a set of $n$ spins in a magnetic field one can rescale the Hamiltonian
|
|
of the system to
|
|
\begin{equation}
|
|
H = -\sum\limits_{i=1}^{n-1} Z_i Z_{i-1} + g\sum\limits_{i=0}^{n-1} X_i
|
|
\end{equation}
|
|
}
|
|
\pause
|
|
\item{
|
|
The time evolution of such a system is given by the transfer matrix
|
|
\begin{equation}
|
|
T := \exp(-itH) \in SU(2^n)
|
|
\end{equation}
|
|
}
|
|
\pause
|
|
\item{
|
|
By associating every qbit with one spin (both are two-level systems)
|
|
one should be able to simulate the behaviour of the spin chain using
|
|
a quantum computer.
|
|
}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
|
|
\begin{itemize}
|
|
\item{
|
|
Trotterizing the matrix exponential
|
|
\begin{equation}
|
|
\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)
|
|
\end{equation}
|
|
}
|
|
\item{
|
|
For $n=3$ spins one gets a circuit
|
|
{\centering\adjustbox{max width=\textwidth}{
|
|
\Qcircuit @C=1em @R=.7em {
|
|
& \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
|
|
& \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} &\qw \\
|
|
& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} &\qw \\
|
|
}
|
|
|
|
}}
|
|
}
|
|
\item{Applying this circuit $N$ times gives an approximation for the time evolution of a state.}
|
|
\end{itemize}
|
|
\end{frame}
|
|
}
|
|
{
|
|
\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
|
|
\begin{figure}[h]
|
|
\begin{center}
|
|
\includegraphics[width=\linewidth]{spin_chain/time_evo_6spin_g3.png}
|
|
\end{center}
|
|
\end{figure}
|
|
\end{frame}
|
|
}
|
|
|
|
\section{Stabilizers}
|
|
|
|
{
|
|
\begin{frame}{The multilocal Pauli Group and the Clifford Group}
|
|
\begin{itemize}
|
|
\item{\textbf{Definition}
|
|
{\itshape
|
|
$P := \{\pm1, \pm i\} \cdot \{X, Y, Z, I\}$ is called the Pauli group.\\
|
|
$P_n := \left\{p_1 \otimes ... \otimes p_n \middle| p_i \in P \right\}$ is called the multilocal Pauli group.
|
|
}}
|
|
\pause
|
|
\item{
|
|
\textbf{Definition}
|
|
{\itshape
|
|
$C_n := \left\{U \in SU(2^n) \middle| \forall p \in P_n: U^\dagger p U \in P_n\right\}$
|
|
is called the Clifford group
|
|
}
|
|
}
|
|
\pause
|
|
\item{
|
|
One can show that $C_n$ is generated by $H, S, CZ_{i,j}$.
|
|
}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Generators of a Group}
|
|
\begin{itemize}
|
|
\item{
|
|
\textbf{Definition}
|
|
{\itshape
|
|
For a finite group $G$ one says $G$ is generated by $g_1, ..., g_N$ iff any $g \in G$ can be expressed
|
|
as a product of $g_1, ..., g_N$. These generators are chosen to be the minimal set for which this
|
|
condition holds.
|
|
|
|
One also writes
|
|
\begin{equation}
|
|
G = \langle g_1,...,g_N \rangle \equiv \langle g_i\rangle_i
|
|
\end{equation}
|
|
}
|
|
}
|
|
\item{
|
|
The generators are not unique. For instance $C_1$ can be generated using $H, S$ or $\sqrt{-iX}, \sqrt{iZ}$.
|
|
}
|
|
\item{
|
|
The generators of a group have some kind of independence property.
|
|
}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Stabilizers and Stabilizer Spaces}
|
|
\begin{itemize}
|
|
\item{
|
|
\textbf{Definition}
|
|
{\itshape
|
|
A finite abelian subgroup $S$ of $P_n$ is called a set of stabilizers iff $-I \notin S$ and
|
|
all elements of $S$ commute.
|
|
}
|
|
}
|
|
\item{
|
|
From $-I \notin S$ follows that all elements of $S$ are hermitian.
|
|
}
|
|
\item{
|
|
From the definition as tensor products of Pauli matrices follows that the
|
|
elements of $S$ have eigenvalues $\pm1$.
|
|
}
|
|
\item{
|
|
These properties together yield that all elements of $S$ can be diagonalized simultaneously.
|
|
Further there exists a vector space $V_S$ with $s \ket{\psi} = +1 \ket{\psi}$ for all $s \in S$
|
|
and all $\ket{\psi} \in V_S$. This space is called the stabilizer space of $S$
|
|
and all $\ket{\psi}$ are called stabilizer states.
|
|
}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\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{
|
|
$\ket{0b00} + \ket{0b11}$ 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 $U \in SU(2^n)$ 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}
|
|
this gives that if $U \in C_n$ $\ket{\psi'}$ 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 $n^2$ time instead of $2^n$.}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
{
|
|
\begin{frame}{Measurements on Stabilizer States}
|
|
\begin{itemize}
|
|
\item{
|
|
}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
}
|
|
|
|
\end{document}
|