138 lines
6.5 KiB
TeX
138 lines
6.5 KiB
TeX
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\section{The Stabilizer Formalism and VOP-Free Graph States}
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\subsection{Stabilizers and Stabilizer States}
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This chapter discusses the stabilizer formalism that was introduced by Gottesman\cite{gottesman1997}
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for quantum error correction but soon proved to be a useful tool to describe a subset of states:
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the stabilizer states which can be simulated in polynomial time \cite{gottesman2008}.
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\begin{definition}
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\begin{equation}
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p \in P_n \Rightarrow p = \bigotimes\limits_{i=0}^n p_i \\
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\forall i: p_i \in P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
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\end{equation}
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Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
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$Y = \left(\begin{array}{cc} 0 & i \\ -i & 0\end{array}\right)$ and
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$Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and
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$I$ is the identity.
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$p \in = P_n$ is called a multi-local Pauli operator.
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\end{definition}
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\begin{definition}
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For a group $G$, $g_1, ..., g_n$ are called the generators of $G$ iff $\forall g \in G: g = \prod\limits_{i \in I} g_i$ for a
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subsed $I$ of $\{1, ..., n\}$. We write $G = \langle g_i \rangle_i$ if G is generated by the $g_i$. The generators $g_i$ are chosen
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to be the smallest set of generators of $G$.
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\end{definition}
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\begin{definition}
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\label{def:stabilizer}
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For a $n$ qbit state $\ket{\psi}$ $\langle S_i \rangle_i$ is called the stabilizer of $\ket{\psi}$ if
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\begin{enumerate}
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\item{$\forall i = 1, ..., n$ $S_i \in P_n$}
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\item{$\forall i,j = 1, ..., n$ $[S_i, S_j] = 0$ $S_i$ and $S_j$ commute}
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\item{$\forall i = 1, ..., n$ $S_i\ket{\psi} = +1 \ket{\psi}$}
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\end{enumerate}
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\end{definition}
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\begin{lemma}
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For every $\langle S_i \rangle_i$ fulfilling the first two conditions in definition \ref{def:stabilizer} there exists
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a (up to a global phase) unique state $\ket{\psi}$ fulfilling the third condition.
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\end{lemma}
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\begin{proof}
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All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as they
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commute they have a common set of eigenstates. Because each $S_i$ has eigenvalues $+1, -1$, there
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exist $2^n$ eigenstates, one state $\ket{\psi}$ with eigenvalue $+1$ for all $S_i$. As the dimension of $n$ qbits is $2^n$
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the state $\ket{psi}$ is unique up to a global phase.
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\end{proof}
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\subsection{The Vertex Operator-Free Graph States}
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In order to understand some essential transformations of graph states it is necessary
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to study the vertex operator-free graph states first, partially because the graph states as used in this paper
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were derived from the vertex operator-free graph states.
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\begin{definition}
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\label{def:vop_free_g_state}
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A $n$ qbit vertex operator-free graph state $\ket{\overline{G}}$ is associated with a graph $(V, E)$
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by the $n$ operators
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\begin{equation}
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K^{(i)}_G := X_i \left(\prod\limits_{\{i, j\} \in E} Z_j\right)
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\end{equation}
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for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit.
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A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$.
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\end{definition}
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\begin{corrolary}
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All $K^{(i)}_G$ commute and are hermitian. Therefore they have a common set of eigenstates
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(in particular definition \ref{def:vop_free_g_state} is well defined).
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In terms of quantum mechanics $K^{(i)}_G$ are observables.
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Further as $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$ which are
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multi-local Pauli operators, $\{K^{(i)}_G | i \in \{0, ..., n-1\}\}$ is the stabilizer
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of $\ket{\overline{G}}$ and $\ket{\overline{G}}$ is a stabilizer state.
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\end{corrolary}
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\begin{proof}
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As $X_i$ and $Z_i$ are hermitian their product is hermitian.
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Consider the case $\{i,j\} \notin E$ first:
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\begin{equation}
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\begin{aligned}
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\left[K^{(i)}_G, K^{(j)}_G\right] = \left[X_i \prod\limits_{\{i, n\} \in E} Z_n, X_j \prod\limits_{\{j, m\} \in E} Z_m\right] = 0
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\end{aligned}
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\end{equation}
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As operators acting on different qbits commute. The case $\{i,j\} \in E$ is slightly less trivial:
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\begin{equation}
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\begin{aligned}
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\left[K^{(i)}_G, K^{(j)}_G\right] &= \left[X_i \left(\prod\limits_{\{i, n\} \in E, n \neq j} Z_n\right) Z_j, X_j \left(\prod\limits_{\{j, m\} \in E, m \neq i} Z_m\right) Z_i\right] \\
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&= \left[X_i Z_j \prod\limits_n Z_n, X_j Z_i \prod\limits_m Z_m\right]\\
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&= \left(X_i Z_j X_j Z_i - X_j Z_i X_i Z_j\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
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&= \left(Z_j X_j X_i Z_i - X_j Z_j Z_i X_i\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
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&= \left((-1)^2X_j Z_j Z_i X_i - X_j Z_j Z_i X_i\right) \prod\limits_n Z_n \prod\limits_m Z_m \\
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&= 0
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\end{aligned}
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\end{equation}
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as $X$, $Z$ anticommute.
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\end{proof}
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\begin{lemma}
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\begin{equation}
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\ket{\overline{G}} = \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right) \left(\prod\limits_{l \in V} H_l\right) \ket{0}
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\end{equation}
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In particular definitions \ref{def:vop_free_g_state} and \ref{def:graph_state} are consistent, when there are no
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vertex operators on the graph state $\ket{G}$.
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\end{lemma}
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\begin{proof}
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Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before. Note that for any $X_i$ $X_i \ket{+} = +1 \ket{+}$.
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Set $\ket{\tilde{G}} := \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right)\ket{+}$.
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\begin{equation}
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\begin{aligned}
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K_G^{(i)} \ket{\tilde{G}} & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\left(\prod\limits_{\{l,j\} \in E} CZ_{l,j} \right) \ket{+} \\
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& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
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& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) X_i \ket{+} \\
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& = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{2\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
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& = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
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& = +1 \ket{\tilde{G}}
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\end{aligned}
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\end{equation}
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as $X, Z$ anticommute and $Z\ket{1} = -1\ket{1}$.
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\end{proof}
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These insights can be used to understand how measurement works on the vop-free graph state \cite{nielsen_chuang_2010}:
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Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a hermitian $g$ that is to be measured.
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