From 8622bba374a4a1c8ec05e93de0362cb037e9bc5c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Fri, 6 Dec 2019 12:58:44 +0100 Subject: [PATCH] added chapter about stabilizers --- thesis/chapters/graph_simulator.tex | 105 --------------------- thesis/chapters/stabilizer.tex | 137 ++++++++++++++++++++++++++++ thesis/main.bib | 9 ++ thesis/main.tex | 2 + 4 files changed, 148 insertions(+), 105 deletions(-) create mode 100644 thesis/chapters/stabilizer.tex diff --git a/thesis/chapters/graph_simulator.tex b/thesis/chapters/graph_simulator.tex index 08a0445..e426add 100644 --- a/thesis/chapters/graph_simulator.tex +++ b/thesis/chapters/graph_simulator.tex @@ -5,27 +5,6 @@ The following definitions and lemmata are required to understand both how the graph formalism works and how the simulator handles gates. -\begin{definition} - \begin{equation} - p \in P_n \Rightarrow p = \bigotimes\limits_{i=0}^n p_i \\ - \forall i: p_i \in P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\} - \end{equation} - -Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$, - $Y = \left(\begin{array}{cc} 0 & i \\ -i & 0\end{array}\right)$ and - $Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and - $I$ is the identity. -\end{definition} - - -\begin{definition} - Let $p_i \in P_n \forall i = 1, ..., n$, $[p_i, p_j] = 0 \forall i,j$ be commuting multi-local Pauli operators. - Then a $n$ qbit state $\ket{\psi}$ is called a stabilizer state iff - - \begin{equation} - \forall i: p_i\ket{\psi} = +1\ket{\psi} - \end{equation} -\end{definition} %A $n$ qbit graph or stabilizer state is a $+1$ eigenstate of some $ p \in P_n$ where $P_n$ is the Pauli group\cite{andersbriegel2005}. \begin{definition} @@ -187,90 +166,6 @@ Where every $o_i$ acts on the $i$-th qbit. One can show that any stabilizer state can be realized as a graph state (for instance in \cite{schlingenmann2001}). -\subsubsection{The Vertex Operator-Free Graph States} - -In order to understand some essential transformations of graph states it is necessary -to study the vertex operator-free graph states first, partially because the graph states as used in this paper -were derived from the vertex operator-free graph states. - -\begin{definition} - \label{def:vop_free_g_state} - A $n$ qbit vertex operator-free graph state $\ket{\overline{G}}$ is associated with a graph $(V, E)$ - by the $n$ operators - - \begin{equation} - K^{(i)}_G := X_i \left(\prod\limits_{\{i, j\} \in E} Z_j\right) - \end{equation} - - for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit. - - A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$. -\end{definition} - -\begin{corrolary} - All $K^{(i)}_G$ commute and are hermitian. Therefore they have a common set of eigen states - (in particular definition \ref{def:vop_free_g_state} is well defined). - In terms of quantum mechanics $K^{(i)}_G$ are observables. - - Further as $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$ which are - multi-local Pauli operators, $\{K^{(i)}_G | i \in \{0, ..., n-1\}\}$ is the stabilizer - of $\ket{\overline{G}}$ and $\ket{\overline{G}}$ is a stabilizer state. -\end{corrolary} - -\begin{proof} - As $X_i$ and $Z_i$ are hermitian their product is hermitian. - - Consider the case $\{i,j\} \notin E$ first: - \begin{equation} - \begin{aligned} - \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 - \end{aligned} - \end{equation} - - As operators acting on different qbits commute. The case $\{i,j\} \in E$ is slightly less trivial: - \begin{equation} - \begin{aligned} - \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] \\ - &= \left[X_i Z_j \prod\limits_n Z_n, X_j Z_i \prod\limits_m Z_m\right]\\ - &= \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 \\ - &= \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 \\ - &= \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 \\ - &= 0 - \end{aligned} - \end{equation} - - as $X$, $Z$ anticommute. -\end{proof} - -\begin{lemma} - \begin{equation} - \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} - \end{equation} - - In particular definitions \ref{def:vop_free_g_state} and \ref{def:graph_state} are consistent, when there are no - vertex operators on the graph state $\ket{G}$. -\end{lemma} -\begin{proof} - Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before. Note that for any $X_i$ $X_i \ket{+} = +1 \ket{+}$. - Set $\ket{\tilde{G}} := \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right)\ket{+}$. - - \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,j\} \in E} CZ_{l,j} \right) \ket{+} \\ - & = \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{+} \\ - & = \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{+} \\ - & = \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{+} \\ - & = \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{+} \\ - & = +1 \ket{\tilde{G}} - \end{aligned} - \end{equation} - - as $X, Z$ anticommute and $Z\ket{1} = -1\ket{1}$. -\end{proof} - -These insights can be used to understand how measurement works on the vop-free graph state \cite{nielsen_chuang_2010}: -Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a hermitian $g$ that is to be measured. - diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex new file mode 100644 index 0000000..a1ed329 --- /dev/null +++ b/thesis/chapters/stabilizer.tex @@ -0,0 +1,137 @@ +\section{The Stabilizer Formalism and VOP-Free