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added chapter about stabilizers

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105
thesis/chapters/graph_simulator.tex
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@ -5,27 +5,6 @@
The following definitions and lemmata are required to understand both how the
graph formalism works and how the simulator handles gates.
\begin{definition}
\begin{equation}
p \in P_n \Rightarrow p = \bigotimes\limits_{i=0}^n p_i \\
\forall i: p_i \in P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
\end{equation}
Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
$Y = \left(\begin{array}{cc} 0 & i \\ -i & 0\end{array}\right)$ and
$Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and
$I$ is the identity.
\end{definition}
\begin{definition}
Let $p_i \in P_n \forall i = 1, ..., n$, $[p_i, p_j] = 0 \forall i,j$ be commuting multi-local Pauli operators.
Then a $n$ qbit state $\ket{\psi}$ is called a stabilizer state iff
\begin{equation}
\forall i: p_i\ket{\psi} = +1\ket{\psi}
\end{equation}
\end{definition}
%A $n$ qbit graph or stabilizer state is a $+1$ eigenstate of some $ p \in P_n$ where $P_n$ is the Pauli group\cite{andersbriegel2005}.
\begin{definition}
@ -187,90 +166,6 @@ Where every $o_i$ acts on the $i$-th qbit.
One can show that any stabilizer state can be realized as a graph state (for instance in \cite{schlingenmann2001}).
\subsubsection{The Vertex Operator-Free Graph States}
In order to understand some essential transformations of graph states it is necessary
to study the vertex operator-free graph states first, partially because the graph states as used in this paper
were derived from the vertex operator-free graph states.
\begin{definition}
\label{def:vop_free_g_state}
A $n$ qbit vertex operator-free graph state $\ket{\overline{G}}$ is associated with a graph $(V, E)$
by the $n$ operators
\begin{equation}
K^{(i)}_G := X_i \left(\prod\limits_{\{i, j\} \in E} Z_j\right)
\end{equation}
for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit.
A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$.
\end{definition}
\begin{corrolary}
All $K^{(i)}_G$ commute and are hermitian. Therefore they have a common set of eigen states
(in particular definition \ref{def:vop_free_g_state} is well defined).
In terms of quantum mechanics $K^{(i)}_G$ are observables.
Further as $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$ which are
multi-local Pauli operators, $\{K^{(i)}_G | i \in \{0, ..., n-1\}\}$ is the stabilizer
of $\ket{\overline{G}}$ and $\ket{\overline{G}}$ is a stabilizer state.
\end{corrolary}
\begin{proof}
As $X_i$ and $Z_i$ are hermitian their product is hermitian.
Consider the case $\{i,j\} \notin E$ first:
\begin{equation}
\begin{aligned}
\left[K^{(i)}_G, K^{(j)}_G\right] = \left[X_i \prod\limits_{\{i, n\} \in E} Z_n, X_j \prod\limits_{\{j, m\} \in E} Z_m\right] = 0
\end{aligned}
\end{equation}
As operators acting on different qbits commute. The case $\{i,j\} \in E$ is slightly less trivial:
\begin{equation}
\begin{aligned}
\left[K^{(i)}_G, K^{(j)}_G\right] &= \left[X_i \left(\prod\limits_{\{i, n\} \in E, n \neq j} Z_n\right) Z_j, X_j \left(\prod\limits_{\{j, m\} \in E, m \neq i} Z_m\right) Z_i\right] \\
&= \left[X_i Z_j \prod\limits_n Z_n, X_j Z_i \prod\limits_m Z_m\right]\\
&= \left(X_i Z_j X_j Z_i - X_j Z_i X_i Z_j\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
&= \left(Z_j X_j X_i Z_i - X_j Z_j Z_i X_i\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
&= \left((-1)^2X_j Z_j Z_i X_i - X_j Z_j Z_i X_i\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
&= 0
\end{aligned}
\end{equation}
as $X$, $Z$ anticommute.
