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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
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 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
in \ref{ref:code_example_graphs}.

103
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
encode quantum information such that it is protected against noise. The
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.
It was only later that Gottesman and Knill discovered that stabilizer states
can be simulated in polynomial time on a classical machine
\cite{gottesman2008}. This performance has since been improved to $n\log(n)$
time on average \cite{andersbriegel2005}.
It was only later that Gottesman and Knill realized that stabilizer states can
be simulated in polynomial time on a classical machine \cite{gottesman2008}.
This performance has since been improved to $n\log(n)$ time on average
\cite{andersbriegel2005}.
\section{Stabilizers and Stabilizer States}
@ -39,8 +39,8 @@ either commute or anticommute.
is called the multilocal Pauli group on $n$ qbits \cite{andersbriegel2005}.
\end{definition}
The group property of $P_n$ and the (anti-)commutator relationships follow
directly from its definition via the tensor product.
The group property of $P_n$ and the (anti-)commutator relationships can be
deduced from its definition via the tensor product.
%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 $p = -I$.
@ -59,20 +59,19 @@ The discussion below follows the argumentation given in \cite{nielsen_chuang_201
\end{definition}
\begin{lemma}
If $S$ is a set of stabilizers, the following statements follow
directly:
If $S$ is a set of stabilizers, these statements follow directly:
\begin{enumerate}
\item{$\pm iI \notin S$}
\item{$(S^{(i)})^2 = I$ for all $i$}
\item{$S^{(i)}$ are hermitian for all $i$ }
\item{$(S^{(i)})^2 = I$ $\forall i$}
\item{$S^{(i)}$ are hermitian $\forall i$ }
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\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
$\tilde{p}_j \in \{X, Y, Z, I\}$ 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.
@ -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
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.
\begin{definition}
@ -99,20 +98,19 @@ group.
$g_i$ and $m$ is the smallest integer for which these statements hold.
\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
its generators.
\subsection{Stabilizer States}
\label{ref:stab_states}
One important basic insight from quantum mechanics is that hermitian operators
have real eigenvalues and eigenspaces which are associated with these
eigenvalues. Finding these eigenvalues and eigenvectors is what one calls
solving a quantum mechanical system. One of the most fundamental insights of
quantum mechanics is that commuting operators have a common set of
eigenvectors, i.e. they can be diagonalized simultaneously. This motivates and
justifies the following definition.
One important property of hermitian operators is that they have real
eigenvalues and eigenspaces which are associated with these eigenvalues.
Finding these eigenvalues and eigenvectors is what one calls solving a quantum
mechanical system. It is fundamental for quantum mechanics that commuting
operators have a common set of eigenvectors, i.e. they can be diagonalized
simultaneously. This motivates and justifies the following definition.
\begin{definition}
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
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
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
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
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}
\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
describes the dynamics of the system, i.e.
\begin{equation} \ket{\psi'} = U \ket{\psi} \end{equation}
It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$
anymore. There are however some statements that can be made
anymore. Under some constraints there are statements that can be made
\cite{nielsen_chuang_2010}:
\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\}
\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}.
\end{definition}
@ -190,8 +188,9 @@ a set of stabilizers.
and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i
\\ -i & 1 \end{array}\right)$.
Also $C_L$ is generated by a product of at most $5$ matrices
$\sqrt{iZ}$, $\sqrt{-iX}$. }
Also $C_L$ is generated by $\sqrt{iZ}$, $\sqrt{-iX}$. When
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 $CX$.}
\end{enumerate}
\end{theorem}
@ -211,17 +210,17 @@ a set of stabilizers.
\end{enumerate}
\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
stabilized by $S'$ one can consider the dynamics of the stabilizers instead of
the actual state. This is considerably more efficient as only $n$ stabilizers
have to be modified, each being just the tensor product of $n$ Pauli matrices.
This has led to the simulation using stabilizer tableaux
the actual state. Updating the $n$ stabilizers is considerably more efficient
as each stabilizer the tensor product of $n$ Pauli matrices. This has led to
the simulation using stabilizer tableaux
\cite{gottesman_aaronson2008}\cite{CHP}.
\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 $g_a \in \{\pm X_a, \pm Y_a,
\pm Z_a\}$ 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}.
\end{equation}
If now $g_a$ commutes with all $S^{(i)}$ a result of $s=0$ is measured with
probability $1$ and the stabilizers are left unchanged:
If now $g_a$ commutes with all $S^{(i)}$ a result of $s=0,1$ (depending on
whether $g_a \in S$ or $-g_a \in S$) is measured with probability $1$ and the
stabilizers are left unchanged:
\begin{equation}
\begin{aligned}
@ -244,7 +244,7 @@ probability $1$ and the stabilizers are left unchanged:
As the state that is stabilized by $S$ is unique $\ket{\psi'} = \ket{\psi}$
\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.
\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|\\
&= P(s=1)
\end{aligned}
\notag
\end{equation}
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)^sg_a}{2}\ket{\psi}
\end{aligned}
\notag
\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 $\ket{\psi'}$
\cite{nielsen_chuang_2010}.
\end{proof}
\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 the following section about
graph states.
are called vertex operator-free will be clear in \ref{ref:sec_g_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
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
in this definition.
here.
\begin{definition}
For a graph $G = (V = \{0, ..., n-1\}, E)$ the associated stabilizers are
@ -324,7 +321,7 @@ in this definition.
\end{definition}
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{aligned}
@ -346,12 +343,13 @@ commute is clear if $\{a,b\} \notin E$. If $\{a, b\} \in E$
\end{equation}
This definition of a graph state might not seem to be straight forward but
recalling Theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$ is
unique. The following lemma will provide a way to construct this state from the
This definition of a graph state might not seem too helpful but recalling
Theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$ is unique.
Lemma \ref{lemma:g_bar} will provide a way to construct this state from the
graph.
\begin{lemma}
\label{lemma:g_bar}
For a graph $G = (V, E)$ the associated state $\ket{\bar{G}}$ is
constructed using \cite{hein_eisert_briegel2008}
@ -472,13 +470,13 @@ Another transformation on the VOP-free graph states for a vertex $a \in V$ is
\end{equation}
This transformation toggles the neighbourhood of $a$ which is an operation
that will be used later\cite{andersbriegel2005}.
that will be used later \cite{andersbriegel2005}.
\begin{lemma}
\label{lemma:M_a}
When applying $M_a$ to a state $\ket{\bar{G}}$ the new state
$\ket{\bar{G}'}$ is again a VOP-free graph state and the
graph is updated according to\cite{andersbriegel2005}:
graph is updated according to \cite{andersbriegel2005}:
\begin{equation}
\begin{aligned}
n_a' &= n_a \\
@ -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{-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)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{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
$\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$ it is clear that
$\forall j \notin n_a$ $K_{G'}^{(j)} = K_G^{(j)}$. To construct the
$K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a =: \{j\} \cup F$ and $n_j =:
\{a\} \cup D$. Then follows:
$K_{G'}^{(i)}$ choose a $j \in n_a$ and partition the neighbourhoods $n_a
=: \{j\} \cup F$ and $n_j =: \{a\} \cup D$. Then follows:
\begin{equation}
\begin{aligned}
@ -551,6 +549,7 @@ Therefore the associated graph is changed as given in the third equation.
\end{proof}
\section{Graph States}
\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
@ -644,7 +643,7 @@ 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 true power of this formalism is seen when
studying its dynamics. The simplest case is a local Clifford operator $c_j$
acting on a qbit $j$: The stabilizers are changed to $\langle c_j S^{(i)}
c_j^\dagger\rangle_i$. Using the definition of the graphical representation one