Graph States} + +\subsection{Stabilizers and Stabilizer States} + +This chapter discusses the stabilizer formalism that was introduced by Gottesman\cite{gottesman1997} +for quantum error correction but soon proved to be a useful tool to describe a subset of states: +the stabilizer states which can be simulated in polynomial time \cite{gottesman2008}. + + + +\begin{definition} + \begin{equation} + p \in P_n \Rightarrow p = \bigotimes\limits_{i=0}^n p_i \\ + \forall i: p_i \in P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\} + \end{equation} + +Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$, + $Y = \left(\begin{array}{cc} 0 & i \\ -i & 0\end{array}\right)$ and + $Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and + $I$ is the identity. + + $p \in = P_n$ is called a multi-local Pauli operator. +\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 $G$. +\end{definition} + +\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 + + \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{$\forall i = 1, ..., n$ $S_i\ket{\psi} = +1 \ket{\psi}$} + \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 + a (up to a global phase) unique state $\ket{\psi}$ fulfilling the third condition. +\end{lemma} + +\begin{proof} + All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as they + 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} + +\subsection{The Vertex Operator-Free Graph States} + +In order to understand some essential transformations of graph states it is necessary +to study the vertex operator-free graph states first, partially because the graph states as used in this paper +were derived from the vertex operator-free graph states. + +\begin{definition} + \label{def:vop_free_g_state} + A $n$ qbit vertex operator-free graph state $\ket{\overline{G}}$ is associated with a graph $(V, E)$ + by the $n$ operators + + \begin{equation} + K^{(i)}_G := X_i \left(\prod\limits_{\{i, j\} \in E} Z_j\right) + \end{equation} + + for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit. + + A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$. +\end{definition} + +\begin{corrolary} + All $K^{(i)}_G$ commute and are hermitian. Therefore they have a common set of eigenstates + (in particular definition \ref{def:vop_free_g_state} is well defined). + In terms of quantum mechanics $K^{(i)}_G$ are observables. + + Further as $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$ which are + multi-local Pauli operators, $\{K^{(i)}_G | i \in \{0, ..., n-1\}\}$ is the stabilizer + of $\ket{\overline{G}}$ and $\ket{\overline{G}}$ is a stabilizer state. +\end{corrolary} + +\begin{proof} + As $X_i$ and $Z_i$ are hermitian their product is hermitian. + + Consider the case $\{i,j\} \notin E$ first: + \begin{equation} + \begin{aligned} + \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 + \end{aligned} + \end{equation} + + As operators acting on different qbits commute. The case $\{i,j\} \in E$ is slightly less trivial: + \begin{equation} + \begin{aligned} + \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] \\ + &= \left[X_i Z_j \prod\limits_n Z_n, X_j Z_i \prod\limits_m Z_m\right]\\ + &= \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 \\ + &= \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 \\ + &= \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 \\ + &= 0 + \end{aligned} + \end{equation} + + as $X$, $Z$ anticommute. +\end{proof} + +\begin{lemma} + \begin{equation} + \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} + \end{equation} + + In particular definitions \ref{def:vop_free_g_state} and \ref{def:graph_state} are consistent, when there are no + vertex operators on the graph state $\ket{G}$. +\end{lemma} +\begin{proof} + Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before. Note that for any $X_i$ $X_i \ket{+} = +1 \ket{+}$. + Set $\ket{\tilde{G}} := \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right)\ket{+}$. + + \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,j\} \in E} CZ_{l,j} \right) \ket{+} \\ + & = \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{+} \\ + & = \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{+} \\ + & = \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{+} \\ + & = \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{+} \\ + & = +1 \ket{\tilde{G}} + \end{aligned} + \end{equation} + + as $X, Z$ anticommute and $Z\ket{1} = -1\ket{1}$. +\end{proof} + +These insights can be used to understand how measurement works on the vop-free graph state \cite{nielsen_chuang_2010}: +Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a hermitian $g$ that is to be measured. + diff --git a/thesis/main.bib b/thesis/main.bib index f0f8801..0d6ce99 100644 --- a/thesis/main.bib +++ b/thesis/main.bib @@ -99,3 +99,12 @@ publisher={CAMBRIDGE UNIVERSITY PRESS}, note={www.cambridge.org/9781107002173} } + +@article{ + gottesman2008, + title={The Heisenberg Representation of Quantum Computers}, + year=2008, + author={Daniel Gottesman}, + note={https://arxiv.org/abs/quant-ph/9807006} +} + diff --git a/thesis/main.tex b/thesis/main.tex index beda6df..9b2ef2e 100644 --- a/thesis/main.tex +++ b/thesis/main.tex @@ -48,6 +48,8 @@ Simulator with a Focus on Simulation in the Graph Formalism } \include{chapters/naive_simulator} +\include{chapters/stabilizer} + \include{chapters/graph_simulator} \section{Appendix}