From 98444d6f0d171a1510a97547463169afbca5d2a3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Wed, 29 Jan 2020 11:19:13 +0100 Subject: [PATCH] some work --- thesis/chapters/quantum_computing.tex | 19 +++--- thesis/chapters/stabilizer.tex | 91 ++++++++++++++++++++++++++- 2 files changed, 101 insertions(+), 9 deletions(-) diff --git a/thesis/chapters/quantum_computing.tex b/thesis/chapters/quantum_computing.tex index 57e4d79..3994192 100644 --- a/thesis/chapters/quantum_computing.tex +++ b/thesis/chapters/quantum_computing.tex @@ -56,6 +56,7 @@ and will be used in some calculations later. \end{equation} \subsubsection{Many Qbits} +\label{ref:many_qbits} \begin{postulate} A $N$ qbit quantum mechanical state is the tensor product\cite[Definition 14.3]{wuest1995} of the $N$ single qbit @@ -174,12 +175,14 @@ to time as a measurement can be treated similarely while doing numerics. \subsection{Quantum Circuits} -\textbf{FIXME: citation needed} - -Quantum circuits are a simple and well-readable way to express the application -of several gates on a state. Qbits are represented by horizontal line on which -time goes from left to right. On these lines single qbit gates are -represented by a box: +As mentioned in \ref{ref:many_qbits} one can approximate an arbitrary $n$ +qbit gate $U$ as a product of some single qbit gates and either $CX$ or $CZ$. +Writing (possibly huge) products of matrices is quite unpractical and very much +unreadable. To address this problem quantum circuits have been introduced. +These represent the qbits as a horizontal line, a gate acting on a qbit is +a box with a name on the respective line. Quantum circuits are read from +left to right. This means that a gate $U_i = Z_i X_i H_i$ has the +circuit representation \[ \Qcircuit @C=1em @R=.7em { @@ -198,4 +201,6 @@ the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is } \] -\textbf{TODO: more info about quantum circuits} +\subsection{Quantum Algorithms} + + diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 2f3ae33..5c16891 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -92,7 +92,7 @@ properties to understand the properties of the group. and $m$ is the smallest integer for which these statements hold. \end{definition} -In the following discussions $\rangle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as +In the following discussions $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as the properties of a set of stabilizers that are used in the discussions can be studied using only its generators. @@ -117,4 +117,91 @@ can be diagonalized simultaneously. This motivates and justifies the following d It is clear that it is sufficient to show the stabilization property for the generators of $S$, as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$. -The dimension of $V_S$ is not immediately +The dimension of $V_S$ is not immediately clear. One can however show that +for a set of stabilizers $\langle S^{(i)} \rangle_{i=1, ..., n-m}$ the dimension +$dim V_S = 2^m$ \cite[Chapter 10.5]{nielsen_chuang_2010}. This yields the following important +result: + +\begin{theorem} + For a $n$ qbit system and a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer + space $V_S$ has $dim V_S = 1$, in particular there exists an up to a trivial phase unique + state $\ket{\psi}$ that is stabilized by $S$. + + Without proof. +\end{theorem} + +In the following discussions for $n$ qbits a set $S = \langle S^{(i)} \rangle_{i=1,...,n}$ +of $n$ independent stabilizers will be assumed. + + +\subsubsection{Dynamics of Stabilizer States} + +Consider a $n$ qbit state $\ket{\psi}$ that is the $+1$ eigenstate of $S = \langle S^{(i)} \rangle_{i=1,...,n}$ +and a unitary transformation $U$ that describes the dynamics of the system, i.e. + +\begin{equation} + \ket{\psi'} = U \ket{\psi} +\end{equation} + +It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$ anymore. There are +however some statements that can still be made: + +\begin{equation} + \begin{aligned} + \ket{\psi'} &= U \ket{\psi} \\ + &= U S^{(i)} \ket{\psi} \\ + &= U S^{(i)} U^\dagger U\ket{\psi} \\ + &= U S^{(i)} U^\dagger \ket{\psi'} \\ + &= S^{\prime(i)} \ket{\psi'} \\ + \end{aligned} +\end{equation} + +Note that in \ref{def:stabilizer} it has been demanded that stabilizers are a +subgroup of the multilocal Pauli operators. This does not hold true for an arbitrary +$U$ but there exists a group for which $S'$ will be a set of stabilizers. + +\begin{definition} + For $n$ qbits + \begin{equation} + C_n := \left\{U \in SU(n) | UpU^\dagger \in P_n \forall p \in P_n\right\} + \end{equation} + + is called the Clifford group. $C_1 =: C_L$ is called the local Clifford group. +\end{definition} + +\begin{theorem} + \begin{enumerate} + \item{$C_L$ can be generated using only $H$ and $S$.} + \item{$C_L$ can be generated from $\sqrt{iZ} = \exp(\frac{i\pi}{4}) S^\dagger$ + and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i \\ -i & 1 \end{array}\right)$. + + Also $C_L$ is generated by a product of at most $5$ matrices $\sqrt{iZ}$, $\sqrt{-iX}$. + } + \item{$C_n$ can be generated using $C_L$ and $CZ$ or $CX$.} + \end{enumerate} +\end{theorem} + +\begin{proof} + \begin{enumerate} + \item{See \cite[Theorem 10.6]{nielsen_chuang_2010}} + \item{ + One can easily verify that $\sqrt{iZ} \in C_L$ and $\sqrt{-iX} \in C_L$. + Further one can show easily that (up to a global phase) + $H = \sqrt{iZ} \sqrt{-iX}^3 \sqrt{iZ}$ and $S = \sqrt{iZ}^3$. + + The length of the product can be seen when explicitly calculating + $C_L$. + } + \item{See \cite[Theorem 10.6]{nielsen_chuang_2010}} + \end{enumerate} +\end{proof} + +This is quite an important result: As under a transformation $U \in C_n$ $S'$ is a set of +$n$ independent stabilizers and $\ket{\psi'}$ is stabilized by $S'$ one can consider +the dynamics of the stabilizers instead of the actual state. This is considerably more +efficient as only $n$ stabilizers have to be modified, each being just the tensor +product of $n$ Pauli matrices. This has led to the simulation using stabilizer tableaux +\cite{gottesman_aaronson2008}. + +Interestingly also measurements are dynamics covered by the stabilizers. +When an observable $g_i \in \{\pm X_i, \pm Y_i \pm Z_i\}$ is measured