some work as suggested by Simon

thesis/chapters/appendix.tex

@@ -4,11 +4,11 @@
 \section{Source Code for the Benchmarks}
 \label{ref:code_benchmarks}
 
-The benchmarks used in \ref{ref:performance} are done using this code. Note
+The benchmarks used in \ref{ref:performance} are based on this code. Note
 that the execution time is measured which is inherently noisy. To account for
 the noise several strategies are used:
 
-\begin{enumerate}[1]
+\begin{enumerate}[1.]
     \item{The same circuit is applied to the starting state several times. The
         minimal result is used as the noise must be positive}
     \item{Several circuits are applied to the starting state. The remaining
@@ -30,15 +30,9 @@ The code used to benchmark the three regimes is analogous and not included here.
 
 Because the whole graphs are barely percetible windows have been used
 in Figure \ref{fig:graph_high_linear_regime} and Figure \ref{fig:graph_intermediate_regime}.
-For the sake of completeness the whole graphs are included here in Figure \ref{fig:graph_low_linear_regime_full},
+For the sake of completeness the whole graphs are included here in 
 Figure \ref{fig:graph_intermediate_regime_full} and Figure \ref{fig:graph_high_linear_regime_full}.
 
-\begin{figure}[H]
-    \centering
-    \includegraphics[width=\linewidth]{graphics/graph_low_linear_regime.png}
-    \caption[Typical Graphical State in the Low-Linear Regime]{Typical Graphical State in the Low-Linear Regime}
-    \label{fig:graph_low_linear_regime_full}
-\end{figure}
 
 \begin{figure}[H]
     \centering
@@ -57,9 +51,9 @@ Figure \ref{fig:graph_intermediate_regime_full} and Figure \ref{fig:graph_high_l
 \section{Code to Generate the Example Graphs}
 \label{ref:code_example_graphs}
 
-This code has been used to generate the example graphs used in
+This code has been used to generate the example graphs in \ref{ref:performance}
-\ref{ref:performance}. Note that generating the graph is done using a random
+and \ref{ref:complete_graphs}. Note that generating the graph is done with
-circuit as used in \ref{ref:code_benchmarks}. The generated \lstinline{dot}
+a random circuit as in \ref{ref:code_benchmarks}. The generated \lstinline{dot}
 code is converted to an image using 
 \lstinline{dot i_regime.dot -Tpng -o i_regime.png}.
 
@@ -69,17 +63,17 @@ code is converted to an image using
 \section{Code to Benchmark \lstinline{ufunc} Gates against Python}
 \label{ref:benchmark_ufunc_py}
 
-It has been mentioned several times that the implementation using
+It has been mentioned several times that the implementation with
-\lstinline{ufuncs} as gates is faster than using a \lstinline{python}
+\lstinline{ufuncs} as gates is faster than a pure \lstinline{python}
-implementation.  To support this statement a simple benchmark can be used. The
+implementation.  To support this statement a simple benchmark is written. The
-relatively simple Pauli $X$ is used, more complicated gates like $CX$ or $H$
+relatively simple Pauli $X$ gate us implemented, more complicated gates like $CX$ or $H$
-have worse performance when implemented in \lstinline{python}. The performance
+have worse performance when written in \lstinline{python}. The performance
-improvement when using the \lstinline{ufunc} is a factor around $6.4$ in this tested
+improvement when in this example is a factor around $6.4$.
-case. One must however note that the tested \lstinline{python} code is not
+One must note that the tested \lstinline{python} code is not
-realistic and in a possible applications there would be a significant overhead.
+realistic and in a possible application there would be a significant overhead.
 
 \lstinputlisting[title={Code to Benchmark \lstinline{ufunc} Gates against Python}, language=Python, breaklines=True]{extra_benchmark/benchmark.py}
 
 When using \lstinline{result_py[0::2] = qm_state[1::2]} the result is identical and
-the performance is only increased by a factor around $1.7$. This method is however not
+the performance is only increased by a factor around $1.7$. This method is not
 applicable to general act-qbits and the bit mask has to be used.

thesis/chapters/conclusion.tex

@@ -2,8 +2,8 @@
 \chapter{Conclusion and Outlook}
 
 As seen in \ref{ref:performance} simulation using stabilizers is exponentially
-faster than simulating using state vectors. Using a graphical representation
+faster than simulating with dense state vectors. The graphical representation
-for the stabilizers is on average more efficiently than using a stabilizer
+for stabilizer states is on average more efficiently than the stabilizer
 tableaux. In particular one can simulate more qbits while only applying
 Clifford gates.
 
@@ -13,7 +13,7 @@ stabilizer code requires $5$ qbits to encode one logical qbit
 \cite{nielsen_chuang_2010}.  Several layers of data encoding increase the
 number of required qbits exponentially.
 
-Simulating in the stabilizer formalism is rather uninteresting from a physical
+Simulating in the stabilizer formalism is uninteresting from a physical
 point of view as basically no physically interesting simulations can be
 performed: As shown in \ref{ref:meas_stab} probability amplitudes have to be
 $0, \frac{1}{2}, 1$; this leaves very few points in time that could be
@@ -21,7 +21,7 @@ simulated by applying a transfer matrix. Algorithms like the quantum fourier
 transform also require non-Clifford gates for qbit counts $n \neq 2, 4$.
 
 The basic idea of not simulating a state but (after imposing some conditions on
-the Hilbert space) other objects that describe the state is extremely
+the Hilbert space) other objects that describe the state is 
 interesting for physics as often the exponentially large or infinitely large
 Hilbert spaces cannot be mapped to a classical (super) computer.  One key idea
 to take from the stabilizer formalism is to simulate the Hamiltonian instead of
@@ -34,14 +34,14 @@ ground state of this Hamiltonian.
 
