some final work
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1.500000000000000000e+03 1.000000000000000000e+02 4.152314034017763611e-02 5.716930792484469599e-04
|
||||
1.550000000000000000e+03 1.000000000000000000e+02 4.438926930015441030e-02 6.721290882549972113e-04
|
||||
1.600000000000000000e+03 1.000000000000000000e+02 4.552266089023760537e-02 7.453956208500090320e-04
|
||||
1.650000000000000000e+03 1.000000000000000000e+02 5.021009291991504475e-02 6.862840120510182822e-04
|
||||
1.700000000000000000e+03 1.000000000000000000e+02 5.453786087025946222e-02 6.546641481410810049e-04
|
||||
1.750000000000000000e+03 1.000000000000000000e+02 5.628588498999306799e-02 8.462562766021738741e-04
|
||||
1.800000000000000000e+03 1.000000000000000000e+02 6.095474819019727764e-02 7.517515934752839373e-04
|
||||
1.850000000000000000e+03 1.000000000000000000e+02 6.366028950997133784e-02 8.152459203989234496e-04
|
||||
1.900000000000000000e+03 1.000000000000000000e+02 6.785009805011213424e-02 7.698881522351951924e-04
|
||||
1.950000000000000000e+03 1.000000000000000000e+02 7.188723056995513505e-02 6.833998803979537658e-04
|
||||
2.000000000000000000e+03 1.000000000000000000e+02 7.566429144997527390e-02 8.365863514799617290e-04
|
||||
2.050000000000000000e+03 1.000000000000000000e+02 7.861056976002146757e-02 8.193592148539533947e-04
|
||||
2.100000000000000000e+03 1.000000000000000000e+02 8.247058204007771953e-02 8.197200493282149258e-04
|
||||
2.150000000000000000e+03 1.000000000000000000e+02 8.493876021024333867e-02 8.342453122990786934e-04
|
||||
2.200000000000000000e+03 1.000000000000000000e+02 8.813758615971892252e-02 7.911501815949151164e-04
|
||||
2.250000000000000000e+03 1.000000000000000000e+02 9.468871829980343713e-02 9.350049573202011063e-04
|
||||
2.300000000000000000e+03 1.000000000000000000e+02 9.786318559981736775e-02 9.314812240165508146e-04
|
||||
2.350000000000000000e+03 1.000000000000000000e+02 1.018545777700273908e-01 9.261618059141366391e-04
|
||||
2.400000000000000000e+03 1.000000000000000000e+02 1.051895230298396239e-01 8.616776153947947308e-04
|
||||
2.450000000000000000e+03 1.000000000000000000e+02 1.070327199899838849e-01 9.274632293436604977e-04
|
||||
2.500000000000000000e+03 1.000000000000000000e+02 1.112921274800100918e-01 8.472622784928632040e-04
|
||||
2.550000000000000000e+03 1.000000000000000000e+02 1.145994285400956936e-01 9.315154484987448174e-04
|
||||
2.600000000000000000e+03 1.000000000000000000e+02 1.193636358499861605e-01 9.910600547399727421e-04
|
||||
2.650000000000000000e+03 1.000000000000000000e+02 1.218935546997090558e-01 8.861179673363818541e-04
|
||||
2.700000000000000000e+03 1.000000000000000000e+02 1.261190436399556303e-01 9.270747298007727191e-04
|
||||
2.750000000000000000e+03 1.000000000000000000e+02 1.291786062698520310e-01 9.713837766685412545e-04
|
||||
4.000000000000000000e+02 5.000000000000000000e+01 1.707738740005879585e-03 7.100286700408404613e-06
|
||||
4.500000000000000000e+02 5.000000000000000000e+01 1.906502640049438820e-03 9.853300832137104791e-06
|
||||
5.000000000000000000e+02 5.000000000000000000e+01 2.194008100050268747e-03 2.225048323246775191e-05
|
||||
5.500000000000000000e+02 5.000000000000000000e+01 2.483741079940955578e-03 2.902929435898853839e-05
|
||||
6.000000000000000000e+02 5.000000000000000000e+01 2.970475479905871751e-03 5.357547401616545894e-05
|
||||
6.500000000000000000e+02 5.000000000000000000e+01 3.360749460098304622e-03 7.760192059341477707e-05
|
||||
7.000000000000000000e+02 5.000000000000000000e+01 4.184713580034440245e-03 1.257844672870958591e-04
|
||||
7.500000000000000000e+02 5.000000000000000000e+01 5.004965839962096513e-03 1.756539085744744525e-04
|
||||
8.000000000000000000e+02 5.000000000000000000e+01 5.734667459946649927e-03 1.910387597965613972e-04
|
||||
8.500000000000000000e+02 5.000000000000000000e+01 7.131355419896862752e-03 1.934052327071289392e-04
|
||||
9.000000000000000000e+02 5.000000000000000000e+01 8.802635380070568394e-03 3.408876648940773542e-04
|
||||
9.500000000000000000e+02 5.000000000000000000e+01 1.094996251993506954e-02 3.780652528111945248e-04
|
||||
1.000000000000000000e+03 5.000000000000000000e+01 1.176883709997127853e-02 3.875201443014548758e-04
|
||||
1.050000000000000000e+03 5.000000000000000000e+01 1.426043346002188635e-02 4.682970383390439679e-04
|
||||
