From 40b979835b4265357284b690960c0d49478503d4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Tue, 17 Dec 2019 11:19:36 +0100 Subject: [PATCH] some corrections in the notation --- thesis/chapters/stabilizer.tex | 39 ++++++++++++++++++---------------- 1 file changed, 21 insertions(+), 18 deletions(-) diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index edda92d..55bbf3f 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -23,31 +23,34 @@ Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$, \end{definition} \begin{definition} - 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 - 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 - to be the smallest set of generators of $G$. + For a group $G$, $g_1, ..., g_n$ are called the generators of $G$ iff $\forall g \in G: g = \prod\limits_{i \in J} g_i$ for a + subsed $J$ of $I = \{1, ..., n\}$. We write $G = \langle g_i \rangle_i$ if G is generated by the $g_i$. The generators $g_i$ are chosen + to be the smallest set of generators of $G$. \end{definition} +The notation $\langle g_i \rangle_i \equiv \langle g_i \rangle_{i \in I}$ is used used as a shorthand +notation for $\langle \{g_i\}_{i \in I} \rangle$. + \begin{definition} \label{def:stabilizer} - For a $n$ qbit state $\ket{\psi}$ $\langle S_i \rangle_i$ is called the stabilizer of $\ket{\psi}$ if + For a $n$ qbit state $\ket{\psi}$ $\langle S^{(i)} \rangle_{i}$ is called the stabilizer of $\ket{\psi}$ if \begin{enumerate} - \item{$\forall i = 1, ..., n$ $S_i \in P_n$} - \item{$\forall i,j = 1, ..., n$ $[S_i, S_j] = 0$ $S_i$ and $S_j$ commute} - \item{$\forall i = 1, ..., n$ $S_i\ket{\psi} = +1 \ket{\psi}$} + \item{$\forall i = 1, ..., n$ $S^{(i)} \in P_n$} + \item{$\forall i,j = 1, ..., n$ $[S^{(i)}, S^{(j)}] = 0$ $S^{(i)}$ and $S^{(j)}$ commute} + \item{$\forall i = 1, ..., n$ $S^{(i)}\ket{\psi} = +1 \ket{\psi}$} \end{enumerate} \end{definition} \begin{lemma} - For every $\langle S_i \rangle_i$ fulfilling the first two conditions in definition \ref{def:stabilizer} there exists + For every $\langle S^{(i)} \rangle_i$ fulfilling the first two conditions in definition \ref{def:stabilizer} there exists a (up to a global phase) unique state $\ket{\psi}$ fulfilling the third condition. This state is called stabilizer state. \end{lemma} \begin{proof} - All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as the $S_i$ - commute they have a common set of eigenstates. Because each $S_i$ has eigenvalues $+1, -1$, there - exist $2^n$ eigenstates, one state $\ket{\psi}$ with eigenvalue $+1$ for all $S_i$. As the dimension of $n$ qbits is $2^n$ + All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as the $S^{(i)}$ + commute they have a common set of eigenstates. Because each $S^{(i)}$ has eigenvalues $+1, -1$, there + exist $2^n$ eigenstates, one state $\ket{\psi}$ with eigenvalue $+1$ for all $S^{(i)}$. As the dimension of $n$ qbits is $2^n$ the state $\ket{psi}$ is unique up to a global phase. \end{proof} @@ -67,13 +70,13 @@ transformations that map stabilizers to other stabilizers: the Clifford group. The properties of this group will be discussed later, for the time being is existence is enough. \begin{lemma} - Let $\ket{\psi}$ be stabilized by $\langle S_i \rangle_i$, then $U\ket{\psi}$ is stabilized - by $\langle US_iU^\dagger \rangle_i$. + Let $\ket{\psi}$ be stabilized by $\langle S^{(i)} \rangle_i$, then $U\ket{\psi}$ is stabilized + by $\langle US^{(i)}U^\dagger \rangle_i$. \end{lemma} \begin{proof} -$$ U\ket{\psi} = US_i\ket{\psi} = US_iU^\dagger U\ket{\psi}$$ -So $U\ket{\psi}$ is a $+1$ eigenstate of $US_iU^\dagger$. +$$ U\ket{\psi} = US^{(i)}\ket{\psi} = US^{(i)}U^\dagger U\ket{\psi}$$ +So $U\ket{\psi}$ is a $+1$ eigenstate of $US^{(i)}U^\dagger$. \end{proof} This is an important insight that is used for simulations\cite{gottesman_aaronson2008}, as @@ -91,14 +94,14 @@ measuring a qbit is given by the following lemma: \begin{lemma} \label{lemma:stab_measurement} - Let $J := \{ S_i | [g_a, S_i] \neq 0\}$. Then + Let $J := \{ S^{(i)} | [g_a, S^{(i)}] \neq 0\}$. Then \begin{enumerate} \item{If $J = \{\}$, one value is measured with probability $1$ and the stabilizers are unchanged.} \item{If $J \neq \{\}$, $1$ and $0$ are measured with probability $\frac{1}{2}$ and after choosing a $j \in J$ the new state $\ket{\psi'}$ is stabilized by \begin{equation} - \langle \{(-1)^s g_a\} \cup \{S_i S_j | i \in J \setminus \{j\} \} \cup \{S_i | i \in J^c\}\rangle + \langle \{(-1)^s g_a\} \cup \{S^{(i)} S^{(j)} | S^{(i)} \in J \setminus \{S^{(j)}\} \} \cup J^c\rangle \end{equation}} \end{enumerate} \end{lemma} @@ -108,7 +111,7 @@ measuring a qbit is given by the following lemma: \item{As $g_a$ commutes with all stabilizers, $\ket{\psi}$ is an eigenstate of $g_a$, so the result of the measurement is deterministic and $\ket{\psi}$ is left unchanged.} \item{As $g_a$ is a Pauli operator and $S^{(i)} \in J$ are multi-local Pauli operators, - $S_i$ and $g_a$ anticommute. Choose a $S^{(j)} \in J$. Then + $S^{(i)}$ and $g_a$ anticommute. Choose a $S^{(j)} \in J$. Then \begin{equation} \begin{aligned}