 While trying to extend the stabilizer formalism one inevitably hits the
 question:\\ \textit{Why is there a constraint on the $R_\phi$ angle? Why is it
-$\frac{\pi}{2}$?}\\ The answer to this question is hidden in the Clifford
+$\frac{\pi}{2}$?}\\ The answer to this question can be found by taking a look
-group. Recalling Definition \ref{def:clifford_group} the Clifford group is not
+at the Clifford group. Recalling Definition \ref{def:clifford_group} the
-defined to be generated by $H, S, CZ$ but by its property of normalizing the
+Clifford group is not defined to be generated by $H, S, CZ$, but by its property
-multilocal Pauli group. Storing and manipulating the multilocal Pauli group is
+of normalizing the multilocal Pauli group. Storing and manipulating the
-only so efficient (or possible) because it is the tensor product of Pauli
+multilocal Pauli group is only so efficient (or possible) because it is the
-matrices. A general unitary on $n$ qbits would be a $2^{n} \times 2^{n}$ matrix
+tensor product of Pauli matrices. A general unitary on $n$ qbits would be
-which requires more space than a dense state vector. The Clifford group is
+a $2^{n} \times 2^{n}$ matrix which requires more space than a dense state
-a group preserving this tensor product property.
+vector. The Clifford group is a group preserving this tensor product property.
 
 %{{{
 %When lifting the constraint that $\ket{\varphi}$ is stabilized by the multilocal Pauli
@@ -103,10 +103,10 @@ a group preserving this tensor product property.
 The stabilizer formalism as introduced in \ref{ref:stab_states} has since been
 generalized to normalizers of a finite Abelian group over the Hilbert space
 \cite{bermejovega_lin_vdnest2015}\cite{bermejovega_vdnest2018}\cite{vandennest2019}\cite{vandennest2018}.
-This allows to simulate more classes of circuits efficiently on classical computers 
+This allows to simulate more classes of circuits efficiently on classical
-including the Quantum Fourier Transforms which is often believed to be
+computers including the Quantum Fourier Transforms which is often believed to
-responsible for exponential speedups. One must note that in the definition of
+be responsible for exponential speedups. One must note that in the definition
-the QFTs as in \cite{vandennest2018} the QFT depends on the Abelian group it
+of the QFTs as in \cite{vandennest2018} the QFT depends on the Abelian group it
 acts on. In particular the QFT on the group that decomposes the Hilbert space
 as seen in this paper ($Z_2^n$) is just the tensor product of the $H$ gates.
 The QFT as used in \ref{ref:quantum_algorithms} for the phase estimation
@@ -117,6 +117,6 @@ the Clifford group \cite{vandennest2018}.
 
 The exponential speedup of quantum computing is often attributed to
 entanglement, superposition and interference effects
-\cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states
+\cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states show
-however show both entanglement, superposition and interference effects;
+entanglement, superposition and interference effects; as do computations using
-as do computations done using general normalizers \cite{vandennest2018}.
+general normalizers \cite{vandennest2018}.

thesis/chapters/implementation.tex

@@ -2,13 +2,13 @@
 \chapter{Implementation}
 
 This chapter discusses how the concepts introduced before are implemented into
-a simulator. Futher the infrastructure around the simulation and some tools are
+a simulator. Further the infrastructure around the simulation and some tools are
 explained.
 
 The implementation is written as the \lstinline{python3} package \lstinline{pyqcs} \cite{pyqcs}. This allows
 users to quickly construct circuits, apply them to states and measure
-amplitudes.  Full access to the state (including intermediate states) has been
+amplitudes.  Full access to the states (including intermediate states) has been
-priorized over execution speed. To keep the simulation speed as high as
+prioritized over execution speed. To keep the simulation speed as high as
 possible under these constraints some parts are implemented in \lstinline{C}.
 
 \section{Dense State Vector Simulation}
@@ -23,30 +23,29 @@ useful features when it comes to computations:
     \item{The projection on the integer states is trivial.}
     \item{For any qbit $j$ and $0 \le i \le 2^n-1$ the coefficient $c_i$ is part of the $\ket{1}_j$ amplitude iff
         $i \& (1 << j)$ and part of the $\ket{0}_j$ amplitude otherwise.}
-    \item{For a qbit $j$ the coefficients $c_i$ and $c_{i \hat{ } (1 << j)}$ are the conjugated coefficients.}
+    \item{For a qbit $j$ the coefficients $c_i$ and $c_{i \wedge (1 << j)}$ are the conjugated coefficients.}
 \end{itemize}
 
-Where $\hat{}$ is the binary XOR, $\&$ the binary AND and $<<$ the binary
+Where $\wedge$ is the binary XOR, $\&$ the binary AND and $<<$ the binary
 leftshift operator.
 
-While implementing the dense state vectors two key points were allowing
+While implementing the dense state vectors two key points were a simple way to
-a simple and readable way to use them and simple access to the states by users
+use them and easy access to the underlaying data.  To meet both requirements
-that want more information than an abstracted view could allow. To meet both
+the states are implemented as Python objects that provide abstract features such
-requirements the states are implemented as Python objects providing abstract
+as normalization checking, checking for sufficient qbit number when applying
-features such as normalization checking, checking for sufficient qbit number
+a circuit, computing overlaps with other states, a stringify method and stored
-when applying a circuit, computing overlaps with other states, a stringify
+measurement results.  To store the measurement results a NumPy \lstinline{int8}
-method and stored measurement results.  To store the measurement results
+array \cite{numpy_array} is used; this is called the classical state.  The
-a NumPy \lstinline{int8} array \cite{numpy_array} is used; this is called the
+Python states also have a NumPy \lstinline{cdouble} array that stores the
-classical state.  The Python states also have a NumPy \lstinline{cdouble} array
+quantum mechanical state.  Using NumPy arrays has the advantage that access to
-that stores the quantum mechanical state.  Using NumPy arrays has the advantage
+the data is simple and safe while operations on the states can be implemented
-that access to the data is simple and safe while operations on the states can
+in \lstinline{C} \cite{numpy_ufunc} providing a considerable speedup
-be implemented in \lstinline{C} \cite{numpy_ufunc} providing a considerable
+\ref{ref:benchmark_ufunc_py}.
-speedup \ref{ref:benchmark_ufunc_py}.
 
