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some fixos

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This commit is contained in:
Daniel Knüttel 2020-03-11 10:48:51 +01:00
parent ba73d1a449
commit bbbac43c82
1 changed files with 16 additions and 10 deletions
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presentation/main.tex
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@ -486,10 +486,10 @@
To compute the probability to measure a result of $s=0$ one can use the trace formula
\begin{equation}
\begin{aligned}
P(s=0) &= \Tr(\frac{I + g_a}{2} \ket{\psi}\bra{\psi}) \\
&= \Tr(\frac{I + g_a}{2} S^{(j)} \ket{\psi}\bra{\psi}) \\
&= \Tr(S^{(j)}\frac{I - g_a}{2} \ket{\psi}\bra{\psi}) \\
&= \Tr(\frac{I - g_a}{2} \ket{\psi}\bra{\psi}) \\
P(s=0) &= \left|\Tr(\frac{I + g_a}{2} \ket{\psi}\bra{\psi})\right| \\
&= \left|\Tr(\frac{I + g_a}{2} S^{(j)} \ket{\psi}\bra{\psi})\right| \\
&= \left|\Tr(S^{(j)}\frac{I - g_a}{2} \ket{\psi}\bra{\psi})\right| \\
&= \left|\Tr(\frac{I - g_a}{2} \ket{\psi}\bra{\psi})\right| \\
&= P(s=1)\\
\end{aligned}
\end{equation}
@ -498,7 +498,6 @@
trace and absorbed into the $\bra{\psi}$.
}
\end{itemize}
\end{frame}
}
@ -773,19 +772,26 @@
\begin{equation}
\begin{aligned}
P_a \ket{G} &= \left(\prod\limits{i \neq a} o_i\right) P_a o_a \ket{\bar{G}} \\
&= \left(\prod\limits{i \neq a} o_i\right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
&= \left(\prod\limits{i} o_i \right) \tilde{P}_a \ket{\bar{G}}\\
P_a \ket{G} &= \left(\prod\limits_{i \neq a} o_i\right) P_a o_a \ket{\bar{G}} \\
&= \left(\prod\limits_{i \neq a} o_i\right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
&= \left(\prod\limits_{i} o_i \right) \tilde{P}_a \ket{\bar{G}}\\
\end{aligned}
\end{equation}
With a new projector $\tilde{P}_a = \frac{I + g_a}{2}$.
}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Measurements on Graph States}
\begin{itemize}
\item{The anticommuting stabilizers are given by
\begin{itemize}
\item{$A_X = \{K_G^{(i)} | i \in n_a\}$ for $g_a = \pm X_a$,}
\item{$A_Y = \{K_G^{(i)} | i \in n_a \cup a\}$ for $g_a = \pm Y_a$ and}
\item{$A_Z = \{K_G^{(a)}$ for $g_a = Z_a$.}
\item{$A_Y = \{K_G^{(i)} | i \in n_a \cup \{a\}\}$ for $g_a = \pm Y_a$ and}
\item{$A_Z = \{K_G^{(a)}\}$ for $g_a = \pm Z_a$.}
\end{itemize}}
\item{This can be used to compute the probability amplitudes and update $(G,V,E)$ after
the measurement.}