diff --git a/thesis/chapters/implementation.tex b/thesis/chapters/implementation.tex index 89575fd..4e058a2 100644 --- a/thesis/chapters/implementation.tex +++ b/thesis/chapters/implementation.tex @@ -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}. diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index e253dcc..17f5ec7 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -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