121 lines
4.7 KiB
TeX
121 lines
4.7 KiB
TeX
% vim: ft=tex
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\section{The Stabilizer Formalism}
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The stabilizer formalism was originally introduced by Gottesman\cite{gottesman1997}
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for quantum error correction and is a useful tool to encode quantum information
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such that it is protected against noise. The prominent Shor code \cite{shor1995}
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is an example of a stabilizer code (although it was discovered before the stabilizer
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formalism was discovered), as are the 3-qbit bit-flip and phase-flip codes.
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It was only later that Gottesman and Knill discovered that stabilizer states can
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be simulated in polynomial time on a classical machine \cite{gottesman2008}. This
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performance has since been improved to $n\log(n)$ time on average \cite{andersbriegel2005}.
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\subsection{Stabilizers and Stabilizer States}
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\subsubsection{Local Pauli Group and Multilocal Pauli Group}
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\begin{definition}
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\begin{equation}
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P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
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\end{equation}
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Is called the Pauli group.
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\end{definition}
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The group property of $P$ can be verified easily. Note that
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the elements of $P$ either commute or anticommute.
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\begin{definition}
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For $n$ qbits
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\begin{equation}
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P_n := \{\bigotimes\limits_{i=0}^{n-1} p_i | p_i \in P\}
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\end{equation}
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is called the multilocal Pauli group on $n$ qbits.
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\end{definition}
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The group property of $P_n$ follows directly from its definition
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via the tensor product as do the (anti-)commutator relationships.
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%Further are $p \in P_n$ hermitian and have the eigenvalues $\pm 1$ for
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%$p \neq \pm I$, $+1$ for $p = I$ and $-1$ for $p = -I$.
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\subsubsection{Stabilizers}
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\begin{definition}
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\label{def:stabilizer}
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An abelian subgroup $S = \{S^{(0)}, ..., S^{(N)}\}$ of $P_n$ is called a set of stabilizers iff
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\begin{enumerate}
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\item{$\forall i,j = 1, ..., N$ $[S^{(i)}, S^{(j)}] = 0$ $S^{(i)}$ and $S^{(j)}$ commute}
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\item{$-I \notin S$}
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\end{enumerate}
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\end{definition}
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\begin{lemma}
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If $S$ is a set of stabilizers, the following statements are follow
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directly
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\begin{enumerate}
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\item{$\pm iI \notin S$}
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\item{$(S^{(i)})^2 = I$ for all $i$}
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\item{$S^{(i)}$ are hermitian for all $i$}
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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\begin{enumerate}
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\item{$(iI)^2 = (-iI)^2 = -I$. Which contradicts the definition of $S$.}
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\item{From the definition of $S$ ($G_n$ respectively) follows that any
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$S^{(i)} \in S$ has the form $\pm i^l (\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$ where
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$\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$
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is hermitian $(S^{(i)})^2$ is either $+I$ or $-I$. As $-I \notin S$ $(S^{(i)})^2 = I$ follows directly.
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}
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\item{Following the argumentation above $(S^{(i)})^2 = -I \Leftrightarrow l=1$
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therefore $(S^{(i)})^2 = -I \Leftrightarrow (S^{(i)})^\dagger \neq (S^{(i)})$.}
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\end{enumerate}
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\end{proof}
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As considering all elements of a group can be unpractical for some calculations
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the generators of a group are introduced. It is usually enough to discuss the generator's
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properties to understand the properties of the group.
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\begin{definition}
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For a finite group $G$ and some $m \in \mathbb{N}$ one denotes the generators
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of G
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$$ \langle g_1, ..., g_m \rangle \equiv \langle g_i \rangle_{i=1,...,m}$$
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where $g_i \in G$, every element in $G$ can be written as a product of the $g_i$
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and $m$ is the smallest integer for which these statements hold.
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\end{definition}
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In the following discussions $\rangle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as
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the properties of a set of stabilizers that are used in the discussions can be studied using only its
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generators.
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\subsubsection{Stabilizer States}
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One important basic property of quantum mechanics is that hermitian operators have real eigenvalues
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and eigenspaces associated with these eigenvalues. Finding these eigenvalues and eigenvectors
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is what one calls solving a quantum mechanical system. One of the most fundamental insights of
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quantum mechanics is that operators that commute have a common set of eigenvectors, i.e. they
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can be diagonalized simultaneously. This motivates and justifies the following definition
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\begin{definition}
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For a set of stabilizers $S$ the vector space
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\begin{equation}
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V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = +1\ket{\psi} \forall S^{(i)} \in S\}
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\end{equation}
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is called the space of stabilizer states associated with $S$ and one says
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$\ket{\psi}$ is stabilized by $S$.
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\end{definition}
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It is clear that it is sufficient to show the stabilization property for the generators of
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$S$, as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$.
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The dimension of $V_S$ is not immediately
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