diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 4587a1d..e6f7dbc 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -311,14 +311,31 @@ graph from graph theory that are implied in this definition. $\langle K_G^{(i)} \rangle_{i = 0, ..., n-1}$ \cite{hein_eisert_briegel2008}. \end{definition} -It is clear that the $K_G^{(i)}$ are multilocal Pauli operators. That they commute -follows from the fact that $\{i,j\} \in E \Leftrightarrow \{j,i\} \in E$ therefore for two -operators $K_G^{(i)}$ and $K_G^{(j)}$ either $\{i, j\} \notin E$ so they commute trivially. -If $\{i,j\} \in E$ $X_i$, $Z_j$ and $X_j$, $Z_i$ anticommute which yields that the -operators commute. +It is clear that the $K_G^{(i)}$ are multilocal Pauli operators. That they commute +is clear if $\{a,b\} \notin E$. If $\{a, b\} \in E$ + +\begin{equation} +\begin{aligned} + K_G^{(a)} K_G^{(b)} &= X_a \left(\prod\limits_{i \in n_a} Z_i\right) + X_b \left(\prod\limits_{j\in n_b} Z_j\right)\\ + &= X_a \left(\prod\limits_{i \in \setminus \{b\}} Z_i\right) Z_b + X_b \left(\prod\limits_{j\in n_b\setminus \{b\}} Z_j\right) Z_a\\ + &= X_a Z_b X_b Z_a + \left(\prod\limits_{j\in n_b\setminus \{b\}} Z_j\right) + \left(\prod\limits_{i \in \setminus \{b\}} Z_i\right)\\ + &= -X_b Z_b X_a Z_a + \left(\prod\limits_{j\in n_b\setminus \{b\}} Z_j\right) + \left(\prod\limits_{i \in \setminus \{b\}} Z_i\right)\\ + &= X_b Z_a X_a Z_b + \left(\prod\limits_{j\in n_b\setminus \{b\}} Z_j\right) + \left(\prod\limits_{i \in \setminus \{b\}} Z_i\right)\\ + &= K_G^{(b)} K_G^{(a)}.\\ +\end{aligned} +\end{equation} + This definition of a graph state might not seem to be straight forward -but recalling theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$ +but recalling Theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$ is unique. The following lemma will provide a way to construct this state from the graph. @@ -413,7 +430,8 @@ that will be used later\cite{andersbriegel2005}. \begin{proof} $\ket{\bar{G}'}$ is stabilized by $\langle M_a K_G^{(i)} M_a^\dagger \rangle_i$, so it is sufficient to study how the $ K_G^{(i)}$ change under $M_a$. - At first note that $[K_G^{(a)}, M_a] = 0$ and $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$, + At first note that $M_a^2 \alpha K_G^{(a)} \Rightarrow [K_G^{(a)}, M_a] = 0$. + Further $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$, so the first two equations follow trivially. For $j \in n_a$ set \begin{equation} @@ -435,19 +453,20 @@ that will be used later\cite{andersbriegel2005}. \end{aligned} \end{equation} - One can now construct a new set of $K_{G'}^{(i)}$ s.t. $M_a \ket{\bar{G}}$ is the $+1$ eigenstate - of the new $K_{G'}^{(i)}$. It is clear that $\forall j \notin n_a$ $K_{G'}^{(j)} = K_G^{(j)}$. - To construct the $K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a = \{j\} \cup I$ and $n_j = \{a\} \cup J$. + One can now construct a new set of $K_{G'}^{(i)}$ such that $M_a \ket{\bar{G}}$ is the $+1$ eigenstate + of the new $K_{G'}^{(i)}$. Because $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$ + it is clear that $\forall j \notin n_a$ $K_{G'}^{(j)} = K_G^{(j)}$. + To construct the $K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a =: \{j\} \cup F$ and $n_j =: \{a\} \cup D$. Then follows: \begin{equation} \begin{aligned} - S^{(j)} &= Z_j X_a X_j Z_a \prod\limits_{l \in J} Z_l\\ - &= Z_j X_a X_j Z_a \left(\prod\limits_{l \in J} Z_l\right) - \left(\prod\limits_{l \in I}Z_l\right) - \left(\prod\limits_{l \in I}Z_l\right) \\ - &= Z_j X_a X_j Z_a \left(\prod\limits_{l \in ((I\cup J) \setminus (I\cap J))} Z_L\right) - \left(\prod\limits_{l \in I}Z_l\right) \\ + S^{(j)} &= Z_j X_a X_j Z_a \prod\limits_{l \in D} Z_l\\ + &= Z_j X_a X_j Z_a \left(\prod\limits_{l \in D} Z_l\right) + \left(\prod\limits_{l \in F}Z_l\right) + \left(\prod\limits_{l \in F}Z_l\right) \\ + &= Z_j X_a X_j Z_a \left(\prod\limits_{l \in ((F\cup D) \setminus (F\cap D))} Z_L\right) + \left(\prod\limits_{l \in F}Z_l\right) \\ &= K_{G'}^{(a)} K_{G'}^{(j)} \\ &= K_{G}^{(a)} K_{G'}^{(j)} \end{aligned} @@ -477,7 +496,7 @@ The answer is straight forward: No. The most simple cases are