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thesis/chapters/implementation.tex
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@ -474,7 +474,7 @@ low-linear, intermediate and high-linear regime can be seen in Figure
and Figure \ref{fig:graph_high_linear_regime}. Due to the great amount of edges 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 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 actual graph. The full images are in \ref{ref:complete_graphs}. Further the
regimes are not clearly visibe for $n>30$ qbits so choosing smaller graphs is 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 not possible. The code that was used to generate these images can be found
in \ref{ref:code_example_graphs}. in \ref{ref:code_example_graphs}.

97
thesis/chapters/stabilizer.tex
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@ -6,13 +6,13 @@ The stabilizer formalism was originally introduced by Gottesman
\cite{gottesman1997} for quantum error correction and is a useful tool to \cite{gottesman1997} for quantum error correction and is a useful tool to
encode quantum information such that it is protected against noise. The encode quantum information such that it is protected against noise. The
prominent Shor code \cite{shor1995} is an example of a stabilizer code prominent Shor code \cite{shor1995} is an example of a stabilizer code
(although it was discovered before the stabilizer formalism was discovered), as (although it was described before the stabilizer formalism was discovered), as
are the 3-qbit bit-flip and phase-flip codes. are the 3-qbit bit-flip and phase-flip codes.
It was only later that Gottesman and Knill discovered that stabilizer states It was only later that Gottesman and Knill realized that stabilizer states can
can be simulated in polynomial time on a classical machine be simulated in polynomial time on a classical machine \cite{gottesman2008}.
\cite{gottesman2008}. This performance has since been improved to $n\log(n)$ This performance has since been improved to $n\log(n)$ time on average
time on average \cite{andersbriegel2005}. \cite{andersbriegel2005}.
\section{Stabilizers and Stabilizer States} \section{Stabilizers and Stabilizer States}
@ -39,8 +39,8 @@ either commute or anticommute.
is called the multilocal Pauli group on $n$ qbits \cite{andersbriegel2005}. is called the multilocal Pauli group on $n$ qbits \cite{andersbriegel2005}.
\end{definition} \end{definition}
The group property of $P_n$ and the (anti-)commutator relationships follow The group property of $P_n$ and the (anti-)commutator relationships can be
directly from its definition via the tensor product. deduced from its definition via the tensor product.
%Further are $p \in P_n$ hermitian and have the eigenvalues $\pm 1$ for %Further are $p \in P_n$ hermitian and have the eigenvalues $\pm 1$ for
%$p \neq \pm I$, $+1$ for $p = I$ and $-1$ for $p = -I$. %$p \neq \pm I$, $+1$ for $p = I$ and $-1$ for $p = -I$.
@ -59,20 +59,19 @@ The discussion below follows the argumentation given in \cite{nielsen_chuang_201
\end{definition} \end{definition}
\begin{lemma} \begin{lemma}
If $S$ is a set of stabilizers, the following statements follow If $S$ is a set of stabilizers, these statements follow directly:
directly:
\begin{enumerate} \begin{enumerate}
\item{$\pm iI \notin S$} \item{$\pm iI \notin S$}
\item{$(S^{(i)})^2 = I$ for all $i$} \item{$(S^{(i)})^2 = I$ $\forall i$}
\item{$S^{(i)}$ are hermitian for all $i$ } \item{$S^{(i)}$ are hermitian $\forall i$ }
\end{enumerate} \end{enumerate}
\end{lemma} \end{lemma}
\begin{proof} \begin{proof}
\begin{enumerate} \begin{enumerate}
\item{$(iI)^2 = (-iI)^2 = -I$. Which contradicts the definition of $S$.} \item{$(iI)^2 = (-iI)^2 = -I$. Which contradicts the definition of $S$.}
\item{From the definition of $S$ ($P_n$ respectively) follows that any \item{From the definition of $S$ ($P_n$ respectively) one sees that any
$S^{(i)} \in S$ has the form $\pm i^l \left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$ where $S^{(i)} \in S$ 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 $\left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$ $\tilde{p}_j \in \{X, Y, Z, I\}$ 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. is hermitian and unitary $(S^{(i)})^2$ is either $+I$ or $-I$. As $-I \notin S$ $(S^{(i)})^2 = I$ follows directly.
@ -85,7 +84,7 @@ The discussion below follows the argumentation given in \cite{nielsen_chuang_201
Considering all the elements of a group can be impractical for some Considering all the elements of a group can be impractical for some
calculations, the generators of a group are introduced. Often it is enough to calculations, the generators of a group are introduced. Often it is enough to
discuss the generator's properties in order to understand the properties of the discuss the generator's properties in order to understand those of the
group. group.
\begin{definition} \begin{definition}
@ -99,20 +98,19 @@ group.