 This quantum mechanical state is the component vector in integer basis
 therefore it has $2^n$ components.  Storing those components is acceptable in
 a range from $1$ to $30$ qbits; above this range the state requires space in
-the order of $1 \mbox{ GiB}$  which is in the range of usual RAM sizes for
+the order of magnitude $1 \mbox{ GiB}$  which is in the range of usual RAM sizes for
 personal computers. For higher qbit numbers moving to high performance
 computers and other simulators is necessary.
 
@@ -72,7 +71,7 @@ A basic set of gates is implemented in PyQCS:
     \item{Hadamard $H$ gate.}
     \item{Pauli $X$ or \textit{NOT} gate.}
     \item{Pauli $Z$ gate.}
-    \item{The $S$ phase gate.}
+    \item{Phase gate $S$.}
     \item{$Z$ rotation $R_\phi$ gate.}
     \item{Controlled $X$ gate: $CX$.}
     \item{Controlled $Z$ gate: $CZ$.}
@@ -98,8 +97,8 @@ new_state = circuit * state
 \end{lstlisting}
 
 
-The elementary gates such as $H, R_\phi, CX$ are implemented as single gate
+The elementary gates are implemented as single gate
-circuits and can be constructing using the built-in generators. The generators
+circuits and can be constructed using the built-in generators. These generators
 take the act-qbit as first argument, parameters such as the control qbit or an
 angle as second argument:
 
@@ -117,7 +116,7 @@ Out[2]: (0.7071067811865476+0j)*|0b0>
         + (0.7071067811865476+0j)*|0b11>
 \end{lstlisting}
 
-Large circuits can be constructed using the binary OR operator \lstinline{|} in
+Large circuits can be built using the binary OR operator \lstinline{|} in
 an analogy to the pipeline operator on many *NIX shells. As usual circuits are
 read from left to right similar to pipelines on *NIX shells:
 
@@ -141,7 +140,7 @@ Out[2]: (0.7071067811865477+0j)*|0b0>
 \end{lstlisting}
 %}
 
-A quick way to generate circuits programatically is to use the \lstinline{list_to_circuit}
+A quick way to generate circuits programmatically is to use the \lstinline{list_to_circuit}
 function:
 
 %\adjustbox{max width=\textwidth}{
@@ -168,7 +167,7 @@ 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 $24$, their order is
+associated with an integer ranging from $0$ to $23$. Their order is
 
 \begin{equation}
 \begin{aligned}
@@ -216,20 +215,20 @@ The edges are stored in an adjacency matrix
 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
+a vertex.  Also one must take care to keep the memory required to store a state
-keeping the required  memory low a linked list-array hybrid is used to store
+low.  To meet both requirements a linked list-array hybrid is used.  For every
-the adjacency matrix. For every vertex the neighbourhood is stored in a sorted
+vertex the neighbourhood is stored in a sorted linked list (which is a sparse
-linked list (which is a sparse representation of a column vector) and these
+representation of a column vector) and these adjacency lists are stored in
-adjacency lists are stored in a length $n$ array.
+a length $n$ array.
 
-Using this storage method all operations including searching and toggling edges
+Using this storage method all operations - including searching and toggling edges -
-inherite their time complexity from the sorted linked list.
+inherit their time complexity from the sorted linked list.
 
 \subsection{Operations on Graphical States}
 
 Operations on Graphical States are divided into three classes: Local Clifford
 operations, the CZ operation and measurements. The graphical states are
-implemented in \lstinline{C} and are exported to python3 in the class
+implemented in \lstinline{C} and are exported to \lstinline{python3} in the class
 \lstinline{RawGraphState}. This class has three main methods to implement the
 three classes of operations. 
 
@@ -249,7 +248,7 @@ three classes of operations.
             returns either $1$ or $0$ as a measurement result.}
 \end{description}
 
-Because this way of modifying the state is rather unconvenient and might lead to many
+Because this way of modifying the state is inconvenient and might lead to many
 errors the \lstinline{RawGraphState} is wrapped by the pure python class\\
 \lstinline{pyqcs.graph.state.GraphState}. It allows the use of circuits as
 described in \ref{ref:pyqcs_circuits} and provides the method
@@ -258,7 +257,7 @@ vector state.
 
 \subsection{Pure C Implementation}
 
-Because python tends to be rather slow \cite{benchmarkgame} and might not run on any architecture
+Because python tends to be slow \cite{benchmarkgame} and might not run on any architecture
 a pure \lstinline{C} implementation of the graphical simulator is also provided.
 It should be seen as a reference implementation that can be extended to the needs
 of the user.
@@ -267,37 +266,43 @@ This implementation reads byte code from a file and executes it. The execution i
 always done in three steps:
 
 \begin{enumerate}[1]
-    \item{Initializing the state according the the header of the bytecode file.}
+    \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).}
-    \item{Sampling the state according the the description given in the header of the byte code 
+    \item{Sampling the state according the description in the header of the byte code 
         file and writing the sampling results to either a file or \lstinline{stdout}. }
 \end{enumerate}
 
 \section{Utilities}
 
-To make both using the simulators more convenient and to help with using them
+The package \lstinline{pyqcs} ships with several utilities that are supposed to
-in as scientific or educational context several utilities have been written.
+make using the simulators more convenient. Some utilities are designed to help
-This chapter explains some of them.
+in a scientific and educational context.  This chapter explains some of them.
 
