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\documentclass[10pt]{beamer}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{amssymb, amsthm}
\usepackage{setspace}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage{url}
\usepackage{geometry}
\usepackage{enumerate}
\usepackage{physics}
\usepackage{listings}
%\usepackage{struktex}
\usepackage{qcircuit}
\usepackage{adjustbox}
\usepackage{tikz}
\usepackage{calc}
\usetikzlibrary{calc}
\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 Graph 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}}
\newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
\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}
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\makeatother
\begin{document}
\maketitle
\section{Introduction}
{
\begin{frame}{Motivation: Exponentially Hard (Physical) Problems}
\begin{itemize}
\item{Some mathematical problems cannot be solved in polynomial time, for instance prime factorization.}
\item{There exist several physical systems which are interesting to study but hard so simulate such as
QCD simulations at finite chemical potential or real time scattering amplitudes in QCD.}
\item{The exponential behaviour in time (and space) complexity brings classical supercomputers to their limits.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{The Universal Quantum Computer}
\begin{itemize}
% 2^n complex vector, arbitrary unitary, measurements
\item{A universal quantum computer is a $2^n$ dimensional complex vector
to which any unitary can be applied with a quantum mechanical measurement process.}
\item{Algorithms using the exponentially large Hilbert space can
solve some classically exponentially hard problems in polynomial time.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Motivation: Using Quantum Computing for Physics}
\begin{itemize}
% Literaturübersicht.
\item{
Possible applications of quantum computing include:
\begin{itemize}
\item{Preparing ground states of non-perturbative systems by adiabatic
time evolution (\textit{https://www.scottaaronson.com/qclec/26.pdf}).\\
Also works in QCD (\textit{arXiv:2001.11145v1}).}
%\item{Use algorithms such as Phase Estimation to analyze the spectrum
% of a system.}
\item{Study the time evolution by applying the transfer matrix (\textit{see later}).}
\item{Compute trace formuli (\textit{arXiv:2001.11145v1}).}
\end{itemize}
}
% Future use & current use.
% (Tensor networks)
\item{
In the near future only small and noisy machines will be available.
Some Hilbert spaces have infinite dimension (bosonic states).
Two main strategies exist to deal with this problem:
\begin{itemize}
\item{Selecting a physically interesting subspace.
% (tensor networks)
}
\item{Dividing the Hilbert space in system and bath, treating
the bath statistically (\textit{Andreas Hackl}).}
\end{itemize}
}
\item{
Long Term: Fault tolerant computing.
}
\end{itemize}
\end{frame}
}
\section{Binary Quantum Computing}
{
\begin{frame}{One Qbit}
\begin{itemize}
\item{One qbit has the Hilbert space $\mathcal{H}$ with basis vectors $\ket{0}$, $\ket{1}$.}
\item{The Pauli matrices $X, Y, Z$ are gates commonly used.
$X$ is also called the $NOT$ gate as it maps $\ket{0}$ to $\ket{1}$ and
vice versa.
}
\item{The Hadamard gate
$H = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)$
transforms from Pauli $Z$ to $X$ basis.}
\item{The $R_\phi = \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)$
gate rotates a state vector around the $Z$ axis.}
\item{The $R_{\frac{\pi}{2}}$ gate transforms from $X$ to $Y$ basis.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Universal Gates}
\begin{itemize}
\item{A quantum computer should be able to simulate any unitary on $\mathcal{H}^{\otimes n}$ the
$n$ qbit Hilbert space.}
\item{Similarly to classical computers a universal set of operations is required.}
\item{\tikzmark{left}One can show that any unitary acting on $\mathcal{H}^{\otimes n}$
can be generated using the $CX$ and universal gates
acting on $\mathcal{H}$ with
\begin{equation}
CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i.