the single qbit stated $\ket{0},\ket{1}$ and $\ket{+_Y}, \ket{-_Y}$. But there is an extension to the VOP-free graph states that allows the representation of an arbitrary stabilizer state. The proof that indeed any state can be represented is -purely constructive. As seen in theorem \ref{thm:clifford_group_approx} any $c \in C_n$ +purely constructive. As seen in Theorem \ref{thm:clifford_group_approx} any $c \in C_n$ can be constructed from $CZ$ and $C_L$. In the following discussion it will become clear that both $C_L$ and $CZ$ can be applied to a general graph state. @@ -544,10 +563,10 @@ $C_L = \langle \sqrt{-iX}, \sqrt{-iZ}\rangle$ which is required in one specific So far the graphical representation of stabilizer states is just another way to store basically a stabilizer tableaux that might require less memory than the tableaux used in -CHP. The true power of this formalism is seen when studying its dynamics. The simplest case -is a local Clifford operator $c_j$ acting on a qbit $j$: The stabilizers of are changed to +CHP\cite{CHP}. The true power of this formalism is seen when studying its dynamics. The simplest case +is a local Clifford operator $c_j$ acting on a qbit $j$: The stabilizers are changed to $\langle c_j S^{(i)} c_j^\dagger\rangle_i$. Using the definition of the graphical representation -it is clear that just the vertex operators are changed and the new vertex operators are given by +one sees that just the vertex operators are changed and the new vertex operators are given by \begin{equation} \begin{aligned} @@ -570,11 +589,12 @@ In this case the CZ can be pulled past the vertex operators and just the edges are changed to $E' = E \Delta \left\{\{a,b\}\right\}$. \textbf{Case 2}(\textit{Sub-Sub-Case 2.2.1})\textbf{:}\\ -The two qbits are isolated: From the definition of the graph state it is clear that +The two qbits are isolated: From the definition of the graph state one can derive that any isolated clique of the graph can be treated independently. Therefore the two isolated qbits can be treated as an independent state and the set of two qbit stabilizer states is finite. An -upper bound to the number of two qbit stabilizer states is given by $2\cdot24^2$: With and without -an edge between the qbits and $24$ Clifford operators on each vertex. +upper bound to the number of two qbit stabilizer states is given by $2\cdot24^2$: +A factor of two for the options with and withoit an edge between the +qbits and $24$ Clifford operators on each vertex. All those states and the resulting state after a $CZ$ application can be computed which leads to another interesting result that will be useful later: If one vertex has the vertex operator $I$ the @@ -582,9 +602,10 @@ resulting state can be chosen such that at least one of the vertex operators is the identity on the vertex can be preserved under the application of a $CZ$. \textbf{Case 3}(\textit{Case 2, but one sub-case has been handled})\textbf{:}\\ -At least one vertex operator does not commute with $CZ$ and at least one vertex -has non-operand neighbours. In this case one can try to clear the vertex operators -which will succeed for at least one qbit. +At least one vertex operator does not commute with $CZ$ and at least one vertex +has non-operand (i.e. neighbours that are neither $a$ nor $b$) neighbours. In +this case one can try to clear the vertex operators which will succeed for at +least one qbit. The transformation given in Lemma \ref{lemma:M_a} is used to "clear" the vertex operators. Recalling that @@ -605,7 +626,7 @@ to the following theorem: As stated in \ref{ref:g_states_vops} $C_L$ is also generated by $\sqrt{-iX}$ and $\sqrt{iZ}$. This yields an algorithm to reduce the vertex operator of a non-isolated qbit $j$ to the identity. The combined operation of toggling the neighbourhood of $j$ and right-multiplying -$M_j^\dagger$ is called $L_j$ transformation and it transforms $(V, E, O)$ into a so-called +$M_j^\dagger$ is called $L_j$ transformation, which transforms $(V, E, O)$ into a so-called local Clifford equivalent graphical representation. The algorithm is given by the following steps: \begin{enumerate} @@ -622,19 +643,22 @@ local Clifford equivalent graphical representation. The algorithm is given by th } \end{enumerate} -This algorithm has the important properties that if the algorithm succeeds -$o_a = I$ and $o_b$ has picked up powers of $\sqrt{iZ}^\dagger$. -If $b$ has non-operand neighbours after clearing the vertex operators on $a$ the vertex operators on $b$ -can be cleared using the same algorithm which gives $o_b = I$ and $o_a = (\sqrt{iZ}^\dagger)^l = S^l \in \mathcal{Z}$ -for some $l \in \{1, ..., 5\}$. Therefore Case 1 can now be applied. +This algorithm has the important properties that if the algorithm succeeds $o_a += I$ and $o_b$ has picked up powers of $\sqrt{iZ}^\dagger$. If $b$ has +non-operand neighbours after clearing the vertex operators on $a$, then the +vertex operators on $b$ can be cleared using the same algorithm which gives +$o_b = I$ and $o_a = (\sqrt{iZ}^\dagger)^l = S^l \in \mathcal{Z}$ for some $l +\in \{1, ..., 5\}$. Therefore Case 1 can now be applied. -If $o_a$ could not be cleared $o_b$ can be cleared using the same procedure and after -clearing $o_b$ one can retry to clear $o_a$. +If $o_a$ could not be cleared $o_b$ can be cleared using the same procedure and +after clearing $o_b$ one can retry to clear $o_a$. -In any case at least one vertex operator has been cleared. If both vertex operators have been -cleared Case 1 will be applied. If there is just one cleared vertex operator it -is the vertex with non-operand neighbours. Using this one can still apply a $CZ$: Without loss of generality -assume that $a$ has non-operand neighbours and $b$ does not. Now the state $\ket{G}$ has the form \cite{andersbriegel2005} +In any case at least one vertex operator has been cleared. If both vertex +operators have been cleared Case 1 will be applied. If there is just one +cleared vertex operator it is the vertex with non-operand neighbours. Using +this one can still apply a $CZ$: Without loss of generality assume that $a$ has +non-operand neighbours and $b$ does not. Now the state $\ket{G}$ has the form +\cite{andersbriegel2005} \begin{equation} \begin{aligned} @@ -653,23 +677,25 @@ As $o_b$ commutes with all operators but $CZ_{a,b}$ and $s \in \{0, 1\}$ indicat \end{aligned} \end{equation} -This allows to re-use the method in Case 2 to apply the $CZ$ while keeping the $o_a = I$. +This allows to re-use the method in Case 2 to apply the $CZ$ while keeping the +$o_a = I$. -As it has been shown how both $CZ$ and $C_L$ act on a graph state $\ket{G}$ and the resulting state -is a graph state as well this proofs constructively that -the graphical representation of a stabilizer state is indeed able to represent any stabilizer state. -If one wants to do computations using this formalism it is however also necessary to perform measurements. +As it has been shown how both $CZ$ and $C_L$ act on a graph state $\ket{G}$ +and the resulting state is a graph state as well this proofs constructively +that the graphical representation of a stabilizer state is indeed able to +represent any stabilizer state. If one wants to do computations using this +formalism it is however also necessary to perform measurements. \subsubsection{Measurements on Graph States} -This is adapted from \cite{andersbriegel2005}; measurement results and updating the graph after -a measurement is described in \cite{hein_eisert_briegel2008}. +This is adapted from \cite{andersbriegel2005}; measurement results and updating +the graph after a measurement is described in \cite{hein_eisert_briegel2008}. -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 -which is a quite expensive computation in theory. It is possible to simplify -the problem by pulling the observable behind the vertex operators. For this consider -the projector $P_{a,s} = \frac{I + (-1)^sZ_a}{2}$ +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 which is a quite expensive computation in theory. It is +possible to simplify the problem by pulling the observable behind the vertex +operators. For this consider the projector $P_{a,s} = \frac{I + (-1)^sZ_a}{2}$ \begin{equation} \begin{aligned}