 \subsection{Sampling and Circuit Generation}
 
 The function \lstinline{pyqcs.sample} provides a simple way to sample from
-a state.  Copies of the state are made when necessary and the results are
+a state.  Copies of the state are made when necessary, and the results are
 returned in a \lstinline{collections.Counter} object. Several qbits can be
 sampled at once; they can be passed to the function either as an integer which
-will be interpreted as a bit mask and the least significant bit will be sampled
+will be interpreted as a bit mask, and the least significant bit will be sampled
 first. When passing the qbits to sample as a list of integers the integers are
 interpreted as qbit indices and are measured in the order they appear.
 
 If the keyword argument \lstinline{keep_states} is \lstinline{True} the
-sampling function will include the resulting states in the result. At the
+sampling function will include the collapsed states in the result. At the
 moment this works for dense vectors only. Checking for equality on graphical
-states has yet to be implemented but has $NP$ computational hardness
+states is not implemented due to the computational hardness.
-\cite{dahlberg_ea2019}.
+\cite{bouchet1991} introduced an algorithm to test whether two VOP-free graph states
+are equivalent with complexity $\mathcal{O}\left(n^4\right)$.
+So checking for equivalency is $c-NP$ complete because the vertex operators have
+to be checked as well. It might be necessary to compute all equivalent graphical
+states. \cite{dahlberg_ea2019} showed that counting equivalent VOP-free 
+graphical states is $\#P$ complete. Showing that two graphical states are 
+equivalent might considerably hard.
 
-Writing circuits out by hand can be rather painful. The function\\
+Writing circuits out by hand can be inconvenient. The function\\
-\lstinline{pyqcs.list_to_circuit} Converts a list of circuits to a circuit.
+\lstinline{pyqcs.list_to_circuit} converts a list of circuits to a circuit.
 This is particularly helpful in combination with python's
 \lstinline{listcomp}:
 
@@ -308,41 +313,40 @@ circuit_H = list_to_circuit([H(i) for i in range(nqbits)])
 The module \lstinline{pyqcs.util.random_circuits} provides the method described
 in \ref{ref:performance} to generate random circuits for both graphical and
 dense vector simulation. Using the module \lstinline{pyqcs.util.random_graphs}
-one can generate random graphical states which is more performant than using
+one can generate random graphical states which is faster than using random
-random circuits.
+circuits.
 
 The function \lstinline{pyqcs.util.to_circuit.graph_state_to_circuit} converts
 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
+By utilization of these circuits the graphical state can be copied or converted to a
 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.
+(see below) which is a way to represent graphical states.
 
 \subsection{Exporting and Flattening Circuits}
 
-Circuits can be drawn using the \LaTeX  package \lstinline{qcircuit}; all
+Circuits can be drawn with the \LaTeX  package \lstinline{qcircuit}; all
-circuits in this documents use \lstinline{qcircuit}. To visualize the circuits
+circuits in this documents use \lstinline{qcircuit}. To visualize 
-built using \lstinline{pyqcs} the function\\
+\lstinline{pyqcs} circuits 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
+\lstinline{qcircuit} code. The diagrams produced by this function are not
-PDFs directly. The diagrams produced by this function are not optimized and the
+optimized, and the diagrams can be unnecessary long.  Usually this can be fixed
-diagrams can be unnecessary long.  Usually this can be fixed easily by editing
+easily by editing the resulting code manually.
-the produced code manually.
 
 The circuits constructed using the \lstinline{|} operator have a tree structure
-which is rather unconvenient when optimizing circuits or exporting them.
+which is rather inconvenient when optimizing circuits or exporting them.
 The function \\
 \lstinline{pyqcs.util.flatten.flatten} converts a circuit
-to a list of single gate circuits that can be analyzed or exported easily.
+to a list of single gate circuits that can be simply analyzed or exported.
 
 
 \section{Performance}
 \label{ref:performance}
 
-To test the performance and compare it to the dense vector simulator the python
+Testing the performance and comparing the graphical with the dense vector
-module is used. Although the pure \lstinline{C} implementation has potential
+simulator is done with the python module. Although the pure \lstinline{C}
-for better performance the python module is better comparable to the dense
+implementation has potential for better performance the python module is better
-vector simulator which is a python module as well.
+comparable to the dense vector simulator which is a python module as well.
 
 For performance tests (and for tests against the dense vector simulator) random
 circuits are used. Length $m$ circuits are generated from the probability space
@@ -351,13 +355,10 @@ circuits are used. Length $m$ circuits are generated from the probability space
 \begin{equation}
     \Omega = \left(\{1, ..., 4n\} \otimes \{1, ..., n-1\} \otimes [0, 2\pi)\right)^{\otimes m}
 \end{equation} 
-
+with the uniform distribution. The continuous part $[0, 2\pi)$ is ignored when
-with the uniform distribution. The continous part $[0, 2\pi)$ is unused when
+generating random circuits for the graphical simulator; in order to generate
-generating random circuits for the graphical simulator; when generating random
+random circuits for dense vector simulations this is used as the argument $\phi$ of the
-circuits for dense vector simulations this is 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 using
 
 \begin{equation}
 \begin{aligned}
@@ -378,7 +379,7 @@ 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 that in both cases the length of the circuits have been scaled linearely
+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
 qbits:
 
@@ -403,10 +404,10 @@ qbits:
     \label{fig:scaling_qbits_log}
 \end{figure}
 
-The reason for this scaling will be clear later; one can observe that the
+The reason for this scaling will be clear later. From simulation data one can
-performance of the graphical simulator increases in some cases with growing
+observe that the performance of the graphical simulator increases in some cases
-number of qbits when the circuit length is constant. The code used to generate the
+with growing number of qbits when the circuit length is constant. The code used
-data for these plots can be found in \ref{ref:code_benchmarks}.
+to generate the data for these plots can be found in \ref{ref:code_benchmarks}.
 
