daknuett/bachelor_thesis
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

a little work

Browse Source
This commit is contained in:
Daniel Knüttel 2019-11-11 11:22:47 +01:00
parent 65406633aa
commit 6cae03db82
1 changed files with 36 additions and 7 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

43
thesis/chapters/graph_simulator.tex
View File

@ -1,13 +1,9 @@
\section{The Graph Simulator}
\subsection{Introduction to the Graph Formalism}
\subsection{Mathematical Prerequisites}
The first step towards the simulation in the graph formalism has been
the discovery of the stabilizer states and stabilizer circuits \cite{gottesman2009}\cite{gottesman1997}.
They led to the faster simulation using stabilizer tableaux\cite{gottesman_aaronson2008} and later
to the graph formalism\cite{schlingenmann2001}\cite{andersbriegel2005}\cite{vandennest_ea2004}.
The following discussion eludicates the graph formalism and explains how the graph simulator works.
Some parts will be kept short as they can be looked up in \cite{andersbriegel2005}.
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}
@ -147,8 +143,40 @@ can be disregarded as discussed above $C_L$ will be used from now on instead of
Therefore $a$ will preserve the (anti-)commutator relations of $P$. Also $P$ is generated by $X,Z$ when disregarding a phase wich
does not matter for anticommutator relations.
This means that $X$ can be mapped to any $p \in P$ which are six elements disregarding
FIXME
\end{proof}
\subsection{Introduction to the Graph Formalism}
The first step towards the simulation in the graph formalism has been
the discovery of the stabilizer states and stabilizer circuits \cite{gottesman2009}\cite{gottesman1997}.
They led to the faster simulation using stabilizer tableaux\cite{gottesman_aaronson2008} and later
to the graph formalism\cite{schlingenmann2001}\cite{andersbriegel2005}\cite{vandennest_ea2004}.
The following discussion eludicates the graph formalism and explains how the graph simulator works.
Some parts will be kept short as they can be looked up in \cite{andersbriegel2005}.
A naive state is just a vector containing the coefficients $c_i$ as defined in \ref{ref:nqbitsystems}.
It is a quite straight forward approach and gates are applied by updating the coefficients according
to the gate's matrix representation. A naive state has the time and space complexity $\mathcal{O}(2^n)$ which limits the number
of qbits drastically. \\
The stabilizer tableaux represent the state by its stabilizers i.e. by those Pauli operators of which the
state is a $+1$ eigenstate. This has a space complexity of $\mathcal{O}(n^2)$ while updating the tableaux
has a time complexity of $\mathcal{O}(n)$ for unitary gates and $mathcal{O}(n^2)$ for measurements.
A graph state now represents the state by the gates that have been applied to it starting from the $\ket{+}$ state:
\begin{equation}
\ket{+} := \bigotimes\limits_{i=0}^{n-1} H_i \ket{0}
\end{equation}
A graph state $\ket{G}$ is a 3-tuple $(V, E, O)$ where $(V = \{0, ..., n-1\}, E)$ is a graph with the vertices $V$, edges $E$
and vertex operators $O = \{o_i | i = 0, ..., n-1; o_i \in C_L \forall i\}$. The vertex operators and edges are defined
by the following relation:
\begin{equation}
\ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+}
\end{equation}
\subsection{Graph Storage}
@ -160,6 +188,7 @@ The following
FIXME
\subsection{Usage}
FIXME