\end{equation}
}
\item{The gates $\{H, R_\phi\}$ are universal on $\mathcal{H}$.\hfill\tikzmark{right}}
\end{itemize}
\DrawBox
\end{frame}
}
{
\begin{frame}{Measurements and Computational Hardness}
\begin{itemize}
\item{When measuring a qbit $i$ the observable $Z_i$ is measured.}
\item{Operations on the $2^n$ dimensional state will have to update $2^n$ complex
coefficients. This cannot be performed in $\mbox{poly}(n)$ time.}
\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 U(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}
\begin{aligned}
\exp(t(A + B)) &= \left(\exp(\frac{t}{2N}A)\exp(\frac{t}{N}B)\exp(\frac{t}{2N}A)\right)^N \\
& + \mathcal{O}\left(\frac{t^3}{N^3}\right)\\
\end{aligned}
\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 Gottesman-Knill Theorem}
\begin{itemize}
\item{When restriciting the $R_\phi$ gate to $\phi = \frac{\pi}{2}$
many states can still be simulated.}
\item{\textbf{Theorem}(\textit{Gottesman-Knill}):
All states that can be reached using $H, R_\frac{\pi}{2}, CZ$ and measurements
starting from the $\ket{0}^{\otimes n}$ state can be
\textbf{simulated} and \textbf{sampled efficiently on a classical computer}, i.e. in $\mbox{poly}(n, m)$ time
where $m$ is the amount of gates/measurements.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{The Gottesman-Knill Theorem}
\begin{itemize}
\item{Note that a general state has $2^n$ complex coefficients.
Computing operations on this state is therefore exponentially hard in
$n$.}
\item{Note that using the $R_\phi$ is universal and $R_\frac{\pi}{4}$
allows rational approximations of universal gates.}
\item{Restricting $\phi$ to $\frac{\pi}{2}$ allows the simulation of
large numbers of qbits on a classical computer.\\
{\bf Goal of this project is to create a simulator that can
simulate several thousand up to millions of qbits.
}}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Graphical States}
\begin{itemize}
\item{
For a set of vertices $V = \{0, ..., n-1\}$, some edges
$E \subset V \otimes V$, and local Clifford operators
$O = \{o_0, ..., o_{n-1}\}$ the tuple $(V, E, O)$
is the graphical representation of a stabilizer state.
}
\item{The state associated with $(V, E, O)$ is given by
\begin{equation}
\ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right) \prod\limits_{\{i,j\} \in E} CZ_{i,j} \ket{+}^{\otimes n}
\end{equation}
where $\ket{+} = H\ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$.
}
\item{One can show that all stabilizer states can be brought into this form.}
\item{For the derivation of the stabilizer and graphical states see my bachelor's thesis.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Example: The $5$ Qbit EPR State}
\begin{itemize}
\item{Start from $\ket{0}^{\otimes 5}$.}
\item{Get to the state $\ket{\psi} = \frac{\ket{0}^{\otimes 5} + \ket{1}^{\otimes 5}}{\sqrt{2}}$.}
\item{Use the circuit \\
\[
\Qcircuit @C=1em @R=.7em {
& \gate{H} & \ctrl{1} & \qw & \ctrl{2} & \qw & \ctrl{3} & \qw & \ctrl{4} &\qw \\
& \qw & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
& \qw & \qw & \qw & \gate{X} & \qw & \qw & \qw & \qw &\qw \\
& \qw & \qw & \qw & \qw & \qw & \gate{X} & \qw & \qw &\qw \\
& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{X} &\qw \\
}.
\]
}
\item{The state has the form
\begin{equation}
\ket{\psi} = \left(\prod\limits_{1 < i < 5} CX_{i,0}\right) H_0 \ket{0}^{\otimes 5}.
\end{equation}}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Example: The $5$ Qbit EPR State}
\begin{itemize}
\item{$CX_{i,j} = H_i CZ_{i,j} H_i$.}
\item{The new circuit is
\[
\Qcircuit @C=1em @R=.7em {
& \gate{H} & \ctrl{1} & \qw & \ctrl{2} & \qw & \ctrl{3} & \qw & \ctrl{4} & \qw &\qw \\
& \gate{H} & \gate{Z} & \qw & \qw & \qw & \qw & \qw & \qw & \gate{H} &\qw \\
& \gate{H} & \qw & \qw & \gate{Z} & \qw & \qw & \qw & \qw & \gate{H} &\qw \\
& \gate{H} & \qw & \qw & \qw & \qw & \gate{Z} & \qw & \qw & \gate{H} &\qw \\
& \gate{H} & \qw & \qw & \qw & \qw & \qw & \qw & \gate{Z} & \gate{H} &\qw \\
}.