\end{proof}
\begin{lemma}
\begin{equation}
\ket{\overline{G}} = \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right) \left(\prod\limits_{l \in V} H_l\right) \ket{0}
\end{equation}
In particular definitions \ref{def:vop_free_g_state} and \ref{def:graph_state} are consistent, when there are no
vertex operators on the graph state $\ket{G}$.
\end{lemma}
\begin{proof}
Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before. Note that for any $X_i$ $X_i \ket{+} = +1 \ket{+}$.
Set $\ket{\tilde{G}} := \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right)\ket{+}$.
\begin{equation}
\begin{aligned}
K_G^{(i)} \ket{\tilde{G}} & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\left(\prod\limits_{\{l,j\} \in E} CZ_{l,j} \right) \ket{+} \\
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) X_i \ket{+} \\
& = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{2\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
& = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
& = +1 \ket{\tilde{G}}
\end{aligned}
\end{equation}
as $X, Z$ anticommute and $Z\ket{1} = -1\ket{1}$.
\end{proof}
These insights can be used to understand how measurement works on the vop-free graph state \cite{nielsen_chuang_2010}:
Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a hermitian $g$ that is to be measured.

137
thesis/chapters/stabilizer.tex Normal file
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@ -0,0 +1,137 @@
\section{The Stabilizer Formalism and VOP-Free Graph States}
\subsection{Stabilizers and Stabilizer States}
This chapter discusses the stabilizer formalism that was introduced by Gottesman\cite{gottesman1997}
for quantum error correction but soon proved to be a useful tool to describe a subset of states:
the stabilizer states which can be simulated in polynomial time \cite{gottesman2008}.
\begin{definition}
\begin{equation}
p \in P_n \Rightarrow p = \bigotimes\limits_{i=0}^n p_i \\
\forall i: p_i \in P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
\end{equation}
Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
$Y = \left(\begin{array}{cc} 0 & i \\ -i & 0\end{array}\right)$ and
$Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and
$I$ is the identity.
$p \in = P_n$ is called a multi-local Pauli operator.
\end{definition}
\begin{definition}
For a group $G$, $g_1, ..., g_n$ are called the generators of $G$ iff $\forall g \in G: g = \prod\limits_{i \in I} g_i$ for a
subsed $I$ of $\{1, ..., n\}$. We write $G = \langle g_i \rangle_i$ if G is generated by the $g_i$. The generators $g_i$ are chosen
to be the smallest set of generators of $G$.
\end{definition}
\begin{definition}
\label{def:stabilizer}
For a $n$ qbit state $\ket{\psi}$ $\langle S_i \rangle_i$ is called the stabilizer of $\ket{\psi}$ if
\begin{enumerate}
\item{$\forall i = 1, ..., n$ $S_i \in P_n$}
\item{$\forall i,j = 1, ..., n$ $[S_i, S_j] = 0$ $S_i$ and $S_j$ commute}
\item{$\forall i = 1, ..., n$ $S_i\ket{\psi} = +1 \ket{\psi}$}
\end{enumerate}
\end{definition}
\begin{lemma}
For every $\langle S_i \rangle_i$ fulfilling the first two conditions in definition \ref{def:stabilizer} there exists
a (up to a global phase) unique state $\ket{\psi}$ fulfilling the third condition.
\end{lemma}
\begin{proof}
All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as they
commute they have a common set of eigenstates. Because each $S_i$ has eigenvalues $+1, -1$, there
exist $2^n$ eigenstates, one state $\ket{\psi}$ with eigenvalue $+1$ for all $S_i$. As the dimension of $n$ qbits is $2^n$
the state $\ket{psi}$ is unique up to a global phase.
\end{proof}
\subsection{The Vertex Operator-Free Graph States}
In order to understand some essential transformations of graph states it is necessary
to study the vertex operator-free graph states first, partially because the graph states as used in this paper
were derived from the vertex operator-free graph states.
\begin{definition}
\label{def:vop_free_g_state}
A $n$ qbit vertex operator-free graph state $\ket{\overline{G}}$ is associated with a graph $(V, E)$
by the $n$ operators
\begin{equation}
K^{(i)}_G := X_i \left(\prod\limits_{\{i, j\} \in E} Z_j\right)
\end{equation}
for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit.
A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$.
\end{definition}
\begin{corrolary}
All $K^{(i)}_G$ commute and are hermitian. Therefore they have a common set of eigenstates
(in particular definition \ref{def:vop_free_g_state} is well defined).
In terms of quantum mechanics $K^{(i)}_G$ are observables.
Further as $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$ which are
multi-local Pauli operators, $\{K^{(i)}_G | i \in \{0, ..., n-1\}\}$ is the stabilizer
of $\ket{\overline{G}}$ and $\ket{\overline{G}}$ is a stabilizer state.
\end{corrolary}
\begin{proof}
As $X_i$ and $Z_i$ are hermitian their product is hermitian.
Consider the case $\{i,j\} \notin E$ first:
\begin{equation}
\begin{aligned}
\left[K^{(i)}_G, K^{(j)}_G\right] = \left[X_i \prod\limits_{\{i, n\} \in E} Z_n, X_j \prod\limits_{\{j, m\} \in E} Z_m\right] = 0
\end{aligned}
\end{equation}
As operators acting on different qbits commute. The case $\{i,j\} \in E$ is slightly less trivial:
\begin{equation}
\begin{aligned}
\left[K^{(i)}_G, K^{(j)}_G\right] &= \left[X_i \left(\prod\limits_{\{i, n\} \in E, n \neq j} Z_n\right) Z_j, X_j \left(\prod\limits_{\{j, m\} \in E, m \neq i} Z_m\right) Z_i\right] \\
&= \left[X_i Z_j \prod\limits_n Z_n, X_j Z_i \prod\limits_m Z_m\right]\\
&= \left(X_i Z_j X_j Z_i - X_j Z_i X_i Z_j\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
&= \left(Z_j X_j X_i Z_i - X_j Z_j Z_i X_i\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
&= \left((-1)^2X_j Z_j Z_i X_i - X_j Z_j Z_i X_i\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
&= 0
\end{aligned}
\end{equation}
as $X$, $Z$ anticommute.
\end{proof}
\begin{lemma}
\begin{equation}
\ket{\overline{G}} = \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right) \left(\prod\limits_{l \in V} H_l\right) \ket{0}
\end{equation}
In particular definitions \ref{def:vop_free_g_state} and \ref{def:graph_state} are consistent, when there are no
vertex operators on the graph state $\ket{G}$.
\end{lemma}
\begin{proof}
Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before. Note that for any $X_i$ $X_i \ket{+} = +1 \ket{+}$.
Set $\ket{\tilde{G}} := \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right)\ket{+}$.
\begin{equation}
\begin{aligned}
K_G^{(i)} \ket{\tilde{G}} & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\left(\prod\limits_{\{l,j\} \in E} CZ_{l,j} \right) \ket{+} \\
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) X_i \ket{+} \\
& = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{2\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
& = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
& = +1 \ket{\tilde{G}}
\end{aligned}
\end{equation}
as $X, Z$ anticommute and $Z\ket{1} = -1\ket{1}$.
\end{proof}
These insights can be used to understand how measurement works on the vop-free graph state \cite{nielsen_chuang_2010}:
Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a hermitian $g$ that is to be measured.

9
thesis/main.bib
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@ -99,3 +99,12 @@
publisher={CAMBRIDGE UNIVERSITY PRESS},
note={www.cambridge.org/9781107002173}
}
@article{
gottesman2008,
title={The Heisenberg Representation of Quantum Computers},
year=2008,
author={Daniel Gottesman},
note={https://arxiv.org/abs/quant-ph/9807006}
}

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thesis/main.tex
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@ -48,6 +48,8 @@ Simulator with a Focus on Simulation in the Graph Formalism }
\include{chapters/naive_simulator}
\include{chapters/stabilizer}
\include{chapters/graph_simulator}
\section{Appendix}