some fixes

thesis/chapters/conclusion.tex

@@ -2,10 +2,10 @@
 \section{Conclusion and Outlook}
 
 As seen in \ref{ref:performance} simulation using stabilizers is exponentially
-faster than simulating using dense state vectors. Using a graphical
+faster than simulating using state vectors. Using a graphical representation
-representation for the stabilizers is on average more efficiently than using
+for the stabilizers is on average more efficiently than using a stabilizer
-a stabilizer tableaux. In particular one can simulate more qbits while only
+tableaux. In particular one can simulate more qbits while only applying
-applying Clifford gates.
+Clifford gates.
 
 This is considerably useful when working on quantum error correcting strategies
 as they often include many qbits; the smallest quantum error correcting
@@ -43,7 +43,7 @@ matrices. A general unitary on $n$ qbits would be a $2^{n} \times 2^{n}$ matrix
 which requires more space than a dense state vector. The Clifford group is
 a group preserving this tensor product property.
 
-When the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
+When lifting the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
 group but using $n$ arbitrary commuting hermitians $\langle h_1, ..., h_n
 \rangle$ that are the tensor product of $2\times 2$ hermitians one quickly
 realizes that one could apply any single-qbit gate to the $h_i$ and preserve

thesis/chapters/implementation.tex

@@ -5,11 +5,11 @@ This chapter discusses how the concepts introduced before are implemented into
 a simulator. Futher the infrastructure around the simulation and some tools are
 explained.
 
-The implementation is written as a \lstinline{python3} module. This allows
+The implementation is written as the \lstinline{python3} package \lstinline{pyqcs} \cite{pyqcs}. This allows
-users to quickly construct circuit, apply them to a state and measure
+users to quickly construct circuits, apply them to states and measure
-amplitudes.  Full access to the state (including intermediate) state has been
+amplitudes.  Full access to the state (including intermediate states) has been
 priorized over execution speed. To keep the simulation speed as high as
-possible under these constraints some parts are implemented in \lstinline{C}
+possible under these constraints some parts are implemented in \lstinline{C}.
 
 \subsection{Dense State Vector Simulation}
 
@@ -76,7 +76,7 @@ A basic set of gates is implemented in PyQCS:
     \item{$Z$ rotation $R_\phi$ gate.}
     \item{Controlled $X$ gate: $CX$.}
     \item{Controlled $Z$ gate: $CZ$.}
-    \item{The measurement "gate" $M$.}
+    \item{The "measurement gate" $M$.}
 \end{itemize}
 
 To allow the implementation of possible hardware related gates the class
@@ -118,8 +118,8 @@ Out[2]: (0.7071067811865476+0j)*|0b0>
 \end{lstlisting}
 
 Large circuits can be constructed using the binary OR operator \lstinline{|} in
-an analogy to the pipeline operator on many *NIX systems. As usual circuits are
+an analogy to the pipeline operator on many *NIX shells. As usual circuits are
-read from left to right similar to pipelines on *NIX systems:
+read from left to right similar to pipelines on *NIX shells:
 
 
 %\adjustbox{max width=\textwidth}{
@@ -195,7 +195,7 @@ their order is
    &\left(\begin{matrix}\frac{1}{2} - \frac{i}{2} & - \frac{1}{2} + \frac{i}{2}\\- \frac{1}{2} + \frac{i}{2} & - \frac{1}{2} + \frac{i}{2}\end{matrix}\right), 
    \left(\begin{matrix}0 & -1\\1 & 0\end{matrix}\right), 
    \left(\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2} i}{2}\end{matrix}\right), 
-   \left(\begin{matrix}\frac{1}{2} - \frac{i}{2} & \frac{i \left(-1 + i\right)}{2}\\- \frac{1}{2} + \frac{i}{2} & \frac{i \left(-1 + i\right)}{2}\end{matrix}\right)
+   \left(\begin{matrix}\frac{1}{2} - \frac{i}{2} & \frac{i \left(-1 + i\right)}{2}\\- \frac{1}{2} + \frac{i}{2} & \frac{i \left(-1 + i\right)}{2}\end{matrix}\right).
 \end{aligned}
 \end{equation}
 
@@ -217,13 +217,13 @@ Recalling some operations on the graph as described in
 \ref{ref:dynamics_graph}, \ref{ref:meas_graph} or Lemma \ref{lemma:M_a} one
 sees that it is important to efficiently access and modify the neighbourhood of
 a vertex.  To ensure good performance when accessing the neighbourhood while
-keeping the memory requirements low a linked list-array hybrid is used to store
+keeping the required  memory low a linked list-array hybrid is used to store
 the adjacency matrix. 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.
 