1.100000000000000000e+03 5.000000000000000000e+01 1.606511461995978676e-02 4.821875787626857496e-04
|
||||
1.150000000000000000e+03 5.000000000000000000e+01 1.810014320004484090e-02 5.439479727767238936e-04
|
||||
1.200000000000000000e+03 5.000000000000000000e+01 2.132428097998854272e-02 5.977411510180347153e-04
|
||||
1.250000000000000000e+03 5.000000000000000000e+01 2.537384307997854124e-02 6.676700647178173504e-04
|
||||
1.300000000000000000e+03 5.000000000000000000e+01 2.700082207993546174e-02 6.834614338333218136e-04
|
||||
1.350000000000000000e+03 5.000000000000000000e+01 2.877831694000633489e-02 6.806464215638255433e-04
|
||||
1.400000000000000000e+03 5.000000000000000000e+01 3.247053898001468070e-02 7.730662439967084590e-04
|
||||
1.450000000000000000e+03 5.000000000000000000e+01 3.598561208000319173e-02 8.005966432160921409e-04
|
||||
1.500000000000000000e+03 5.000000000000000000e+01 3.962553389999811521e-02 9.617593729943524914e-04
|
||||
1.550000000000000000e+03 5.000000000000000000e+01 4.163353030004145888e-02 9.548479191853584576e-04
|
||||
1.600000000000000000e+03 5.000000000000000000e+01 4.562790117995973310e-02 7.835441508272948316e-04
|
||||
1.650000000000000000e+03 5.000000000000000000e+01 4.898105746005967237e-02 8.710963335263738599e-04
|
||||
1.700000000000000000e+03 5.000000000000000000e+01 5.169834830003310067e-02 9.651211683144866882e-04
|
||||
1.750000000000000000e+03 5.000000000000000000e+01 5.374540261993388662e-02 1.000999207215783841e-03
|
||||
1.800000000000000000e+03 5.000000000000000000e+01 6.012160696005594551e-02 9.086338428378314819e-04
|
||||
1.850000000000000000e+03 5.000000000000000000e+01 6.043908248007937717e-02 1.230395679723121118e-03
|
||||
1.900000000000000000e+03 5.000000000000000000e+01 6.464911280001615912e-02 8.604282740948831793e-04
|
||||
1.950000000000000000e+03 5.000000000000000000e+01 6.818929374001526933e-02 1.115434394007077832e-03
|
||||
2.000000000000000000e+03 5.000000000000000000e+01 7.424652839998088782e-02 1.071955559986128735e-03
|
||||
2.050000000000000000e+03 5.000000000000000000e+01 7.574371884000356825e-02 9.012814866758098739e-04
|
||||
2.100000000000000000e+03 5.000000000000000000e+01 8.000210310005058389e-02 1.157114746495429376e-03
|
||||
2.150000000000000000e+03 5.000000000000000000e+01 8.131804838001699398e-02 1.246651522871319254e-03
|
||||
2.200000000000000000e+03 5.000000000000000000e+01 8.642062879996956215e-02 1.198064886081493755e-03
|
||||
2.250000000000000000e+03 5.000000000000000000e+01 9.114948505999563577e-02 8.960646114521502153e-04
|
||||
2.300000000000000000e+03 5.000000000000000000e+01 9.217601664009635043e-02 9.503993848618267530e-04
|
||||
2.350000000000000000e+03 5.000000000000000000e+01 9.563625923990912159e-02 1.108765711978444571e-03
|
||||
2.400000000000000000e+03 5.000000000000000000e+01 1.013627541600544690e-01 1.295486325060567429e-03
|
||||
2.450000000000000000e+03 5.000000000000000000e+01 1.034756118799668861e-01 1.204635265310459924e-03
|
||||
2.500000000000000000e+03 5.000000000000000000e+01 1.065721216400379451e-01 9.677911226791394888e-04
|
||||
2.550000000000000000e+03 5.000000000000000000e+01 1.141797984599725174e-01 1.006250648249219911e-03
|
||||
|
|
|
|||
|
|
|
@ -1 +1 @@
|
|||
{"nstart": 400, "nstop": 2800, "step": 50, "ncircuits": 100, "nqbits0": 100, "nqbits1": 50, "seed": 3735928559}
|
||||
{"nstart": 400, "nstop": 2600, "step": 50, "ncircuits": 50, "nqbits0": 100, "nqbits1": 50, "seed": 3735928559}
|
||||
|
|
@ -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")
|
||||
|
|
|
|||
Binary file not shown.
|
Before Width: | Height: | Size: 152 KiB After Width: | Height: | Size: 160 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 129 KiB After Width: | Height: | Size: 133 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 141 KiB After Width: | Height: | Size: 142 KiB |
|
|
@ -6,6 +6,7 @@
|
|||
\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}
|
||||
|
|
|
|||
|
|
@ -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.
|
||||
|
||||
|
|
|
|||
|
|
@ -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.
|
||||
|
|
|
|||
|
|
@ -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
|
||||
|
|
|
|||
|
|
@ -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}
|
||||
|
|
|
|||
|
|
@ -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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Reference in New Issue