 As described by \cite{andersbriegel2005} the graphical simulator is exponentially
 faster than the dense vector simulator. According to \cite{andersbriegel2005} it
@@ -414,11 +415,12 @@ is considerably faster than a simulator using the straight forward approach simu
 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
+One should be aware that the gate execution time (the time required to apply
-to the state) highly depends on the  state it is applied to. For the dense vector 
+a gate to the state) highly depends on the  state it is applied to. For the
-simulator and CHP this is not true: Gate execution time is constant for all gates
+dense vector simulator and CHP this is not true: Gate execution time is
-and states. Because the graphical simulator has to toggle neighbourhoods the
+constant for all gates and states. Because the graphical simulator has to
-gate execution time of the $CZ$ gate varies greatly. The plot Figure \ref{fig:scaling_circuits_linear}
+toggle neighbourhoods the gate execution time of the $CZ$ gate varies
+significantly. The plot Figure \ref{fig:scaling_circuits_linear}
 shows the circuit execution time for two different numbers of qbits. One can observe three
 regimes:
 
@@ -430,11 +432,11 @@ regimes:
 \end{figure}
 
 \begin{description}
-    \item[Low-Linear Regime] {Here the circuit execution time scales approximately linearely
+    \item[Low-Linear Regime] {\textit{(Ca. $0-10N_q$ gates)} Here the circuit execution time scales approximately linear
         with the number of gates in the circuit (i.e. the $CZ$ gate execution time is approximately constant).
         }
-    \item[Intermediate Regime]{The circuit execution time has a nonlinear dependence on the circuit length.}
+    \item[Intermediate Regime]{\textit{(Ca. $10N_q-20N_q$ gates)} The circuit execution time has a nonlinear dependence on the circuit length.}
-    \item[High-Linear Regime]{This regime shows a linear dependence on the circuit length; the slope is
+    \item[High-Linear Regime]{\textit{(Above ca. $20N_q$ gates)} This regime shows a linear dependence on the circuit length; the slope is
         higher than in the low-linear regime.}
 \end{description}
 
@@ -445,6 +447,19 @@ regimes:
     \label{fig:scaling_circuits_measurements_linear}
 \end{figure}
 
+
+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
+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
+in the intermediate and high-linear regime the pictures show a window of the
+actual graph.  The full images are in \ref{ref:complete_graphs}.  Further the
+regimes are not clearly visible for $n>30$ qbits therefore choosing smaller graphs is
+not possible. The code that was used to generate these images can be found
+in \ref{ref:code_example_graphs}.
+
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{graphics/graph_low_linear_regime.png}
@@ -466,35 +481,18 @@ regimes:
     \label{fig:graph_high_linear_regime}
 \end{figure}
 
-These two 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
-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
-in the intermediate and high-linear regime the pictures show a window of the
-actual graph.  The full images are in \ref{ref:complete_graphs}.  Further the
-regimes are not clearly visible for $n>30$ qbits so choosing smaller graphs is
-not possible. The code that was used to generate these images can be found
-in \ref{ref:code_example_graphs}.
-
 The Figure \ref{fig:scaling_circuits_measurements_linear} brings more substance
 to this interpretation. In this simulation the Pauli $X$ gate has been replaced
 by the measurement gate $M$, .i.e. in every gate drawn from the probability
-space a qbit is measured with probability $\frac{1}{4}$. As described in
+space a qbit is measured with probability $\frac{1}{4}$. Pauli measurements
-\cite{hein_eisert_briegel2008} the Schmidt measure for entropy is bounded from
+decrease the entanglement (and the amount of edges) in a state
-above by Pauli persistency, i.e. the minimal amount of Pauli measurements
-required to disentangle a state. This Pauli persistency is closely related to
-the amount (and structure of) vertices in the graph
-\cite{hein_eisert_briegel2008}. In particular Pauli measurements decrease the
-entanglement (and the amount of edges) in a state
 \cite{hein_eisert_briegel2008}\cite{li_chen_fisher2019}.  The frequent
 measurements in the simulation therefore keeps the amount of edges low thus
 preventing a transition from the low linear regime to the intermediate regime.
 
 
 Because states with more qbits reach the intermediate regime at higher circuit
-lengths it is important to account for this virtual performance boost when
+lengths it is important to compensate this virtual performance boost when
 comparing with other simulation methods. This explains why the circuit length
 in Figure \ref{fig:scaling_qbits_linear} had to be scaled with the qbit number.
 
@@ -502,15 +500,15 @@ 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.  To allow a user
+noisy execution would be an interesting improvement to implement. In order to
-to execute circuits on other machines, including both real hardware and
+allow a user to execute circuits on other machines, including both real
-simulators, a module that exports circuits to OpenQASM \cite{openqasm} seems
+hardware and simulators, a module that exports circuits to OpenQASM
-useful. 
+\cite{openqasm} seems useful. 
 