$g_i$ and $m$ is the smallest integer for which these statements hold. $g_i$ and $m$ is the smallest integer for which these statements hold.
\end{definition} \end{definition}
In the following discussions $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be From now on the generators $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be
used as the required properties of a set of stabilizers that can be studied on used as the required properties of a set of stabilizers that can be studied on
its generators. its generators.
\subsection{Stabilizer States} \subsection{Stabilizer States}
\label{ref:stab_states} \label{ref:stab_states}
One important basic insight from quantum mechanics is that hermitian operators One important property of hermitian operators is that they have real
have real eigenvalues and eigenspaces which are associated with these eigenvalues and eigenspaces which are associated with these eigenvalues.
eigenvalues. Finding these eigenvalues and eigenvectors is what one calls Finding these eigenvalues and eigenvectors is what one calls solving a quantum
solving a quantum mechanical system. One of the most fundamental insights of mechanical system. It is fundamental for quantum mechanics that commuting
quantum mechanics is that commuting operators have a common set of operators have a common set of eigenvectors, i.e. they can be diagonalized
eigenvectors, i.e. they can be diagonalized simultaneously. This motivates and simultaneously. This motivates and justifies the following definition.
justifies the following definition.
\begin{definition} \begin{definition}
For a set of stabilizers $S$ the vector space For a set of stabilizers $S$ the vector space
@ -128,9 +126,9 @@ justifies the following definition.
It is clear that to show the stabilization property of $S$ the proof for the It is clear that to show the stabilization property of $S$ the proof for the
generators is sufficient, as all the generators forming an element in $S$ can generators is sufficient, as all the generators forming an element in $S$ can
be absorbed into $\ket{\psi}$. The dimension of $V_S$ is not immediately be absorbed into $\ket{\psi}$. The dimension of $V_S$ is not immediately
clear. One can however show that for a set of stabilizers $\langle S^{(i)} clear. One can show that for a set of stabilizers $\langle S^{(i)}
\rangle_{i=1, ..., n-m}$ the dimension $\dim V_S = 2^m$ \cite[Chapter \rangle_{i=1, ..., n-m}$ the dimension $\dim V_S = 2^m$ \cite[Chapter
10.5]{nielsen_chuang_2010}. This yields the following important result: 10.5]{nielsen_chuang_2010}. This yields this important result:
\begin{theorem} \label{thm:unique_s_state} For a $n$ qbit system and \begin{theorem} \label{thm:unique_s_state} For a $n$ qbit system and
stabilizers $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer stabilizers $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
@ -146,14 +144,14 @@ In the following discussions for $n$ qbits a set $S = \langle S^{(i)}
\subsection{Dynamics of Stabilizer States} \subsection{Dynamics of Stabilizer States}
\label{ref:dynamics_stabilizer} \label{ref:dynamics_stabilizer}
Consider a $n$ qbit state $\ket{\psi}$ that is the $+1$ eigenstate of $S Consider a $n$-qbit state $\ket{\psi}$ that is the $+1$ eigenstate of $S
= \langle S^{(i)} \rangle_{i=1,...,n}$ and a unitary transformation $U$ that = \langle S^{(i)} \rangle_{i=1,...,n}$ and a unitary transformation $U$ that
describes the dynamics of the system, i.e. describes the dynamics of the system, i.e.
\begin{equation} \ket{\psi'} = U \ket{\psi} \end{equation} \begin{equation} \ket{\psi'} = U \ket{\psi} \end{equation}
It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$ It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$
anymore. There are however some statements that can be made anymore. Under some constraints there are statements that can be made
\cite{nielsen_chuang_2010}: \cite{nielsen_chuang_2010}:
\begin{equation} \begin{equation}
@ -178,7 +176,7 @@ a set of stabilizers.
C_n := \left\{U \in U(2^n) \middle| \forall p \in P_n: UpU^\dagger \in P_n \right\} C_n := \left\{U \in U(2^n) \middle| \forall p \in P_n: UpU^\dagger \in P_n \right\}
\end{equation} \end{equation}
is called the Clifford group. $C_1 =: C_L$ is called the local Clifford is called the Clifford group, $C_1 =: C_L$ the local Clifford
group \cite{andersbriegel2005}. group \cite{andersbriegel2005}.
\end{definition} \end{definition}
@ -190,8 +188,9 @@ a set of stabilizers.
and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i
\\ -i & 1 \end{array}\right)$. \\ -i & 1 \end{array}\right)$.
