some more changes by Simon
This commit is contained in:
parent
5dde04c2ca
commit
a0afd2aeb1
|
|
@ -331,22 +331,38 @@ from the graph.
|
|||
\end{equation}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before. Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$.
|
||||
Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before.
|
||||
Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$.
|
||||
In the following discussion the direction $\prod\limits_{\{l,k\} \in E} := \prod\limits_{\{l,k\} \in E, l < k}$
|
||||
is introduced as the graph is undirected and edges must not be handled twice.
|
||||
Set $\ket{\tilde{G}} := \left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right)\ket{+}$.
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
K_G^{(i)} \ket{\tilde{G}} & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right) \ket{+} \\
|
||||
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
|
||||
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_k \otimes Z_l\right) X_i \ket{+} \\
|
||||
& = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l} + \sum\limits_{\{i,j\} \in E} \delta_{j,k}}\ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
|
||||
K_G^{(i)} \ket{\tilde{G}}
|
||||
& = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)
|
||||
\left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right) \ket{+} \\
|
||||
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,k\} \in E}
|
||||
\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
|
||||
& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,k\} \in E}
|
||||
\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_k \otimes Z_l\right) X_i \ket{+} \\
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
As $X,Z$ anticommute. $X_i$ can now be absorbed into $\ket{+}$. The next step is a bit tricky:
|
||||
A $Z_j$ can be absorbed into a $\ket{0}\bra{0}_j$ giving no phase or into a $\ket{1}\bra{1}_j$ yielding
|
||||
a phase of $-1$. If there is no projector on $j$ the $Z_j$ can be commuted to the next projector.
|
||||
It is guaranteed that a projector on $j$ exists by the definition of $\ket{\tilde{G}}$.
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
K_G^{(i)} \ket{\tilde{G}}
|
||||
& = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l} + \delta_{j,k}}\ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
|
||||
& = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
|
||||
& = +1 \ket{\tilde{G}}
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
as $X, Z$ anticommute and $Z\ket{1} = -1\ket{1}$.
|
||||
|
||||
The $\delta_{i,l} + \delta_{j,k}$ is either $0$ or $2$ by the definitions of $K_G^{(i)}$ and $\ket{\tilde{G}}$.
|
||||
\end{proof}
|
||||
|
||||
\subsubsection{Dynamics of the VOP-free Graph States}
|
||||
|
|
@ -357,8 +373,8 @@ resulting in a multiplication of $Z_j$ to $K_G^{(i)}$ and $Z_i$ to $K_G^{(j)}$.
|
|||
is done by using the symmetric set difference:
|
||||
|
||||
\begin{definition}
|
||||
For to finite sets $A,B$ the symmetric set difference $\Delta$ is
|
||||
defined as
|
||||
For two finite sets $A,B$ the symmetric set difference $\Delta$ is
|
||||
defined as:
|
||||
|
||||
\begin{equation}
|
||||
A \Delta B = (A \cup B) \setminus (A \cap B)
|
||||
|
|
@ -366,7 +382,7 @@ is done by using the symmetric set difference:
|
|||
\end{definition}
|
||||
|
||||
Toggling an edge $\{i, j\}$ updates $E' = E \Delta \left\{\{i,j\}\right\}$.
|
||||
Another transformation on the VOP-free graph states is for a vertex $a \in V$
|
||||
Another transformation on the VOP-free graph states is for a vertex $a \in V$:
|
||||
|
||||
\begin{equation}
|
||||
M_a := \sqrt{-iX_a} \prod\limits_{j\in n_a} \sqrt{iZ_j}
|
||||
|
|
@ -379,7 +395,7 @@ that will be used later.
|
|||
\label{lemma:M_a}
|
||||
When applying $M_a$ to a state $\ket{\bar{G}}$ the new state
|
||||
$\ket{\bar{G}'}$ is again a VOP-free graph state and the
|
||||
graph is updated according to
|
||||
graph is updated according to:
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
n_a' &= n_a \\
|
||||
|
|
@ -413,10 +429,10 @@ that will be used later.
|
|||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
One can now construct a new set of $K_{G'}^{(i)}$ s.t. $M_a \ket{\bar{G}}$ is the $+1$ eigenvalue
|
||||
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 $n_j = \{a\} \cup J$.
|
||||
Then
|
||||
Then follows:
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
|
|
@ -431,7 +447,7 @@ that will be used later.
|
|||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
Using this ine can show that $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$:
|
||||
Using this one can show that $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$:
|
||||
|
||||
\begin{equation}
|
||||
\ket{\bar{G}'} = S^{(j)}\ket{\bar{G}'} = K_{G}^{(a)} K_{G'}^{(j)}\ket{\bar{G}'}
|
||||
|
|
@ -443,7 +459,7 @@ that will be used later.