 The current implementation of some graphical operations can be optimized. While
 clearing VOPs as described in \ref{ref:dynamics_graph} the neighbourhood of
 a vertex is toggled for every $L_a$ transformation. This is the most straight
-forward implementation but often the $L_a$ transformation is performed several
+forward implementation, but often the $L_a$ transformation is performed several
 times on the same vertex. The neighbourhood would have to be toggled either
 once or not at all depending on whether the number of $L_a$ transformations is
 odd or even.

thesis/chapters/introduction.tex

@@ -3,7 +3,7 @@
 
 Quantum computing has been a rapidly growing field over the last years with
 many companies and institutions working on building and using quantum computers
-\cite{ibmq}\cite{intelqc}\cite{microsoftqc}\cite{dwavesys}\cite{lrzqc}\cite{heise25_18}.
+\cite{ibmq}\cite{intelqc}\cite{microsoftqc}\cite{dwavesys}.
 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 \cite{nielsen_chuang_2010}. 

thesis/chapters/quantum_computing.tex

@@ -43,7 +43,7 @@ with
 
 Note that $X = HZH$ and $Z = R_{\pi}$, so the set of $H, R_\phi$ is sufficient. 
 Further note that the basis vectors are chosen s.t. $Z\ket{0} = +\ket{0}$ and $Z\ket{1} = -\ket{1}$;
-transforming to the other Pauli eigenstates is done using $H$ and $SH$
+transforming to the other Pauli eigenstates is done using $H$ and $S$
 
 \begin{equation}
     S := R_{\frac{\pi}{2}} = \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right)
@@ -148,10 +148,10 @@ For a single-qbit gate $U$ and a qbit $j = 0, 1, ..., n - 1$
 
     In particular for $X, Z$
     \begin{equation}\label{eq:CX_pr}
-        CX(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes X_i 
+        CX_{i, j} = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes X_i 
     \end{equation}
     \begin{equation}\label{eq:CZ_pr}
-        CZ(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes Z_i .
+        CZ_{i, j} = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes Z_i .
     \end{equation}
 
     This follows the definition given in \cite{barenco_ea_1995}.
@@ -283,7 +283,8 @@ Where $T\ket{\varphi} = \exp(2\pi i\varphi) \ket{\varphi}$ and the measurement
 result $\tilde{\varphi} = \frac{x}{2^n}$ is an estimation for $\varphi$. If
 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}.
+required \cite{nielsen_chuang_2010}.
+The gate $FT^\dagger$ is the inverse quantum fourier transform as described in \cite{nielsen_chuang_2010}.
 
 %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

thesis/chapters/stabilizer.tex

@@ -98,7 +98,7 @@ group.
     $g_i$ and $m$ is the smallest integer for which these statements hold.
 \end{definition}
 
-From now on the generators $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be
+From now on the generators $\langle S^{(i)} \rangle_{i}$ will be
 used as the required properties of a set of stabilizers that can be studied on
 its generators.
 
@@ -552,14 +552,14 @@ Therefore the associated graph is changed as given in the third equation.
 \label{ref:sec_g_states}
 
 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
+any stabilizer state be described using just a graph?  The answer is no. The
-forward: No. The most simple cases are the single qbit states $\ket{0},\ket{1}$
+most simple cases are the single qbit states $\ket{0},\ket{1}$ and $\ket{+_Y},
-and $\ket{+_Y}, \ket{-_Y}$. But there is an extension to the VOP-free graph
+\ket{-_Y}$. But there is an extension to the VOP-free graph states that allows
-states that allows the representation of an arbitrary stabilizer state. The
+the representation of an arbitrary stabilizer state. The proof that any
-proof that indeed any state can be represented is purely constructive. As seen
+state can be represented is purely constructive. As seen in Theorem
-in Theorem \ref{thm:clifford_group_approx} any $c \in C_n$ can be constructed
+\ref{thm:clifford_group_approx} any $c \in C_n$ can be constructed from $CZ$
-from $CZ$ and $C_L$. In the following discussion it will become clear that both
+and $C_L$. In the following discussion it will become clear that both $C_L$ and
-$C_L$ and $CZ$ can be applied to a general graph state
+$CZ$ can be applied to a general graph state
 \footnote{
     One can show that any stabilizer state is local Clifford equivalent to
     a VOP-free graph state, i.e. only tensor products of $C_L$ matrices are
@@ -586,9 +586,9 @@ $C_L$ and $CZ$ can be applied to a general graph state
     The notation in \cite{andersbriegel2005} is different. Instead of using $(V, E, O)$
     to represent any stabilizer $(V, E)$ is used to represent what is called a VOP-free
     graph state in this paper. Then the state $\ket{\bar{G}}$ is extended with local Clifford
-    gates $c_1, ..., c_n$ to an arbitrary stabilizer state. The state is then denoted as
+    gates $c_1, ..., c_n$ to an arbitrary stabilizer state. The full state is denoted as
     $\ket{\bar{G}; (c_1, ..., c_n)} \equiv \ket{\bar{G}; \vec{c}}$. This paper prefers 
-    the notation using $(V, E, O)$ as it both emphasizes the use of stabilizers and is closer
+    the notation $(V, E, O)$ as it both emphasizes the use of stabilizers and is closer
     to the representation used in the simulator.
     }. 
 \end{definition}
@@ -603,11 +603,12 @@ immediately:
 The great advantage of this representation of a stabilizer state is its space
 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 $n$
+be stored. Theorem \ref{thm:cl24} will improve this even further: Instead of $n$
 matrices it is sufficient to store $n$ integers representing the vertex
-operators:
+operators.
 
 \begin{theorem}
+    \label{thm:cl24}
     $C_L$ has $24$ degrees of freedom disregarding a global phase \cite{andersbriegel2005}.
 \end{theorem}
 \begin{proof} It is clear that $\forall a \in C_L$ a is a group isomorphism $P
@@ -643,9 +644,9 @@ operation on graph states \cite{andersbriegel2005}.
 
 So far the graphical representation of stabilizer states is just another way to
 store basically a stabilizer tableaux that might require less memory than the
-tableaux used in CHP \cite{CHP}. The true power of this formalism is seen when
+tableaux used in CHP \cite{CHP}. The power of this formalism can be seen by
 studying its dynamics. The simplest case is a local Clifford operator $c_j$
-acting on a qbit $j$: The stabilizers are changed to $\langle c_j S^{(i)}
+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
@@ -658,9 +659,9 @@ are given by
 \end{equation}
 
 The action of a $CZ$ gate on the state $(V, E, O)$ is in most cases less
-trivial.  Let $a \neq b$ be two qbits, now consider the action of $CZ_{a,b}$ on
+trivial.  Let $a \neq b$ be two qbits. Consider the action of $CZ_{a,b}$ on
 $(V, E, O)$.  The cases given here follow the implementation of a $CZ$
-application in \cite{pyqcs}, the respective paragraphs from
+application in \cite{pyqcs}. The respective paragraphs from
 \cite{andersbriegel2005} are given in italic.  Most of the discussion follows
 the one given in \cite{andersbriegel2005} closely.
 