 Using this storage method all operations including searching and toggling edges
-are inherite their time complexity from the sorted linked list.
+inherite their time complexity from the sorted linked list.
 
 \subsubsection{Operations on Graphical States}
 
@@ -315,9 +315,9 @@ one can generate random graphical states which is more performant than using
 random circuits.
 
 The function \lstinline{pyqcs.util.to_circuit.graph_state_to_circuit} converts
-graphical states into circuits (mapping the $\ket{0b0..0}$ to this state.
+graphical states to circuits (mapping the $\ket{0b0..0}$ to this state).
 Using these circuits the graphical state can be copied or converted to a
-dense vector state. Further it is a way to optimize circuits to run them on 
+dense vector state. Further it is a way to optimize circuits and later run them on 
 other simulators. Also the circuits can be exported to \lstinline{qcircuit} code
 (see below) which is a relatively readable way to represent graphical states.
 
@@ -328,7 +328,7 @@ circuits in this documents use \lstinline{qcircuit}. To visualize the circuits
 built using \lstinline{pyqcs} the function\\
 \lstinline{pyqcs.util.to_diagram.circuit_to_diagram} can be used to generate
 \lstinline{qcircuit} code that can be used in \LaTeX documents or exported to
-PDFs directly. The diagrams produced by this function is not optimized and the
+PDFs directly. The diagrams produced by this function are not optimized and the
 diagrams can be unnecessary long.  Usually this can be fixed easily by editing
 the produced code manually.
 
@@ -380,9 +380,10 @@ 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
-Figure \ref{fig:scaling_qbits_linear} and Figure \ref{fig:scaling_qbits_log}. Note in both cases the length of the circuits
+Figure \ref{fig:scaling_qbits_linear} and Figure \ref{fig:scaling_qbits_log}.
-have been scaled linearely with the amount of qbits and the measured time was
+Note that in both cases the length of the circuits have been scaled linearely
-divided by the number of qbits:
+with the amount of qbits and the measured time was divided by the number of
+qbits:
 
 \begin{equation}
 \begin{aligned}
@@ -465,8 +466,8 @@ edges increases which makes toggling neighbourhoods harder. 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}. The latter is hardly visible; this is due
-to the great amount of edges in this regime. Further these two regimes are only
+to the great amount of edges in this regime. Further the regimes are not clearly
-visibly for $n>30$ qbits so choosing smaller graphs is not possible. 
+visibe for $n>30$ qbits so choosing smaller graphs is not possible. 
 
 Because states with more qbits reach the intermediate regime at higher circuit
 lengths it is important to account for this virtual performance boost when
@@ -477,10 +478,10 @@ in Figure \ref{fig:scaling_qbits_linear} had to be scaled with the qbit number.
 
 Although the simulator(s) are in a working state and have been tested there is
 still some work that can be done.  A noise model helping to teach and analyze
-noisy execution is one particularly interesting piece of work that should be
+noisy execution is one particularly interesting piece of work.  To allow a user
-done.  To allow a user to execute circuits on other machines, including both
+to execute circuits on other machines, including both real hardware and
-real hardware and simulators, a module that exports circuits to OpenQASM
+simulators, a module that exports circuits to OpenQASM \cite{openqasm} seems
-\cite{openqasm} seems useful. 
+useful. 
 
