From d3e38f6f7e6fd36d45a94ebe742f7f05473df7db Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Thu, 9 Apr 2020 11:46:22 +0200 Subject: [PATCH] some final work --- performance/qbit_scaling_graph.csv | 24 ++--- performance/qbit_scaling_naive.csv | 24 ++--- .../regimes/circuit_scaling_graph0.csv | 92 +++++++++--------- .../regimes/circuit_scaling_graph1.csv | 92 +++++++++--------- performance/regimes/circuit_scaling_meta.json | 2 +- .../regimes/generate_data_scaling_circuits.py | 8 +- .../regimes/scaling_circuits_linear.png | Bin 155902 -> 163906 bytes performance/scaling_qbits_linear.png | Bin 132229 -> 136276 bytes performance/scaling_qbits_log.png | Bin 144486 -> 145030 bytes presentation/main.tex | 59 ++++++++--- thesis/chapters/appendix.tex | 4 +- thesis/chapters/conclusion.tex | 9 +- thesis/chapters/implementation.tex | 32 ++++-- thesis/chapters/stabilizer.tex | 7 +- thesis/main.tex | 14 +++ 15 files changed, 209 insertions(+), 158 deletions(-) diff --git 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50, "ncircuits": 100, "nqbits0": 100, "nqbits1": 50, "seed": 3735928559} \ No newline at end of file +{"nstart": 400, "nstop": 2600, "step": 50, "ncircuits": 50, "nqbits0": 100, "nqbits1": 50, "seed": 3735928559} \ No newline at end of file diff --git a/performance/regimes/generate_data_scaling_circuits.py b/performance/regimes/generate_data_scaling_circuits.py index cb2546d..6c20e78 100644 --- a/performance/regimes/generate_data_scaling_circuits.py +++ b/performance/regimes/generate_data_scaling_circuits.py @@ -51,9 +51,9 @@ def test_scaling_circuits(state_factory if __name__ == "__main__": nstart = 400 - nstop = 2800 + nstop = 2600 step = 50 - ncircuits = 100 + ncircuits = 50 nqbits0 = 100 nqbits1 = 50 seed = 0xdeadbeef @@ -65,7 +65,7 @@ if __name__ == "__main__": , step , nqbits0 , ncircuits - , repeat=10) + , repeat=5) np.random.seed(seed) results_graph1 = test_scaling_circuits(GraphState.new_zero_state , nstart @@ -73,7 +73,7 @@ if __name__ == "__main__": , step , nqbits1 , ncircuits - , repeat=10) + , repeat=4) np.savetxt("circuit_scaling_graph0.csv", results_graph0) print("saved results0 to circuit_scaling_graph0.csv") diff --git a/performance/regimes/scaling_circuits_linear.png b/performance/regimes/scaling_circuits_linear.png index 87eec75c3e009670b09751c0e85d22fa9e2a1934..dbec6ba1a7220a2e2449a3522eef38865859231f 100644 GIT binary patch literal 163906 zcmeFZc{r5s`#(IbDx!s`Y-vN*NOl!c*|KI|5+eIP7?eZ`ZOC4Ptdo78kz~qXl9()G zLQEP<*1=eQ=ji?Ze2?F8e2?dN{(1g-KKF58X3X4k-RE^)=lOcQ&hxq-Uf0psyXVLr z1Ol=5>J>G81cLF9DQwqH__I}twGsYo_r7%1a2FiGyKY6n^W7d-OuZ2ZrTdJZE!~A^ zA9zvLN8QB7z}>;e-`2|>;cDyS;o|P&;&hYW&)&=1$=yv#Oj1lzg#WgWkB7Xt`2YN0 z%-zdTe4hj7VFZF7aaHZ2VL<8xsom|UX`spv;`80>TXu52Ak|Iel&9|!ZQqx!SG-!1FA+HR!|_KE5U^XBbaX7AqLP!H#x(HzeGbX^fSj6`C_eVu>u|{@|`v?eRlBYFlER@X^hH8$m?9^jd*E;olX>qcD zl}yHl#=+OB>oo@`#wrpBBtx-gk$Kwk9A)`?LyQQ1m{jNA=^}F+UmBz09~5-7!HQ7L zhCuwex3)g87c!a{`L=X&B(vDX1baztRsxBIU$VhUTsd?-c*?=Wu%MfL3i-vQ=WWvH 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\usepackage{setspace} \usepackage{amsmath} \usepackage{hyperref} +\usepackage{url} \usepackage{geometry} \usepackage{enumerate} \usepackage{physics} @@ -13,7 +14,7 @@ %\usepackage{struktex} \usepackage{qcircuit} \usepackage{adjustbox} -\usepackage{tikz} +%\usepackage{tikz} \usetheme{metropolis} @@ -71,7 +72,7 @@ \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 classically exponentially hard problems in polynomial time.} + solve some classically exponentially hard problems in polynomial time.} \end{itemize} \end{frame} @@ -275,9 +276,24 @@ starting from the $\ket{0}^{\otimes n}$ state can be simulated and sampled efficiently, 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} @@ -349,13 +365,9 @@ \begin{itemize} \item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state. The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.