@@ -681,7 +682,7 @@ 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
-computed which leads to another interesting result that will be useful later:
+computed which leads to another interesting insight that will be useful later:
 If one vertex has the vertex operator $I$ the resulting state can be chosen
 such that at least one of the vertex operators is $I$ again and in particular
 the identity on the vertex can be preserved under the application of a $CZ$.
@@ -692,7 +693,7 @@ has non-operand (i.e. neighbours that are neither $a$ nor $b$) neighbours. In
 this case one can try to clear the vertex operators which will succeed for at
 least one qbit.
 
-The transformation given in Lemma \ref{lemma:M_a} is used to "clear" the vertex
+The transformation given in Lemma \ref{lemma:M_a} is used to clear the vertex
 operators. Recalling that the transformation $M_j$ toggles the neighbourhood of
 vertex $j$ gives substance to the following theorem:
 
@@ -729,7 +730,7 @@ steps:
         }
 \end{enumerate}
 
-This algorithm has the important properties that if the algorithm succeeds $o_a
+This algorithm has the important properties that if it succeeds $o_a
 = I$ and $o_b$ has picked up powers of $\sqrt{iZ}^\dagger$.  If $b$ has
 non-operand neighbours after clearing the vertex operators on $a$, then the
 vertex operators on $b$ can be cleared using the same algorithm which gives
@@ -755,23 +756,24 @@ non-operand neighbours and $b$ does not. Now the state $\ket{G}$ has the form
 \end{equation}
 
 As $o_b$ commutes with all operators but $CZ_{a,b}$ and $s \in \{0, 1\}$
-indicates whether there is an edge between $a$ and $b$.
+indicates whether there is an edge between $a$ and $b$. Applying the $CZ$
+gives
 
 \begin{equation}
     \begin{aligned}
         CZ_{a,b}\ket{G} &= CZ_{a,b}\left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) o_b (CZ_{a,b})^s \ket{+} \\
-                        &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) \ket{+}_{n-2} \otimes \left(CZ_{a,b} o_b (CZ_{a,b})^s \ket{+}_2\right) \\
+                        &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) \ket{+}_{n-2} \otimes \left(CZ_{a,b} o_b (CZ_{a,b})^s \ket{+}_2\right). \\
     \end{aligned}
 \end{equation}
 
-This allows to re-use the method in Case 2 to apply the $CZ$ while keeping the
+This allows to reuse the method in Case 2 to apply the $CZ$ while keeping the
 $o_a = I$.
 
 As it has been shown how both $CZ$ and $C_L$  act on a graph state $\ket{G}$
 and the resulting state is a graph state as well this proofs constructively
-that the graphical representation of a stabilizer state is indeed able to
+that the graphical representation of a stabilizer state is able to represent
-represent any stabilizer state.  If one wants to do computations using this
+any stabilizer state.  If one wants to do computations using this formalism it
-formalism it is however also necessary to perform measurements.
+is also necessary to perform measurements.
 
 \subsection{Measurements on Graph States}
 \label{ref:meas_graph}
@@ -781,7 +783,7 @@ the graph after a measurement is described in \cite{hein_eisert_briegel2008}.
 
 Recalling \ref{ref:meas_stab} it is clear that one has to compute the
 commutator of the observable $g_a = Z_a$ with the stabilizers to get the
-probability amplitudes which is a quite expensive computation in theory. It is
+probability amplitudes which is a expensive computation in theory. It is
 possible to simplify the problem by pulling the observable behind the vertex
 operators. For this consider the projector $P_{a,s} = \frac{I + (-1)^sZ_a}{2}$
 
@@ -807,11 +809,11 @@ projector as $o_a$ is a Clifford operator
     \end{aligned}
 \end{equation}
 
-Where $\tilde{g}_a \in \{\pm X_a, \pm Y_a, \pm Z_a\}$. Therefore, it is
+Where $\tilde{g}_a \in \{\pm X_a, \pm Y_a, \pm Z_a\}$. Therefore, it is enough
-enough to study the measurements of any Pauli operator on the vertex operator
+to study the measurements of any Pauli operator on the VOP-free graph states.
-free graph states. The commutators of the observable with the $K_G^{(i)}$ are
+The commutators of the observable with the $K_G^{(i)}$ are easy to compute.
-quite easy to compute. Note that Pauli matrices either commute or anticommute
+Note that Pauli matrices either commute or anticommute and it is simpler to
-and it is easier to list the operators that anticommute:
+list the operators that anticommute:
 
 \begin{equation}
     \begin{aligned}
@@ -826,25 +828,26 @@ $\tilde{g}_a = X_a (-X_a)$ is measured the result $s=0(1)$ is obtained with
 probability $1$ and $(V, E, O)$ is unchanged.  In any other case the results
 $s=1$ and $s=0$ have probability $\frac{1}{2}$ and both graph and vertex
 operators have to be updated. Further it is clear that measurements of
-$-\tilde{g}_a$ and $\tilde{g}_a$ are related by just inverting the result $s$.
+$-\tilde{g}_a$ and $\tilde{g}_a$ are related by inverting the result $s$.
 