\]
}
\item{Switching from starting state $\ket{0}^{\otimes 5}$ to $\ket{+}^{\otimes 5}$ gives the
graphical representation.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Example: The $5$ Qbit EPR State}
\noindent\begin{minipage}{0.4\textwidth}
\center
\adjustbox{max width=\textwidth}{
\Qcircuit @C=1em @R=.7em {
& \gate{H} & \ctrl{1} & \qw & \ctrl{2} & \qw & \ctrl{3} & \qw & \ctrl{4} &\qw \\
& \qw & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
& \qw & \qw & \qw & \gate{X} & \qw & \qw & \qw & \qw &\qw \\
& \qw & \qw & \qw & \qw & \qw & \gate{X} & \qw & \qw &\qw \\
& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{X} &\qw \\
}
}\\
\begin{equation}
\begin{aligned}
\ket{\psi} &= \left(\prod\limits_{1 < i < 5} CX_{i,0}\right) H_0 \ket{0}^{\otimes n}\\
\end{aligned}
\end{equation}
\end{minipage} \hfill
\begin{minipage}{0.5\textwidth}
\center
\includegraphics[width=\textwidth, height=\textheight, keepaspectratio=true]{example_graphs/graph_EPR_state.png}
The graphical state with $E =\{\{0,1\}, \{0,2\}, \{0,3\}, \{0,4\}\}$ and $O = (I, H, H, H, H)$.
\begin{equation}
\begin{aligned}
\ket{G} = \left(\prod\limits_{1 < i < 5} H_i\right) \left(\prod\limits_{1 < i < 5} CZ_{i,0}\right) \ket{+}^{\otimes n}
\end{aligned}
\end{equation}
\end{minipage}
\end{frame}
}
{
\begin{frame}{Dynamics of Graphical States}
\begin{itemize}
\item{
\noindent\begin{minipage}{0.4\textwidth}
Applying a local Clifford gate $U_i$ is trivial:
Just the vertex operator is updated to $U o_i$.
\end{minipage} \hfill
\begin{minipage}{0.5\textwidth}
\tikzmark{left}Apply $SH = R_{\frac{\pi}{2}}H$ to qbit $2$:\\
\includegraphics[width=\textwidth, height=\textheight, keepaspectratio=true]{example_graphs/graph_update_VOP.png}\tikzmark{right}
\DrawBox
\end{minipage}
}
\item{
\noindent\begin{minipage}{0.4\textwidth}
If $o_a, o_b \in \{I, Z, R_\frac{\pi}{2}, R_\frac{\pi}{2}^\dagger\}$ applying a $CZ_{a,b}$
just toggles the edge $\{a,b\}$ in $E$.%: $E' = E \Delta \{\{a,b\}\}$.
%With the symmetric set difference $\Delta$.
\end{minipage} \hfill
\begin{minipage}{0.5\textwidth}
\tikzmark{left}Apply $CZ_{2,0}$:\\
\includegraphics[width=\textwidth, height=\textheight, keepaspectratio=true]{example_graphs/graph_apply_CZ.png}\hfill\tikzmark{right}
\DrawBox
\end{minipage}
}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Dynamics of Graphical States}
\begin{itemize}
\item{
\noindent\begin{minipage}{0.4\textwidth}
If the vertices $a, b$ are isolated the resulting state after
applying $CZ_{a,b}$ can be precomputed.
\end{minipage} \hfill
\begin{minipage}{0.5\textwidth}
\tikzmark{left}Before:\\
\includegraphics[width=\textwidth, height=0.3\textheight, keepaspectratio=true]{example_graphs/graph_two_qbit_CZ_before.png}\\
After:\\
\includegraphics[width=0.5\textwidth, height=0.3\textheight, keepaspectratio=true]{example_graphs/graph_two_qbit_CZ_after.png}\hfill\tikzmark{right}
\DrawBox
\end{minipage}
}
\item{
\noindent\begin{minipage}{0.4\textwidth}
When the VOPs do not commute with $CZ$ and the vertices are not isolated one can clear at least one
vertex operator (i.e. transforming it to $I$). This changes the neighbourhood of the vertices.
\end{minipage} \hfill
\begin{minipage}{0.5\textwidth}
\tikzmark{left}The starting state:\\
\includegraphics[width=0.7\textwidth, height=0.5\textheight, keepaspectratio=true]{example_graphs/graph_clear_VOPs_CZ_before.png}\hfill\tikzmark{right}
\DrawBox
\end{minipage}
}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Dynamics of Graphical States}
\begin{itemize}
\item{
If both vertex operators can be cleared $\{a,b\}$ is toggled in $E$.