Also $C_L$ is generated by a product of at most $5$ matrices Also $C_L$ is generated by $\sqrt{iZ}$, $\sqrt{-iX}$. When
$\sqrt{iZ}$, $\sqrt{-iX}$. } using $\sqrt{iZ}, \sqrt{-iX}$ the product has a length not greater
than $5$. }
\item{$C_n$ can be generated using $C_L$ and $CZ$ or $CX$.} \item{$C_n$ can be generated using $C_L$ and $CZ$ or $CX$.}
\end{enumerate} \end{enumerate}
\end{theorem} \end{theorem}
@ -211,17 +210,17 @@ a set of stabilizers.
\end{enumerate} \end{enumerate}
\end{proof} \end{proof}
This is quite an important result: As under a transformation $U \in C_n$ $S' This is an important result: As under a transformation $U \in C_n$ $S'
= U S U^\dagger$ 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 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 the actual state. Updating the $n$ stabilizers is considerably more efficient
have to be modified, each being just the tensor product of $n$ Pauli matrices. as each stabilizer the tensor product of $n$ Pauli matrices. This has led to
This has led to the simulation using stabilizer tableaux the simulation using stabilizer tableaux
\cite{gottesman_aaronson2008}\cite{CHP}. \cite{gottesman_aaronson2008}\cite{CHP}.
\subsection{Measurements on Stabilizer States} \label{ref:meas_stab} \subsection{Measurements on Stabilizer States} \label{ref:meas_stab}
Interestingly also measurements are dynamics covered by the stabilizers Also measurements are dynamics covered by the stabilizers
\cite{nielsen_chuang_2010}. When an observable $g_a \in \{\pm X_a, \pm Y_a, \cite{nielsen_chuang_2010}. When an observable $g_a \in \{\pm X_a, \pm Y_a,
\pm Z_a\}$ acting on qbit $a$ is measured one has to consider the projector \pm Z_a\}$ acting on qbit $a$ is measured one has to consider the projector
@ -229,8 +228,9 @@ Interestingly also measurements are dynamics covered by the stabilizers
P_{g_a,s} = \frac{I + (-1)^s g_a}{2}. P_{g_a,s} = \frac{I + (-1)^s g_a}{2}.
\end{equation} \end{equation}
If now $g_a$ commutes with all $S^{(i)}$ a result of $s=0$ is measured with If now $g_a$ commutes with all $S^{(i)}$ a result of $s=0,1$ (depending on
probability $1$ and the stabilizers are left unchanged: whether $g_a \in S$ or $-g_a \in S$) is measured with probability $1$ and the
stabilizers are left unchanged:
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
@ -244,7 +244,7 @@ probability $1$ and the stabilizers are left unchanged:
As the state that is stabilized by $S$ is unique $\ket{\psi'} = \ket{\psi}$ As the state that is stabilized by $S$ is unique $\ket{\psi'} = \ket{\psi}$
\cite{nielsen_chuang_2010}. \cite{nielsen_chuang_2010}.
If $g_a$ does not commute with all stabilizers the following lemma gives the If $g_a$ does not commute with all stabilizers the Lemma \ref{lemma:stab_measurement} gives the
result of the measurement. result of the measurement.
\begin{lemma} \begin{lemma}
@ -271,7 +271,6 @@ and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing a $j
&= \left|\hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\right|\\ &= \left|\hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\right|\\
&= P(s=1) &= P(s=1)
\end{aligned} \end{aligned}
\notag
\end{equation} \end{equation}
With $P(s=+1) + P(s=-1) = 1$ follows $P(s=+1) = \frac{1}{2} = P(s=-1)$. With $P(s=+1) + P(s=-1) = 1$ follows $P(s=+1) = \frac{1}{2} = P(s=-1)$.
@ -284,18 +283,16 @@ and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing a $j
&= S^{(j)}S^{(i)}\frac{I + (-1)^{s + 2}g_a}{2}\ket{\psi} \\ &= S^{(j)}S^{(i)}\frac{I + (-1)^{s + 2}g_a}{2}\ket{\psi} \\
&= S^{(j)}S^{(i)}\frac{I + (-1)^sg_a}{2}\ket{\psi} &= S^{(j)}S^{(i)}\frac{I + (-1)^sg_a}{2}\ket{\psi}
\end{aligned} \end{aligned}
\notag
\end{equation} \end{equation}
The state after measurement is stabilized by $S^{(j)}S^{(i)}$ $i,j \in J$, The state after measurement is stabilized by $S^{(j)}S^{(i)}$ for $S^{(j)},S^{(i)} \in J$,
and by $S^{(i)} \in J^c$. $(-1)^sg_a$ trivially stabilizes $\ket{\psi'}$ and by $S^{(i)} \in J^c$. $(-1)^sg_a$ trivially stabilizes $\ket{\psi'}$
\cite{nielsen_chuang_2010}. \cite{nielsen_chuang_2010}.