|
|||
multi-local Pauli operators where the $S^{(i)}$ can be generated from the $K_{G'}^{(i)}$
|
||||
and $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$
|
||||
$\langle\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}\rangle$
|
||||
are the stabilizers of $\ket{\bar{G}'}$ and the associated graph is changed as given
|
||||
are the stabilizers of $\ket{\bar{G}'}$. Therefore the associated graph is changed as given
|
||||
in the third equation.
|
||||
\end{proof}
|
||||
|
||||
|
|
@ -451,12 +467,12 @@ that will be used later.
|
|||
|
||||
The definition of a VOP-free graph state above raises an obvious question:
|
||||
Can any stabilizer state be described using just a graph?
|
||||
The answer is quite simple: No. The most simple cases are the single qbit
|
||||
stated $\ket{0},\ket{1}$ and $\ket{+_Y}, \ket{-_Y}$. But there is a simple extension
|
||||
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
|
||||
just constructive. As seen in theorem \ref{thm:clifford_group_approx} any $c \in C_n$
|
||||
can be constructed from $CZ$ and $C_L$ and in the following discussion it will become
|
||||
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.
|
||||
|
||||
\subsubsection{Graph States and Vertex Operators}
|
||||
|
|
@ -467,7 +483,7 @@ clear that both $C_L$ and $CZ$ can be applied to a general graph state.
|
|||
A tuple $(V, E, O)$ is called the graphical representation of a stabilizer state
|
||||
if $(V, E)$ is a graph as in Definition \ref{def:graph} and $O = \{o_1, ..., o_n\}$ where $o_i \in C_L$.
|
||||
|
||||
The state $\ket{G}$ is defined by the eigenvalue relation
|
||||
The state $\ket{G}$ is defined by the eigenvalue relation:
|
||||
|
||||
\begin{equation}
|
||||
+1 \ket{G} = \left(\prod\limits_{j=1}^no_j\right) K_G^{(i)} \left(\prod\limits_{j=1}^no_j\right)^\dagger \ket{G}
|
||||
|
|
@ -485,8 +501,8 @@ Recalling the dynamics of stabilizer states the following relation follows immed
|
|||
The great advantage of this representation of a stabilizer state is its space requirement:
|
||||
Instead of storing $n^2$ $P_1$ matrices only some vertices (which often are implicit),
|
||||
the edges and some vertex operators ($n$ matrices) have to be stored. The following theorem
|
||||
will improve this even further: instead of $n$ matrices it is enough to store $n$ integers
|
||||
representing the vertex operators is enough:
|
||||
will improve this even further: instead of $n$ matrices it is sufficient to store $n$ integers
|
||||
representing the vertex operators:
|
||||
|
||||
\begin{theorem}
|
||||
$C_L$ has $24$ degrees of freedom.
|
||||
|
|
@ -498,14 +514,14 @@ representing the vertex operators is enough:
|
|||
of $X,Z$ only.
|
||||
|
||||
As the transformations are unitary they preserve eigenvalues, so $X$ can be mapped
|
||||
to $\pm X, \pm Y, \pm Z$ which gives $6$ degrees of freedom, the image of $Z$ has to
|
||||
to $\pm X, \pm Y, \pm Z$ which gives $6$ degrees of freedom. Furthermore the image of $Z$ has to
|
||||
anti-commute with the image of $X$ so $Z$ has four possible images under the transformation.
|
||||
This gives another $4$ degrees of freedom and a total of $24$.
|
||||
\end{proof}
|
||||
|
||||
From now on $C_L = \langle H, S \rangle$ (disregarding a global phase) will be used,
|
||||
one can show (by construction) that $H, S$ generate a possible choice of $C_L$, as is
|
||||
$C_L = \langle \sqrt{-iX}, \sqrt{-iZ}\rangle$ which will be used in one specific operation on graph states.
|
||||
From now on $C_L = \langle H, S \rangle$ (disregarding a global phase) will be used.
|
||||
One can show (by construction) that $H, S$ generate a possible choice of $C_L$, as is
|
||||
$C_L = \langle \sqrt{-iX}, \sqrt{-iZ}\rangle$ which is required in one specific operation on graph states.
|
||||
|
||||
\begin{equation}
|
||||
S = \left(\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\right)
|
||||
|
|
@ -525,7 +541,7 @@ basically a stabilizer tableaux that might require less memory than the tableaux
|
|||
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
|
||||
$\langle c_j S^{(i)} c_j^\dagger\rangle_i$. 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
|
||||
it is clear that just the vertex operators are changed and the new vertex operators are given by:
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
|
|
@ -550,11 +566,11 @@ are changed to $E' = E \Delta \left\{\{a,b\}\right\}$.
|
|||
The two qbits are isolated: From the definition of the graph state it is clear 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 or without
|
||||
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.
|
||||
|
||||
All those states and the resulting state after a $CZ$ application can be computed and while doing so one
|
||||
gets another interesting result that will be useful later: If one vertex has the vertex operator $I$ the
|
||||
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
|
||||
resulting state can be chosen such that at least one of the vertex operators is $I$ again and in particular
|
||||
the identity on the vertex can be preserved under the application of a $CZ$.
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user