 The current implementation of some graphical operations can be optimized. While
 clearing VOPs as described in \ref{ref:dynamics_graph} the neighbourhood of

thesis/chapters/introduction.tex

@@ -6,7 +6,7 @@ many companies and institutions working on building and using quantum computers
 \cite{ibmq}\cite{intelqc}\cite{microsoftqc}\cite{dwavesys}\cite{lrzqc}\cite{heise25_18}.
 One important topic in this research is quantum error correction
 \cite{nielsen_chuang_2010}\cite{gottesman2009}\cite{gottesman1997}\cite{shor1995}
-that will allow the execution of arbitrarily long quantum circuits. One
+that will allow the execution of arbitrarily long quantum circuits \cite{nielsen_chuang_2010}. One
 important class of quantum error correction strategies are stabilizer codes
 \cite{gottesman2009}\cite{gottesman1997} that can be simulated exponentially
 faster than general quantum circuits
@@ -20,7 +20,7 @@ This paper describes the development of a quantum computing simulator
 using both the usual dense state vector representation for a general state
 and a graphical representation for stabilizer states. After giving some introduction
 to quantum computing some basic properties of stabilizer states and their
-dynamics are given. Using this the graphical representation is introduced
+dynamics are eludicated. Using this the graphical representation is introduced
 and some operations on the graphical states are explained. Following is
 a chapter describing the implementation of these techniques and some performance
 analysis.

thesis/chapters/quantum_computing.tex

@@ -5,7 +5,7 @@
 \subsubsection{Single Qbits}
 
 \begin{definition}
-    A qbit is a two level quantum mechanical system, i.e. it has the eigenbasis
+    A qbit is a two level quantum mechanical system \cite{nielsen_chuang_2010}, i.e. it has the eigenbasis
     $ \{\ket{\uparrow} \equiv \ket{1}, \ket{\downarrow} \equiv \ket{0}\} $
     with $\braket{\uparrow}{\downarrow} = 0$. In the following this basis will be called
     the $Z$ basis in analogy to the conventions used in spin systems ($\sigma_Z$). For some computations
@@ -94,8 +94,9 @@ with the normation condition
 \end{equation}
 
 The states $\ket{i}$ for $i = 0, ..., 2^{n}-1$ are called integer states. Note
-that integer states are eigenstates of the $Z$ operators. The computational basis
+that integer states are eigenstates of the $Z$ operators.\\
-is $\{\ket{i_0} \otimes ... \otimes \ket{i_{n-1}} | i_0, ..., i_{n-1} = 0, 1\}$.
+The computational basis is 
+$\left\{\ket{i_0} \otimes ... \otimes \ket{i_{n-1}} \middle| i_0, ..., i_{n-1} = 0, 1\right\}$.
 
 \begin{definition}
 For a single-qbit gate $U$ and a qbit $j = 0, 1, ..., n - 1$ 
@@ -116,14 +117,14 @@ is acting on qbit $j$.
     the controlled version of $U$ is defined by
     \begin{equation}
         \begin{aligned}\label{eq:CU}
-            CU_{i, j} &:= &\ket{0}\bra{0}_j\otimes I_i 
+            CU_{i, j} &:= \ket{0}\bra{0}_j\otimes I_i 
                         + \ket{1}\bra{1}_j \otimes U_i \\
-                    &:= &\left(\bigotimes\limits_{l < j} I\right) 
+                    &:= \left(\bigotimes\limits_{l < j} I\right) 
                             \otimes \ket{0}\bra{0} 
                             \otimes \left(\bigotimes\limits_{j < l < i} I\right) 
                             \otimes I 
                             \otimes \left(\bigotimes\limits_{j > l} I \right)\\
-                        & &+ \left(\bigotimes\limits_{l < j} I\right) 
+                    &\mbox{  } + \left(\bigotimes\limits_{l < j} I\right) 
                             \otimes \ket{1}\bra{1} 
                             \otimes \left(\bigotimes\limits_{j < l < i} I\right) 
                             \otimes U 
@@ -147,7 +148,7 @@ In Definition \ref{def:CU} $i$ is called the act-qbit and $j$ the control-qbit.
 $CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1}$ state.
 
 One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough
-to generate an arbitrary $n$-qbit gate \cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}.
+to generate an arbitrary $n$-qbit gate \cite{nielsen_chuang_2010}\cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}.
 The matrix representation of $CX$ and $CZ$ for two qbits is given by
 
 \begin{equation}
@@ -232,17 +233,18 @@ Several qbits can be abbreviated by writing a slash on the qbit line:
 
