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\section { The Graph Simulator}
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\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
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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} .
\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}
\begin { equation}
C_ n := \{ U \in SU(2) | UpU^ \dagger \in P_ n \forall p \in P_ n\}
\end { equation}
is called the Clifford group on $ n $ qbits.
$ C _ 1 = : C _ L $ is called the local Clifford group.
\end { definition}
One can show that the Clifford group $ C _ n $ can be generated using the elements of $ C _ L $ acting on all qbits and
the controlled phase gate $ CZ $ between all qbits\cite { andersbriegel2005} . It is worth noting that the $ CX $ gate can be
generated using $ CZ $ and $ C _ L $ gates.
\begin { lemma}
Let $ a \in C _ L $ then $ \forall \phi \in [ 0 , 2 \pi ) $ also $ \exp ( i \phi ) a \in C _ L $ .
\textbf { Note} : This is also true for $ C _ n \forall n > = 1 $ .
\end { lemma}
\begin { proof}
Let $ a' : = \exp ( i \phi ) a $ . $ a' \in C _ L $ iff $ a'pa ^ { \prime \dagger } \in P \forall p \in P $ .
\begin { equation}
\begin { aligned}
a'pa^ { \prime \dagger } & = (\exp (i\phi )a)p(\exp (i\phi )a)^ \dagger \\
& = \exp (i\phi )ap\exp (-i\phi )a^ \dagger \\
& = \exp (i\phi )\exp (-i\phi ) apa^ \dagger \\
& = apa^ \dagger \in P
\end { aligned}
\end { equation}
\end { proof}
\begin { lemma}
One cannot measure phases by projecting states.
\end { lemma}
\begin { proof}
Let $ \ket { \psi } $ be a state, $ \ket { \varphi } \bra { \varphi } $ a projector. $ \ket { \psi ' } : = \exp ( i \phi ) \ket { \psi } $ for some $ \phi \in [ 0 , 2 \pi ) $ .
\begin { equation}
\begin { aligned}
\bra { \psi '} \ket { \varphi } \bra { \varphi } \ket { \psi '} & = \exp (-i\phi )\bra { \psi } \ket { \varphi } \bra { \varphi } \exp (i\phi )\ket { \psi } \\
& = \exp (-i\phi )\exp (i\phi )\bra { \psi } \ket { \varphi } \bra { \varphi } \ket { \psi } \\
& = \bra { \psi } \ket { \varphi } \bra { \varphi } \ket { \psi }
\end { aligned}
\end { equation}
\end { proof}
\begin { definition}
A phase $ \phi \in [ 0 , 2 \pi ) $ is called qbit-global, if for some qbit states $ \ket { \psi } , \ket { \varphi } $ $ \ket { \psi } = \exp ( i \phi ) \ket { \varphi } $ .
\end { definition}
\begin { lemma}
When entangling qbits via projections one can disregard qbit-global phases.
Two qbits are entangled via projection, if for some single qbit gates $ M,N $
and two orthonormal states $ \ket { a } , \ket { b } $
\begin { equation}
C^ { M,N} (i,j) = \ket { a} \bra { a} _ j \otimes M_ i + \ket { b} \bra { b} _ j \otimes N_ i
\end { equation}
\textbf { Remark.}
In particular when entangling states using $ CX $ and $ CZ $ one can disregard qbit-global phases.
This is immideatly clear when recalling \eqref { eq:CX_ pr} and \eqref { eq:CZ_ pr} .
\end { lemma}
\begin { proof}
Let $ \alpha , \beta \in [ 0 , 2 \pi ) $ be some phases, $ \ket { \psi } , \ket { \varphi } , \ket { \psi ' } : = \exp ( i \alpha ) \ket { \psi } , \ket { \varphi ' } : = \exp ( i \beta ) \ket { \varphi } $ some single qbit states,
$ M, N, \ket { a } , \ket { b } , C ^ { M,N } ( i,j ) $ as defined above.
\begin { equation}
\begin { aligned}
C^ { M,N} (1, 0) (\ket { \psi '} \otimes \ket { \varphi '} ) & = \ket { a} \braket { a} { \varphi '} \otimes M\ket { \psi '} + \ket { b} \braket { b} { \varphi '} \otimes N\ket { \psi '} \\
& = \exp (i\beta )\ket { a} \braket { a} { \varphi } \otimes \exp (i\alpha )M\ket { \phi } + \exp (i\beta )\ket { b} \braket { b} { \varphi } \otimes \exp (i\alpha )N\ket { \phi } \\
& = \exp (i(\beta + \alpha ))(\ket { a} \braket { a} { \varphi } \otimes M\ket { \psi } + \ket { b} \braket { b} { \varphi } \otimes N\ket { \psi } )\\
& = \exp (i(\beta + \alpha ))C^ { M,N} (1, 0) (\ket { \psi } \otimes \ket { \varphi } )
\end { aligned}
\end { equation}
Where $ \exp ( i ( \beta + \alpha ) ) $ is a multi-qbit-global phase which can be (following the above argumentation) disregarded.
\end { proof}
\begin { corrolary}
One can disregard global phases of elements of the $ C _ L $ group.
\end { corrolary}
\begin { proof}
As it has been shown above a quantum computer cannot measure global phases. Also
the entanglement gates $ CX, CZ $ map qbit-global phases to multi-qbit-global phases which cannot
be measured. It has been shown above that one can choose the $ C _ L $ operators such that they do not yield
a phase.
\end { proof}
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\subsection { Graph Storage}
One of the gread advantages of simulating in the graph formalism is a great increase
in simulation performance and a lower memory requirement. The simulation of
at least $ 10 ^ 6 $ qbits on a common desktop computer should be possible\cite { andersbriegel2005} .
Therefore one has to take care when choosing a representation of the graph state.
The following
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\subsection { Usage}
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\subsection { Performance}
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