\end{proof} \end{proof}
\section{The VOP-free Graph States} \section{The VOP-free Graph States}
This section will discuss the vertex operator (VOP)-free graph states. Why they This section will discuss the vertex operator (VOP)-free graph states. Why they
are called vertex operator-free will be clear in the following section about are called vertex operator-free will be clear in \ref{ref:sec_g_states}.
graph states.
\subsection{VOP-free Graph States} \subsection{VOP-free Graph States}
@ -311,7 +308,7 @@ called the neighbourhood of $i$ \cite{hein_eisert_briegel2008}.
This definition of a graph is way less general than the definition of a graph This definition of a graph is way less general than the definition of a graph
in graph theory. Using this definition will however allow to avoid an in graph theory. Using this definition will however allow to avoid an
extensive list of constraints on the graph from graph theory that are implied extensive list of constraints on the graph from graph theory that are implied
in this definition. here.
\begin{definition} \begin{definition}
For a graph $G = (V = \{0, ..., n-1\}, E)$ the associated stabilizers are For a graph $G = (V = \{0, ..., n-1\}, E)$ the associated stabilizers are
@ -324,7 +321,7 @@ in this definition.
\end{definition} \end{definition}
It is clear that the $K_G^{(i)}$ are multilocal Pauli operators. That they It is clear that the $K_G^{(i)}$ are multilocal Pauli operators. That they
commute is clear if $\{a,b\} \notin E$. If $\{a, b\} \in E$ commute is trivial for $\{a,b\} \notin E$. If $\{a, b\} \in E$
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
@ -346,12 +343,13 @@ commute is clear if $\{a,b\} \notin E$. If $\{a, b\} \in E$
\end{equation} \end{equation}
This definition of a graph state might not seem to be straight forward but This definition of a graph state might not seem too helpful but recalling
recalling Theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$ is Theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$ is unique.
unique. The following lemma will provide a way to construct this state from the Lemma \ref{lemma:g_bar} will provide a way to construct this state from the
graph. graph.
\begin{lemma} \begin{lemma}
\label{lemma:g_bar}
For a graph $G = (V, E)$ the associated state $\ket{\bar{G}}$ is For a graph $G = (V, E)$ the associated state $\ket{\bar{G}}$ is
constructed using \cite{hein_eisert_briegel2008} constructed using \cite{hein_eisert_briegel2008}
@ -509,7 +507,7 @@ that will be used later\cite{andersbriegel2005}.
&= \sqrt{iZ_j} X_j\sqrt{iZ_j}^\dagger\left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right) &= \sqrt{iZ_j} X_j\sqrt{iZ_j}^\dagger\left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)
\sqrt{-iX_a} Z_a \sqrt{-iX_a}^\dagger \\ \sqrt{-iX_a} Z_a \sqrt{-iX_a}^\dagger \\
&= (-1)^2 Y_j Y_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)\\ &= (-1)^2 Y_j Y_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)\\
&= (-1)i^2 Z_j X_a X_j Z_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right) &= (-1)i^2 Z_j X_a X_j Z_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right).
\end{aligned} \end{aligned}
\end{equation} \end{equation}
@ -517,8 +515,8 @@ that will be used later\cite{andersbriegel2005}.
\ket{\bar{G}}$ is the $+1$ eigenstate of the new $K_{G'}^{(i)}$. Because \ket{\bar{G}}$ is the $+1$ eigenstate of the new $K_{G'}^{(i)}$. Because
$\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$ it is clear that $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$ it is clear that
$\forall j \notin n_a$ $K_{G'}^{(j)} = K_G^{(j)}$. To construct the $\forall j \notin n_a$ $K_{G'}^{(j)} = K_G^{(j)}$. To construct the
$K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a =: \{j\} \cup F$ and $n_j =: $K_{G'}^{(i)}$ choose a $j \in n_a$ and partition the neighbourhoods $n_a
\{a\} \cup D$. Then follows: =: \{j\} \cup F$ and $n_j =: \{a\} \cup D$. Then follows:
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
@ -551,6 +549,7 @@ Therefore the associated graph is changed as given in the third equation.
\end{proof} \end{proof}
\section{Graph States} \section{Graph States}
\label{ref:sec_g_states}
The definition of a VOP-free graph state above raises an obvious question: Can 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 straight