 \subsection{Quantum Algorithms}
 
-The great hope behind quantum computing is that it will speed up some computations
+The great hope behind quantum computing is that it will speed up some
-exponentially using algorithms that utilize the laws of quantum mechanics. Current 
+computations exponentially using algorithms that utilize the laws of quantum
-algorithms are based upon quantum search and quantum fourier transform \cite{nielsen_chuang_2010}.
+mechanics. Current algorithms are based upon quantum search and quantum fourier
-The latter is particular interesting for physical problems as it is used in the phase estimation
+transform \cite{nielsen_chuang_2010}.  The latter is particular interesting for
-algorithm that can be used to analyze the spectrum of the transfer matrix:
+physical problems as it is a key component in the phase estimation algorithm
+that can be used to analyze the spectrum of the transfer matrix:
 
 \begin{equation}
     T = \exp(itH)
 \end{equation}
 
-The eigenvalue of $T$ can now be estimated by using the phase estimation circuit
+The spectrum of $T$ can now be estimated by using the phase estimation circuit
 
 \[
 \Qcircuit @C=1em @R=.7em {
@@ -251,7 +253,12 @@ The eigenvalue of $T$ can now be estimated by using the phase estimation circuit
 }
 \]
 
-Where $T\ket{\varphi} = \exp(2\pi i\varphi) \ket{\varphi}$ and the measurement result $\tilde{\varphi} = \frac{x}{2^n}$ is an
+Where $T\ket{\varphi} = \exp(2\pi i\varphi) \ket{\varphi}$ and the measurement
-estimation for $\varphi$. If a success rate of $1-\epsilon$ and an accuracy of $| \varphi - \tilde{\varphi} | < 2^{-m}$
+result $\tilde{\varphi} = \frac{x}{2^n}$ is an estimation for $\varphi$. If
-is wanted, then $N = m + \log_2(2 + \frac{1}{2\epsilon})$ qbits are required \cite{nielsen_chuang_2010}\cite{lehner2019}.
+a success rate of $1-\epsilon$ and an accuracy of $| \varphi - \tilde{\varphi}
+| < 2^{-m}$ is wanted, then $N = m + \log_2(2 + \frac{1}{2\epsilon})$ qbits are
+required \cite{nielsen_chuang_2010}\cite{lehner2019}.
 
+%Another possible way to use quantum computers in physics is to simulate a system's
+%time evolution using the transfer matrix. By mapping an interesting system to
+%the qbit space and applying the 

thesis/chapters/stabilizer.tex

@@ -72,7 +72,7 @@ The discussion below follows the argumentation given in \cite{nielsen_chuang_201
 
     \begin{enumerate}
         \item{$(iI)^2 = (-iI)^2 = -I$. Which contradicts the definition of $S$.}
-        \item{From the definition of $S$ ($G_n$ respectively) follows that any
+        \item{From the definition of $S$ ($P_n$ respectively) follows that any
             $S^{(i)} \in S$ has the form $\pm i^l \left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$ where
             $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $\left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$
             is hermitian and unitary $(S^{(i)})^2$ is either $+I$ or $-I$. As $-I \notin S$ $(S^{(i)})^2 = I$ follows directly. 
@@ -153,7 +153,7 @@ describes the dynamics of the system, i.e.
 \begin{equation} \ket{\psi'} = U \ket{\psi} \end{equation}
 
 It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$
-anymore. There are however some statements that can still be made
+anymore. There are however some statements that can be made
 \cite{nielsen_chuang_2010}:
 
 \begin{equation}
@@ -212,18 +212,18 @@ a set of stabilizers.
 \end{proof}
 
 This is quite an important result: As under a transformation $U \in C_n$ $S'
-= U^\dagger S U$ is a set of $n$ independent stabilizers and $\ket{\psi'}$ is
+= U S U^\dagger$ is a set of $n$ independent stabilizers and $\ket{\psi'}$ is
 stabilized by $S'$ one can consider the dynamics of the stabilizers instead of
 the actual state. This is considerably more efficient as only $n$ stabilizers
 have to be modified, each being just the tensor product of $n$ Pauli matrices.
 This has led to the simulation using stabilizer tableaux
-\cite{gottesman_aaronson2008}.
+\cite{gottesman_aaronson2008}\cite{CHP}.
 
