bachelor_thesis/thesis/chapters/stabilizer.tex
2020-01-29 11:19:13 +01:00

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\section{The Stabilizer Formalism}
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 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}.
\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 $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ 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 clear. One can however 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:
\begin{theorem}
For a $n$ qbit system and a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
space $V_S$ has $dim V_S = 1$, in particular there exists an up to a trivial phase unique
state $\ket{\psi}$ that is stabilized by $S$.
Without proof.
\end{theorem}
In the following discussions for $n$ qbits a set $S = \langle S^{(i)} \rangle_{i=1,...,n}$
of $n$ independent stabilizers will be assumed.
\subsubsection{Dynamics of Stabilizer States}
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 still be made:
\begin{equation}
\begin{aligned}
\ket{\psi'} &= U \ket{\psi} \\
&= U S^{(i)} \ket{\psi} \\
&= U S^{(i)} U^\dagger U\ket{\psi} \\
&= U S^{(i)} U^\dagger \ket{\psi'} \\
&= S^{\prime(i)} \ket{\psi'} \\
\end{aligned}
\end{equation}
Note that in \ref{def:stabilizer} it has been demanded that stabilizers are a
subgroup of the multilocal Pauli operators. This does not hold true for an arbitrary
$U$ but there exists a group for which $S'$ will be a set of stabilizers.
\begin{definition}
For $n$ qbits
\begin{equation}
C_n := \left\{U \in SU(n) | UpU^\dagger \in P_n \forall p \in P_n\right\}
\end{equation}
is called the Clifford group. $C_1 =: C_L$ is called the local Clifford group.
\end{definition}
\begin{theorem}
\begin{enumerate}
\item{$C_L$ can be generated using only $H$ and $S$.}
\item{$C_L$ can be generated from $\sqrt{iZ} = \exp(\frac{i\pi}{4}) S^\dagger$
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 $\sqrt{iZ}$, $\sqrt{-iX}$.
}
\item{$C_n$ can be generated using $C_L$ and $CZ$ or $CX$.}
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}
\item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
\item{
One can easily verify that $\sqrt{iZ} \in C_L$ and $\sqrt{-iX} \in C_L$.
Further one can show easily that (up to a global phase)
$H = \sqrt{iZ} \sqrt{-iX}^3 \sqrt{iZ}$ and $S = \sqrt{iZ}^3$.
The length of the product can be seen when explicitly calculating
$C_L$.
}
\item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
\end{enumerate}
\end{proof}
This is quite an important result: As under a transformation $U \in C_n$ $S'$ 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
\cite{gottesman_aaronson2008}.
Interestingly also measurements are dynamics covered by the stabilizers.
When an observable $g_i \in \{\pm X_i, \pm Y_i \pm Z_i\}$ is measured