From 1c4430dd580b1b1292c4c43ff77ca4f8bb9e1337 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Tue, 18 Feb 2020 13:27:37 +0100 Subject: [PATCH] basically finished the theory on the stabilizer formalism --- thesis/chapters/stabilizer.tex | 56 ++++++++++++++++++++++++++++++---- thesis/main.bib | 8 +++++ 2 files changed, 58 insertions(+), 6 deletions(-) diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 8763c5d..36aec57 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -634,15 +634,15 @@ Recalling \ref{ref:meas_stab} it is clear that one has to compute the commutator 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 $P_a = \frac{I + (-1)^sZ_a}{2}$: +the projector $P_{a,s} = \frac{I + (-1)^sZ_a}{2}$: \begin{equation} \begin{aligned} - P_a \ket{\psi} &= P_a \left(\prod\limits_{o_i \in O} o_i \right) \ket{\bar{G}} \\ + P_{a,s} \ket{\psi} &= P_{a,s} \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}} \\ + &= \left(\prod\limits_{o_i \in O} o_i \right) \tilde{P}_{a,s} \ket{\bar{G}} \\ \end{aligned} \end{equation} @@ -651,7 +651,7 @@ as $o_a$ is a Clifford operator: \begin{equation} \begin{aligned} - \tilde{P}_a &= o_a^\dagger P_a o_a \\ + \tilde{P}_{a,s} &= 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} \\ @@ -671,6 +671,50 @@ so it is easier to list the operators that anticommute: \end{aligned} \end{equation} -This gives one immediate result: if a qbit $a$ is isolated and the operator $\tilde{g}_a = X_a$ -is measured the result $+1$ is obtained with probability $1$ and $(V, E, O)$ is unchanged. +This gives one immediate result: if a qbit $a$ is isolated and the operator $\tilde{g}_a = X_a (-X_a)$ +is measured the result $s=0(1)$ is obtained with probability $1$ and $(V, E, O)$ is unchanged. +In any other case the results $s=1$ and $s=0$ both have probability $\frac{1}{2}$ and both +graph and vertex operators have to be updated. Further it is clear that measurement of $-\tilde{g}_a$ +and $\tilde{g}_a$ are related by just inverting the result $s$. + +The calculations to obtain the transformation on graph and vertex operators are lengthy and follow +the scheme of Lemma \ref{lemma:M_a}. \cite[Section IV]{hein_eisert_briegel2008} also contains +the steps required to obtain the following results: + +\begin{equation} + \begin{aligned} + U_{Z,s} &= \left(\prod\limits_{b \in n_a} Z_b^s\right) X_a^s H_a \\ + U_{Y,s} &= \prod\limits_{b \in n_a} \sqrt{(-1)^{1-s} iZ_b} \sqrt{(-1)^{1-s} iZ_a}\\ + \end{aligned} +\end{equation} + +These transformations split it two parts: the first is a result of Lemma \ref{lemma:stab_measurement} +and the second part makes sure that the qbit $a$ is diagonal in the correct state of the measured state. +When comparing with Lemma \ref{lemma:stab_measurement} in both cases $Y,Z$ the anticommuting stabilizer +$K_G^{(a)}$ is chosen. The graph is changed according to + +\begin{equation} + \begin{aligned} + E'_{Z} &= E \setminus \left\{\{i,a\} | i \in V\right\}\\ + E'_{Y} &= E \Delta (n_a \otimes n_a) \setminus \left\{\{i,a\} | i \in V\right\}\\ + \end{aligned} +\end{equation} + + +For $g_a = X_a$ one has to choose a $b \in n_a$ and the transformations are + +\begin{equation} + \begin{aligned} + U_{X,0} &= \sqrt{iY_b} \prod\limits_{c \in n_a \setminus n_b \setminus \{b\}} Z_c \\ + U_{X,1} &= \sqrt{-iY_b} \prod\limits_{c \in n_b \setminus n_a \setminus \{a\}} Z_c \\ + \end{aligned} +\end{equation} +\begin{equation} + \begin{aligned} + E'_{X} = E &\Delta (n_b \otimes n_a) \\ + & \Delta ((n_b \cap n_a) \otimes (n_b \cap n_a)) \\ + & \Delta (\{b\} \otimes (n_a \setminus \{b\})) \\ + & \setminus \left\{\{i,a\} | i \in V\right\}\\ + \end{aligned} +\end{equation} diff --git a/thesis/main.bib b/thesis/main.bib index cc3f5b0..afacd94 100644 --- a/thesis/main.bib +++ b/thesis/main.bib @@ -139,3 +139,11 @@ year=2020, note={https://github.com/daknuett/pyqcs}, } + +@article{ + hein_eisert_briegel2008, + title={Multi-party entanglement in graph states}, + year=2008, + author={M. Hein, J. Eisert, H.J. Briegel}, + note={https://arxiv.org/abs/quant-ph/0307130v7} +}