diff --git a/thesis/chapters/quantum_computing.tex b/thesis/chapters/quantum_computing.tex index bc998eb..9e21b80 100644 --- a/thesis/chapters/quantum_computing.tex +++ b/thesis/chapters/quantum_computing.tex @@ -21,19 +21,13 @@ common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ wit \label{ref:singleqbitgates} \begin{equation} - X := \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right) -\end{equation} -\begin{equation} - Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) -\end{equation} -\begin{equation} - H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) -\end{equation} -\begin{equation} - R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right) -\end{equation} -\begin{equation} - I := \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right) + \begin{aligned} + & X := \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right) \\ + & Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) \\ + & H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) \\ + & R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)\\ + & I := \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right) \\ + \end{aligned} \end{equation} Note that $X = HZH$ and $Z = R_{\pi}$, so the set of $H, R_\phi$ is sufficient. @@ -47,6 +41,20 @@ transforming to the other Pauli eigenstates is done using $H$ and $SH$: S H Z H^\dagger S^\dagger = S X S^\dagger = Y \end{equation} +The following states are the $\pm 1$ eigenstates of the $X$, $Y$, $Z$ operators +and will be used in some calculations later. + +\begin{equation} + \begin{aligned} + &\ket{+_X} \equiv \ket{+} := \frac{1}{\sqrt{2}} (\ket{0} + \ket{1}) \\ + &\ket{-_X} \equiv \ket{-} := \frac{1}{\sqrt{2}} (\ket{0} - \ket{1}) \\ + &\ket{+_Y} := \frac{1}{\sqrt{2}} (\ket{0} + i\ket{1}) \\ + &\ket{-_Y} := \frac{1}{\sqrt{2}} (\ket{0} - i\ket{1}) \\ + &\ket{+_Z} := \ket{0} \\ + &\ket{-_Z} := \ket{1} \\ + \end{aligned} +\end{equation} + \subsubsection{Many Qbits} \begin{postulate} @@ -130,8 +138,6 @@ matrices to the basis states) \subsubsection{Measurements} -\textbf{FIXME} I don't like this at all. - \begin{postulate} Let $$\ket{\psi} = \alpha\ket{\phi_1} \otimes \ket{1}_j + \beta\ket{\phi_0} \otimes \ket{0}_j$$ @@ -145,6 +151,9 @@ matrices to the basis states) \end{postulate} Measuring a qbit will also yield a classical result $0$ or $1$ with the respective probabilities. +Measurements are always performed in the computational basis, i.e. for a qbit +$i$ $Z_i$ is measured. With the eigenstate of the $+1$ eigenvalue being the $\ket{0}$ +and $\ket{1}$ for the $-1$ eigenvalue of $Z$. \begin{corrolary} In general the measurement of a qbit is not invertible, in particular it cannot be represented as a @@ -162,27 +171,31 @@ As a measurement is not unitary it is not a gate as in the definition above. In the following discussion the term \textit{measurement gate} will be used from time to time as a measurement can be treated similarely while doing numerics. -Measurements are always performed in the computational basis, i.e. for a qbit -$i$ $Z_i$ is measured. Let the state to be measured be - -\begin{equation} - \ket{\psi} = \alpha\ket{0}\otimes\ket{\psi_0} + \beta\ket{1}\otimes\ket{\psi_1} -\end{equation} - -then a result $0$ is measured with probability $|\alpha|^2$ and $1$ with probability -$|\beta|^2 = 1 - |\alpha|^2$. The wave function is then collapsed to - -\begin{equation} - \begin{aligned} - \ket{\psi'} = \left\{\begin{array}{c}\ket{0}\otimes\ket{\psi_0}, \mbox{ for a result 0 } \\ - \ket{1}\otimes\ket{\psi_1}, \mbox{ for a result 1 } - \end{array}\right\} - \end{aligned} -\end{equation} \subsection{Quantum Circuits} -Quantum circuits are a simple and well-readable way to express the application -of several gates on a state. +\textbf{FIXME: citation needed} -\textbf{TODO} +Quantum circuits are a simple and well-readable way to express the application +of several gates on a state. Qbits are represented by horizontal line on which +time goes from left to right. On these lines single qbit gates are +represented by a box: + +\[ +\Qcircuit @C=1em @R=.7em { +& \gate{H} & \gate{X} & \gate{Z} &\qw \\ +} +\] + +The controlled gates (such as $CX$ and $CZ$) have a vertical line from the control qbit to +the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is + +\[ +\Qcircuit @C=1em @R=.7em { +& \qw & \ctrl{2} & \qw & \qw &\qw \\ +& \qw & \qw & \qw & \ctrl{1} &\qw \\ +& \qw & \gate{X} & \qw & \gate{Z} &\qw \\ +} +\] + +\textbf{TODO: more info about quantum circuits} diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 5cf70ea..24e3411 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -14,3 +14,107 @@ performance has since been improved to $n\log(n)$ time on average \cite{andersbr \subsection{Stabilizers and Stabilizer States} +\subsubsection{Local Pauli Group and Multilocal Pauli Group} + +\begin{definition} + \begin{equation} + P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\} + \end{equation} + + Is called the Pauli group. +\end{definition} + +The group property of $P$ can be verified easily. Note that +the elements of $P$ either commute or anticommute. + +\begin{definition} + For $n$ qbits + + \begin{equation} + P_n := \{\bigotimes\limits_{i=0}^{n-1} p_i | p_i \in P\} + \end{equation} + + is called the multilocal Pauli group on $n$ qbits. +\end{definition} + +The group property of $P_n$ follows directly from its definition +via the tensor product as do the (anti-)commutator relationships. +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 $p = -I$. + +\subsubsection{Stabilizers} + +\begin{definition} + \label{def:stabilizer} + An abelian subgroup $S = \{S^{(0)}, ..., S^{(N)}\}$ of $P_n$ is called a set of stabilizers iff + \begin{enumerate} + \item{$\forall i,j = 1, ..., N$ $[S^{(i)}, S^{(j)}] = 0$ $S^{(i)}$ and $S^{(j)}$ commute} + \item{$-I \notin S$} + \end{enumerate} +\end{definition} + +\begin{lemma} + If $S$ is a set of stabilizers, the following statements are follow + directly + + \begin{enumerate} + \item{$\pm iI \notin S$} + \item{$(S^{(i)})^2 = I$ for all $i$} + \item{$S^{(i)}$ are hermitian for all $i$} + \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 $S$ ($G_n$ respectively) follows that any + $S^{(i)} \in S$ has the form $\pm i^l (\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$ where + $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$ + is hermitian $(S^{(i)})^2$ is either $+I$ or $-I$. As $-I \notin S$ $(S^{(i)})^2 = I$ follows directly. + } + \item{Following the argumentation above $(S^{(i)})^2 = -I \Leftrightarrow l=1$ + therefore $(S^{(i)})^2 = -I \Leftrightarrow (S^{(i)})^\dagger \neq (S^{(i)})$.} + \end{enumerate} + +\end{proof} + +As considering all elements of a group can be unpractical for some calculations +the generators of a group are introduced. It is usually enough to discuss the generator's +properties to understand the properties of the group. + +\begin{definition} + For a finite group $G$ and some $m \in \mathbb{N}$ one denotes the generators + of G + + $$ \langle g_1, ..., g_m \rangle \equiv \langle g_i \rangle_{i=1,...,m}$$ + + where $g_i \in G$, every element in $G$ can be written as a product of the $g_i$ + and $m$ is the smallest integer for which these statements hold. +\end{definition} + +In the following discussions $\rangle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as +the properties of a set of stabilizers that are used in the discussions can be studied using only its +generators. + +\subsubsection{Stabilizer States} + +One important basic property of quantum mechanics is that hermitian operators have real eigenvalues +and eigenspaces 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 operators that commute 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 + + \begin{equation} + V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = +1\ket{\psi} \forall S^{(i)} \in S\} + \end{equation} + + is called the space of stabilizer states associated with $S$ and one says + $\ket{\psi}$ is stabilized by $S$. +\end{definition} + +It is clear that it is sufficient to show the stabilization property for the generators of +$S$, as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$. +The dimension of $V_S$ is not immediately