} - \item{A Pauli observable $g_a \in \{X_a, Y_a, Z_a\}$ will either commute with all stabilizers - (in this case $g_a$ is a stabilizer, the measurement is deterministic and the stabilizers are unchanged) or - or anticommute with at least one stabilizer.} - \item{If $g_a$ anticommutes with a subset $A := \{S^{(i)} | \{S^{(i)}, g_a\} = 0 \}$ - the probability to measure $+1$ or $-1$ is $\frac{1}{2}$ and the stabilizers $A$ - are changed.} - \item{When going from $+g_a$ to $-g_a$ the results are changed from $+1$ to $-1$ and vice versa.} + \item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.} + \item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are + possible.} \end{itemize} \end{frame} } @@ -363,14 +375,13 @@ { \begin{frame}{Graphical States} \begin{itemize} - \item{The graphical representation of stabilizer states is an optimized way to write + \item{The graphical representation is an optimized way to write the stabilizers.} \item{ - $(V, E, O)$ is called the graphical representation of - a stabilizer state if $V = \{0, ..., n-1\}$, - $E \subset \{\{i,j\} | i,j \in V, i \neq j \}$ and - $O = \{o_0, ..., o_{n-1}\}$ where $o_i \in C_1$. - $G = (V, E)$ is a graph, $O$ are called vertex operators. + 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} @@ -570,8 +581,20 @@ { \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} } @@ -680,4 +703,10 @@ \end{itemize} \end{frame} } + +{ +\begin{frame}{References} + See the bibliography of my bachelor thesis. +\end{frame} +} \end{document} diff --git a/thesis/chapters/appendix.tex b/thesis/chapters/appendix.tex index 1d54c73..9613ba4 100644 --- a/thesis/chapters/appendix.tex +++ b/thesis/chapters/appendix.tex @@ -66,9 +66,9 @@ code is converted to an image using It has been mentioned several times that the implementation with \lstinline{ufuncs} as gates is faster than a pure \lstinline{python} implementation. To support this statement a simple benchmark is written. The -relatively simple Pauli $X$ gate us implemented, more complicated gates like $CX$ or $H$ +relatively simple Pauli $X$ gate is implemented, more complicated gates like $CX$ or $H$ have worse performance when written in \lstinline{python}. The performance -improvement when in this example is a factor around $6.4$. +improvement in this example is a factor around $6.4$. One must note that the tested \lstinline{python} code is not realistic and in a possible application there would be a significant overhead. diff --git a/thesis/chapters/conclusion.tex b/thesis/chapters/conclusion.tex index 4f6135d..b298dc3 100644 --- a/thesis/chapters/conclusion.tex +++ b/thesis/chapters/conclusion.tex @@ -3,9 +3,9 @@ As seen in \ref{ref:performance} simulation using stabilizers is exponentially faster than simulating with dense state vectors. The graphical representation -for stabilizer states is on average more efficiently than the stabilizer -tableaux. In particular one can simulate more qbits while only applying -Clifford gates. +for stabilizer states is in realistic cases more efficiently than the +stabilizer tableaux \cite{andersbriegel2005}. In particular one can simulate +more qbits while only applying Clifford gates. This is considerably useful when working on quantum error correcting strategies as they often include many qbits; the smallest quantum error correcting @@ -119,4 +119,5 @@ The exponential speedup of quantum computing is often attributed to entanglement, superposition and interference effects \cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states show entanglement, superposition and interference effects; as do computations using -general normalizers \cite{vandennest2018}. +general normalizers \cite{vandennest2018}. The question why quantum computing +can speed up computations exponentially is non-trivial. diff --git a/thesis/chapters/implementation.tex b/thesis/chapters/implementation.tex index 0b726c6..5435e4c 100644 --- a/thesis/chapters/implementation.tex +++ b/thesis/chapters/implementation.tex @@ -172,7 +172,13 @@ For the graphical state $(V, E, O)$ the list of vertices $V$ can be stored impli by demanding $V = \{0, ..., n - 1\}$. This leaves two components that have to be stored: The edges $E$ and the vertex operators $O$. Storing the vertex operators is done using a \lstinline{uint8_t} array. Every local Clifford operator is -associated with an integer ranging from $0$ to $23$. Their order is +associated with an integer ranging from $0$ to $23$ +\footnote{ + \cite{andersbriegel2005} also uses integers from $0$ to $23$ to represent + the Clifford operators. The authors do not describe the mapping from integer + to operator and the order might be different. +} +. Their order is \begin{equation} \begin{aligned} @@ -224,7 +230,8 @@ a vertex. Also one must take care to keep the memory required to store a state low. To meet both requirements a linked list-array hybrid is used. For every vertex the neighbourhood is stored in a sorted linked list (which is a sparse representation of a column vector) and these adjacency lists are stored in -a length $n$ array. +a length $n$ array. \cite{andersbriegel2005} uses the same representation for +the graph. Using this storage method all operations - including searching and toggling edges - inherit their time complexity from the sorted linked list. @@ -240,7 +247,7 @@ three classes of operations. \begin{description} \item[\hspace{-1em}]{\lstinline{RawGraphState.apply_C_L}\\ This method implements local Clifford gates. It takes the qbit index - and the index of the local Clifford operator (ranging form $0$ to $23$).} + and the index of the local Clifford operator (ranging from $0$ to $23$).} \item[\hspace{-1em}]{\lstinline{RawGraphState.apply_CZ}\\ Applies the $CZ$ gate to the state. The first argument is the act-qbit, the second the control qbit (note that this is just for @@ -270,7 +277,7 @@ of the user. This implementation reads byte code from a file and executes it. The execution is always done in three steps: -\begin{enumerate}[1] +\begin{enumerate}[1.] \item{Initializing the state according the header of the byte code file.} \item{Applying operations given by the byte code to the state. This includes local Clifford gates, $CZ$ gates and measurements (the measurement outcome is ignored).} @@ -363,7 +370,7 @@ circuits are used. Length $m$ circuits are generated from the probability space with the uniform distribution. The continuous part $[0, 2\pi)$ is ignored when generating random circuits for the graphical simulator; in order to generate random circuits for dense vector simulations this is used as the argument $\phi$ of the -$R_\phi$ gate. For $m=1$ an outcome is mapped to a gate where +$R_\phi$ gate. For $m=1$ an outcome is mapped to a gate with \begin{equation} \begin{aligned} @@ -382,7 +389,12 @@ dense vector simulator $S$ can be replaced by $R_\phi$ with the parameter $x$. Using this method circuits are generated and applied both to graphical and dense vector states and the time required to execute the operations -\cite{timeit} is measured. The resulting graph can be seen in +\cite{timeit} is measured\footnote{ + The used computer had an \lstinline{Intel(R) Core(TM) i5-4590 CPU @ 3.30GHz} processor, + \lstinline{7.7 GiB} RAM, and \lstinline{27 GiB} SSD swap space (which was unused). + The operating system was \lstinline{Linux 4.19.0-8-amd64 #1 SMP Debian 4.19.98-1 (2020-01-26) x86_64 GNU/Linux}. + +}. The resulting graph can be seen in Figure \ref{fig:scaling_qbits_linear} and Figure \ref{fig:scaling_qbits_log}. Note that in both cases the length of the circuits have been scaled linearly with the amount of qbits and the measured time was divided by the number of @@ -421,11 +433,11 @@ the stabilizer tableaux like CHP \cite{CHP} with an average runtime behaviour of $\mathcal{O}\left(n\log(n)\right)$ instead of $\mathcal{O}\left(n^2\right)$. One should be aware that the gate execution time (the time required to apply -a gate to the state) highly depends on the state it is applied to. For the +a gate to the state) highly depends on the state it is applied to \cite{andersbriegel2005}. For the dense vector simulator and CHP this is not true: Gate execution time is constant for all gates and states. Because the graphical simulator has to -toggle neighbourhoods the gate execution time of the $CZ$ gate varies -significantly. The plot Figure \ref{fig:scaling_circuits_linear} +toggle neighbourhoods the gate execution time of the $CZ$ and $M$ gates vary +significantly \cite{andersbriegel2005}. The plot Figure \ref{fig:scaling_circuits_linear} shows the circuit execution time for two different numbers of qbits. One can observe three regimes: @@ -455,7 +467,7 @@ regimes: These three regimes can be explained when considering the graphical states that typical live in these regimes. With increased circuit length the amount of -edges increases which makes toggling neighbourhoods harder. Graphs from the +edges increases which makes toggling neighbourhoods harder \cite{andersbriegel2005}. Graphs from the low-linear, intermediate and high-linear regime can be seen in Figure \ref{fig:graph_low_linear_regime}, Figure \ref{fig:graph_intermediate_regime} and Figure \ref{fig:graph_high_linear_regime}. Due to the great amount of edges diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 96a9d55..f082683 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -16,6 +16,8 @@ This performance has since been improved to $n\log(n)$ time on average \section{Stabilizers and Stabilizer States} +The discussion below follows the argumentation given in \cite{nielsen_chuang_2010}. + \subsection{Local Pauli Group and Multilocal Pauli Group} \begin{definition} @@ -46,7 +48,6 @@ deduced from its definition via the tensor product. \subsection{Stabilizers} -The discussion below follows the argumentation given in \cite{nielsen_chuang_2010}. \begin{definition} \label{def:stabilizer} @@ -292,6 +293,8 @@ and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing a $S^{( \section{The VOP-free Graph States} This section will discuss the vertex operator (VOP)-free graph states. Why they are called vertex operator-free will be clear in \ref{ref:sec_g_states}. +Most of the discussion here is adapted from \cite{hein_eisert_briegel2008} +whith some parts from \cite{andersbriegel2005}. \subsection{VOP-free Graph States} @@ -650,7 +653,7 @@ studying its dynamics. The simplest case is a local Clifford operator $c_j$ acting on a qbit $j$ changing the stabilizers to $\langle c_j S^{(i)} c_j^\dagger\rangle_i$. Using the definition of the graphical representation one sees that just the vertex operators are changed and the new vertex operators -are given by +are given by \cite{andersbriegel2005} \begin{equation} \begin{aligned} diff --git a/thesis/main.tex b/thesis/main.tex index 2d678ad..2dcbd82 100644 --- a/thesis/main.tex +++ b/thesis/main.tex @@ -88,6 +88,20 @@ Supervised by Prof. Dr. Christoph Lehner} \bibliographystyle{unsrt} \bibliography{main}{} +\newpage +{\Large\textbf{Acknowledgements}} +\\ +\\ +I want to thank Prof. Dr. Christoph Lehner for offering me the possibility to +write this thesis in his workgroup. He also found the time to answer my +questions and help me when I got stuck while keeping an inspiring atmosphere.\\ +Further I own a big thank you to Simon Feyrer for proof-reading my thesis. \\ +Also I want to thank Andreas Hackl for proof-reading my thesis and helping out +with several technical problems.\\ +Thanks to Shirley Galbaw for helping to fix some of my terrible ingris grammar. + + + \newpage \textbf{Erklärung zur Anfertigung:}\\ Ich habe die Arbeit selbständig verfasst, keine anderen als die angegebenen

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