 \subsubsection{Measurements on Stabilizer States} \label{ref:meas_stab}
 
-Interestingly also measurements are dynamics covered by the stabilizers.  When
+Interestingly also measurements are dynamics covered by the stabilizers
-an observable $g_a \in \{\pm X_a, \pm Y_a, \pm Z_a\}$ acting on qbit $a$ is
+\cite{nielsen_chuang_2010}.  When an observable $g_a \in \{\pm X_a, \pm Y_a,
-measured one has to consider the projector
+\pm Z_a\}$ acting on qbit $a$ is measured one has to consider the projector
 
 \begin{equation} 
 P_{g_a,s} = \frac{I + (-1)^s g_a}{2}.  
@@ -396,7 +396,7 @@ graph.
                 &\mbox{   }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
                 &=           \left(\ket{1}\bra{1}_j \otimes Z_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{1}\bra{1}_j \otimes Z_i\right)\\
                 &\mbox{   }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
-                &= \left(\left(-\ket{1}\bra{1}_j + \ket{1}\bra{1}_j\right) \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \\
+            &= \left(\left(-\ket{1}\bra{1}_j + \ket{0}\bra{0}_j\right) \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \\
                 &= \left(Z_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \\
                 &= X_i \prod\limits_{l \in n_i} Z_l \\
                 &= K_G^{(i)}
@@ -554,7 +554,7 @@ Therefore the associated graph is changed as given in the third equation.
 
 The definition of a VOP-free graph state above raises an obvious question: Can
 any stabilizer state be described using just a graph?  The answer is straight
-forward: No. The most simple cases are the single qbit stated $\ket{0},\ket{1}$
+forward: No. The most simple cases are the single qbit states $\ket{0},\ket{1}$
 and $\ket{+_Y}, \ket{-_Y}$. But there is an extension to the VOP-free graph
 states that allows the representation of an arbitrary stabilizer state. The
 proof that indeed any state can be represented is purely constructive. As seen
@@ -588,7 +588,7 @@ immediately:
 \end{equation}
 
 The great advantage of this representation of a stabilizer state is its space
-requirement: Instead of storing $n^2$ $P$ matrices only some vertices (which
+requirement: Instead of storing $n^2$ Pauli matrices only some vertices (which
 often are implicit), the edges and some vertex operators ($n$ matrices) have to
 be stored. The following theorem will improve this even further: instead of $n$
 matrices it is sufficient to store $n$ integers representing the vertex
@@ -664,7 +664,7 @@ derive that any isolated clique of the graph can be treated independently.
 Therefore the two isolated qbits can be treated as an independent state and the
 set of two qbit stabilizer states is finite. An upper bound to the number of
 two qbit stabilizer states is given by $2\cdot24^2$: A factor of two for the
-options with and withoit an edge between the qbits and $24$ Clifford operators
+options with and without an edge between the qbits and $24$ Clifford operators
 on each vertex.
 
 All those states and the resulting state after a $CZ$ application can be
@@ -794,17 +794,17 @@ projector as $o_a$ is a Clifford operator
     \end{aligned}
 \end{equation}
 
-Where $\tilde{g}_a \in \{+1, -1\}\cdot\{X_a, Y_a, Z_a\}$. Therefore, it is
+Where $\tilde{g}_a \in \{\pm X_a, \pm Y_a, \pm Z_a\}$. Therefore, it is
 enough to study the measurements of any Pauli operator on the vertex operator
 free graph states. The commutators of the observable with the $K_G^{(i)}$ are
 quite easy to compute. Note that Pauli matrices either commute or anticommute
-and it is easier to list the operators that anticommute
+and it is easier to list the operators that anticommute:
 
 \begin{equation}
     \begin{aligned}
-        A_{\pm X_a} &= \left\{j \middle| \{j, a\} \in E\right\}\\
+        A_{\pm X_a} &= n_a \\
-        A_{\pm Y_a} &= \left\{j \middle| \{j, a\} \in E\right\} \cup \{a\} \\
+        A_{\pm Y_a} &= n_a \cup \{a\} \\
-        A_{\pm Z_a} &= \{a\}.\\
+        A_{\pm Z_a} &= \{a\}\\
     \end{aligned}
 \end{equation}
 

thesis/main.tex

@@ -16,6 +16,7 @@
 %\usepackage{courier}
 %\lstset{basicstyle=\fontfamily{qcr}\selectfont}
 \lstset{basicstyle=\ttfamily}
+\setlength{\parindent}{0cm}
 
 \geometry{left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm}