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\begin{frame}{Stabilizers and Stabilizer Spaces}
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\begin{frame}{Stabilizers and Stabilizer Spaces}
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\begin{itemize}
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\begin{itemize}
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\item{
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\item{
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Choose a finite Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
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Choose a finite commuting Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
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One can show that all elements of $S$ are hermitian.
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One can show that all elements of $S$ are hermitian.
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}
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}
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\item{
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\item{
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}
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}
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\item{If the vertices $a, b$ are isolated the resulting state after
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\item{If the vertices $a, b$ are isolated the resulting state after
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applying $CZ_{a,b}$ can be precomputed. }
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applying $CZ_{a,b}$ can be precomputed. }
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\item{When none of the above is possible one can clear at least one
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Dynamics of Graphical States}
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\begin{itemize}
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\item{When the VOPs do not commute with $CZ$ and the vertices are not isolated one can clear at least one
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vertex operator (i.e. transforming it to $I$).}
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vertex operator (i.e. transforming it to $I$).}
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\item{If both vertex operators can be cleared $\{a,b\}$ is toggled in $E$.
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\item{If both vertex operators can be cleared $\{a,b\}$ is toggled in $E$.
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}
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\item{
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If just one vertex operator has been cleared the other vertex is isolated
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If just one vertex operator has been cleared the other vertex is isolated
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and one can precompute all resulting states.}
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and one can precompute all resulting states.}
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\end{itemize}
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\end{itemize}
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