\noindent\begin{minipage}{0.4\textwidth}
\tikzmark{left}After clearing vertices $1,2$:\\
\includegraphics[width=0.7\textwidth, height=0.5\textheight, keepaspectratio=true]{example_graphs/graph_clear_VOPs_CZ_cleared.png}\hfill\tikzmark{right}
\DrawBox
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
\tikzmark{left}Applying $CZ_{2, 1}$:\\
\includegraphics[width=0.7\textwidth, height=0.5\textheight, keepaspectratio=true]{example_graphs/graph_clear_VOPs_CZ_after.png}\tikzmark{right}
\DrawBox
\end{minipage}
}
\item{
If just one vertex operator has been cleared the other vertex is isolated
and one can precompute all resulting states.}
\item{One can show that the probability amplitudes when measuring a qbit of a graphical state
are either $0$, $1$ or $\frac{1}{2}$.}
\end{itemize}
\end{frame}
}
\section{Implementation and Performance}
{
\begin{frame}{Implementation}
\noindent\begin{minipage}{0.5\textwidth}
\center
\includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio=true]{screenshot_github.png}
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
{\small
\begin{itemize}
\item{The code is GPLv3.0 licensed.}
\item{GitHub repository: \url{https://github.com/daknuett/pyqcs}}
\item{Install it via PyPI: \lstinline{pip3 install pyqcs}}
\item{The code is tested automatically on TravisCI.}
\end{itemize}
}
\end{minipage}
\end{frame}
}
{
\begin{frame}{Implementation}
\begin{itemize}
\item{Both a dense vector simulator and a simulator using the graphical
representation have been implemented in the \lstinline{python3} package
\lstinline{pyqcs}.}
\item{To increase simulation efficiency the core of both simulators has been
implemented in \lstinline{C}.}
\item{The dense vector states are stored in \lstinline{numpy} arrays.}
\item{The graph is stored in an length $n$ array of linked lists.}
\item{The vertex operators are local Clifford operators. The local Clifford
group as $24$ elements, they are represented by integers
stored in a \lstinline{uint8_t} array.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
\includegraphics[width=\textwidth]{../performance/scaling_qbits_linear.png}
\end{frame}
}
%{
%\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
% \includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png}
%\end{frame}
%}
{
\begin{frame}{Performance: Circuit Length on Graphical Representation}
\includegraphics[width=\textwidth]{../performance/regimes/scaling_circuits_linear.png}
\end{frame}
}
%{
%\begin{frame}{Performance: Circuit Length on Graphical Representation}
% \begin{itemize}
% \item{There seem to be three regimes: Low-linear regime, intermediate regime and high-linear regime.}
% \item{In the low-linear regime only few VOPs have to be cleared.
% The length of this regime increases with the number of qbits.
% }
% \item{In the high-linear regime the neighbourhoods are big; the probability that VOPs
% must be cleared is high. Clearing VOPs involves many vertices.}
% \item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$
% no linear behaviour.}
% \end{itemize}
%
%\end{frame}
%}
%{
%\begin{frame}{Graph in the Low-Linear Regime}
% \includegraphics[width=\textwidth]{../thesis/graphics/graph_low_linear_regime.png}
%\end{frame}
%}
%{
%\begin{frame}{Window in a Graph in the Intermediate Regime}
% \includegraphics[width=\textwidth]{../thesis/graphics/graph_intermediate_regime_cut.png}
%\end{frame}
%}
%{
%\begin{frame}{Window in a Graph in the High-Linear Regime}
% \includegraphics[width=\textwidth]{../thesis/graphics/graph_high_linear_regime_cut.png}
%\end{frame}
%}
%{
%\begin{frame}{Performance: Circuit Length on Graphical Representation}
% \begin{itemize}
% \item{Pauli measurements reduce entanglement entropy.}
% \item{Pauli measurements reduce the amount of edges in the graph.}
% \item{When adding measurement to the random circuits the regimes
% do not show up.}
% \end{itemize}
%\end{frame}
%}
{
\begin{frame}{Performance: Circuit Length on Graphical Representation}
\includegraphics[width=\textwidth]{../performance/regimes/scaling_circuits_measurements_linear.png}
\end{frame}
}
\section{Conclusion and Outlook}
{
\begin{frame}{Performance}
\begin{itemize}
\item{Simulation using stabilizers (in particular using the graphical representation)
is polynomial hard.}
\item{Simulation using dense state vectors is exponentially hard.}
\item{The performance of the graphical simulator depends on the state/circuit.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Physical Applications}
\begin{itemize}
\item{The graphical simulator cannot be used for physical applications because
interesting problems require real (or at least rational) probability amplitudes.}
\item{Stabilizer states do not span a vector space.}
\item{The idea of finding a more efficient representation for a subset of
states can be used in physics.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{References}
See the bibliography of my bachelor thesis.
\end{frame}
}
\end{document}
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