some work

This commit is contained in:
Daniel Knüttel 2019-12-06 18:27:36 +01:00
parent 8622bba374
commit 47cd875151
3 changed files with 34 additions and 11 deletions

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@ -5,6 +5,7 @@ bibtex=bibtex
chapters = chapters/introduction.tex \
chapters/naive_simulator.tex \
chapters/introduction_qc.tex \
chapters/stabilizer.tex \
chapters/graph_simulator.tex
all: main.pdf

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@ -7,14 +7,6 @@ graph formalism works and how the simulator handles gates.
%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}
\begin{equation}
C_n := \{U \in SU(2) | UpU^\dagger \in P_n \forall p \in P_n\}
\end{equation}
is called the Clifford group on $n$ qbits.
$C_1$ is called the local Clifford group.
\end{definition}
One can show that the Clifford group $C_n$ can be generated using the elements of $C_1$ acting on all qbits and
the controlled phase gate $CZ$ between all qbits\cite{andersbriegel2005}. It is worth noting that the $CX$ gate can be
generated using $CZ$ and $C_1$ gates.

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@ -19,7 +19,7 @@ Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
$Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and
$I$ is the identity.
$p \in = P_n$ is called a multi-local Pauli operator.
$p \in P_n$ is called a multi-local Pauli operator.
\end{definition}
\begin{definition}
@ -41,16 +41,46 @@ Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
\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.
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 they
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}
One can study the dynamics of stabilizer states using only the stabilizers\cite{nielsen_chuang_2010}. Two important
cases are the unitary transformation of a state and the measurement of a qbit. When applying a unitary gate to a stabilizer
state $\ket{\psi}$ the resulting state will in general be no stabilizer state anymore, however there exists a group of
transformations that map stabilizers to other stabilizers: the Clifford group.
\begin{definition}
\begin{equation}
C_n := \{U \in SU(2) | UpU^\dagger \in P_n \forall p \in P_n\}
\end{equation}
is called the Clifford group on $n$ qbits.
$C_1$ is called the local Clifford group.
\end{definition}
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 $\langle US_iU^\dagger \rangle_i$.
\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 $US_iU^\dagger$.
\end{proof}
This is an important insight that is used for simulations\cite{gottesman_aaronson2008}, as
updating the $n$ stabilizers that are a tensor product of $n$ Pauli matrices scales with roughly
$\mathcal{O}(n^2)$ instead of $\mathcal{O}(2^n)$ for the state vector approach.
\subsection{The Vertex Operator-Free Graph States}
In order to understand some essential transformations of graph states it is necessary