 The calculations to obtain the transformation on graph and vertex operators are
 lengthy and follow the scheme of Lemma \ref{lemma:M_a}. \cite[Section
 IV]{hein_eisert_briegel2008} also contains the steps required to obtain the
-following results
+following results:
 
 \begin{equation}
     \begin{aligned}
         U_{Z,s} &= \left(\prod\limits_{b \in n_a} Z_b^s\right) X_a^s H_a \\
-        U_{Y,s} &= \prod\limits_{b \in n_a} \sqrt{(-1)^{1-s} iZ_b} \sqrt{(-1)^{1-s} iZ_a}.\\
+        U_{Y,s} &= \prod\limits_{b \in n_a} \sqrt{(-1)^{1-s} iZ_b} \sqrt{(-1)^{1-s} iZ_a}\\
     \end{aligned}
 \end{equation}
 
-These transformations split it two parts: the first is a result of Lemma
+These transformations split it in two parts: the first is a result of Lemma
 \ref{lemma:stab_measurement}.  The second part makes sure that the qbit $a$ is
-diagonal in measured observable and has the correct eigenvalue.  When comparing
+diagonal in the measured observable and has the correct eigenvalue.  When comparing
-with Lemma \ref{lemma:stab_measurement} in both cases $Y,Z$ the anticommuting
+with Lemma \ref{lemma:stab_measurement} in both cases $Y$ and $Z$ the anticommuting
 stabilizer $K_G^{(a)}$ is chosen. The graph is changed according to
+\footnote{$n_a \otimes n_a$ should be read as $\{\{i,j\} | i,j \in n_a, i \neq j\}$.}
 
 \begin{equation}
     \begin{aligned}
@@ -854,7 +857,7 @@ stabilizer $K_G^{(a)}$ is chosen. The graph is changed according to
 \end{equation}
 
 
-For $g_a = X_a$ one has to choose a $b \in n_a$ and the transformations are
+For $g_a = X_a$ one has to choose a neighbour $b \in n_a$ and the transformations are
 
 \begin{equation}
     \begin{aligned}

thesis/main.bib

@@ -190,11 +190,13 @@
     urldate={19.09.2019},
     title={IBM Q - What is quantum computing?},
     author={IBM},
+    note={https://www.ibm.com/quantum-computing/learn/what-is-quantum-computing/},
     year=2019
 }
 @online{
     intelqc,
     url={https://newsroom.intel.com/press-kits/quantum-computing/\#gs.2s0dux},
+    note={https://newsroom.intel.com/press-kits/quantum-computing/\#gs.2s0dux},
     urldate={19.09.2019},
     title={Intel Press Kit: Quantum Computing},
     author={Intel},
@@ -203,6 +205,7 @@
 @online{
     microsoftqc,
     url={https://www.microsoft.com/en-us/quantum/default.aspx},
+    note={https://www.microsoft.com/en-us/quantum/default.aspx},
     urldate={19.09.2019},
     title={Quantum Computing | Microsoft},
     author={Microsoft},
@@ -211,6 +214,7 @@
 @online{
 	dwavesys,
 	url={https://www.dwavesys.com/quantum-computing},
+	note={https://www.dwavesys.com/quantum-computing},
     urldate={19.09.2019},
     title={Quantum Computing | D-Wave Systems},
     author={D-Wave Systems Inc}
@@ -218,6 +222,7 @@
 @online{
 	lrzqc,
 	url={https://www.lrz.de/wir/newsletter/2019-08/\#LRZ_bereit_fuer_bayerische_Quantechnologie},
+	note={https://www.lrz.de/wir/newsletter/2019-08/\#LRZ_bereit_fuer_bayerische_Quantechnologie},
     urldate={19.09.2019},
     title={LRZ-Newsletter Nr. 08/2019 vom 01.08.2019},
     author={S. Vieser},
@@ -226,6 +231,7 @@
 @online{
 	heise25_18,
 	url={https://www.heise.de/select/ct/2018/25/1544250249368810},
+	note={https://www.heise.de/select/ct/2018/25/1544250249368810},
     urldate={19.09.2019},
     title={Europa entfesselt Quanten-Power},
     author={Arne Grävemeyer},
@@ -287,3 +293,10 @@
     title={The Computer Language Benchmarks Game},
     note={https://benchmarksgame-team.pages.debian.net/benchmarksgame/}
 }
+@article{
+    bouchet1991,
+    author={Andrk Bouchet},
+    title={Recognizing locally equivalent graphs},
+    year=1991,
+    note={https://pdf.sciencedirectassets.com/271536/1-s2.0-S0012365X00X02166/1-s2.0-0012365X9390357Y/main.pdf}
+}

thesis/main.tex

@@ -90,11 +90,12 @@ Supervised by Prof. Dr. Christoph Lehner}
 
 \newpage
 \textbf{Erklärung zur Anfertigung:}\\
-Ich habe die Arbeit selbständig verfasst, keine anderen als die angegebenen Quellen und Hilfsmittel be-
+Ich habe die Arbeit selbständig verfasst, keine anderen als die angegebenen
-nutzt und bisher keiner anderen Prüfungsbehörde vorgelegt. Außerdem bestätige ich hiermit, dass die
+Quellen und Hilfsmittel benutzt und bisher keiner anderen Prüfungsbehörde
-vorgelegten Druckexemplare und die vorgelegte elektronische Version der Arbeit identisch sind, dass ich
+vorgelegt. Außerdem bestätige ich hiermit, dass die vorgelegten Druckexemplare
-über wissenschaftlich korrektes Arbeiten und Zitieren aufgeklärt wurde und dass ich von den in § 24 Abs.
+und die vorgelegte elektronische Version der Arbeit identisch sind, dass ich
-5 vorgesehenen Rechtsfolgen Kenntnis habe.
+über wissenschaftlich korrektes Arbeiten und Zitieren aufgeklärt wurde und dass
+ich von den in § 24 Abs.  5 vorgesehenen Rechtsfolgen Kenntnis habe.
 \\
 \\
 Unterschrift: