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added something about measurements

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Daniel Knüttel 2020-02-17 10:18:04 +01:00
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thesis/chapters/stabilizer.tex
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@ -205,6 +205,9 @@ efficient as only $n$ stabilizers have to be modified, each being just the tenso
product of $n$ Pauli matrices. This has led to the simulation using stabilizer tableaux
\cite{gottesman_aaronson2008}.
\subsubsection{Measurements on Stabilizer States}
\label{ref:meas_stab}
Interestingly also measurements are dynamics covered by the stabilizers.
When an observable $g_a \in \{\pm X_a, \pm Y_a \pm Z_a\}$ acting on qbit $a$ is measured
one has to consider the projector
@ -626,3 +629,34 @@ the graphical representation of a stabilizer state is indeed able to represent a
If one wants to do computations using this formalism it is however also necessary to perform measurements.
\subsubsection{Measurements on Graph States}
Recalling \ref{ref:meas_stab} it is clear that one has to compute the commutator of
the observable $g_a = Z_a$ with the stabilizers to get the probability amplitudes.
This is a quite expensive computation in theory however it is possible to simplify
the problem by pulling the observable behind the vertex operators. For this consider
the projector $P_a = \frac{I + (-1)^sZ_a}{2}$:
\begin{equation}
\begin{aligned}
P_a \ket{\psi} &= P_a \left(\prod\limits_{o_i \in O} o_i \right) \ket{\bar{G}} \\
&= \left(\prod\limits_{o_i \in O \setminus o_a}o_i \right)P_a o_a \ket{\bar{G}} \\
&= \left(\prod\limits_{o_i \in O \setminus o_a}o_i \right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
&= \left(\prod\limits_{o_i \in O} o_i \right) o_a^\dagger P_a o_a \ket{\bar{G}} \\
&= \left(\prod\limits_{o_i \in O} o_i \right) \tilde{P}_a \ket{\bar{G}} \\
\end{aligned}
\end{equation}
This transformed projector has the important property that it still is a Pauli projector
as $o_a$ is a Clifford operator:
\begin{equation}
\begin{aligned}
\tilde{P}_a &= o_a^\dagger P_a o_a \\
&= o_a^\dagger \frac{I + (-1)^sZ_a}{2} o_a \\
&= \frac{I + (-1)^s o_a^\dagger Z_a o_a}{2} \\
&= \frac{I + (-1)^s \tilde{g}_a}{2} \\
\end{aligned}
\end{equation}
Where $\tilde{g}_a \in \{+1, -1\}\cdot\{X_a, Y_a, Z_a\}$. It is therefore enough to study the measurements
of any Pauli operator on the vertex operator free graph states.