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presentation/main.tex
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@ -22,7 +22,7 @@
\setbeamercolor{background canvas}{bg=white!20}
\title{An Efficient Quantum Computing Simulator using a Graphical Description for Many-Qbit Systems}
\subtitle{Simulation in the Stabilizer Formalism}
\subtitle{Simulation in the Graph Formalism}
\author{Daniel Knüttel}
\date{21.02.2020}
\institute{Universität Regensburg}
@ -107,41 +107,45 @@
the bath statistically (\textit{Andreas Hackl}).}
\end{itemize}
}
\item{
Long Term: Fault tolerant computing.
}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Quantum Errors and Quantum Error Correction}
\begin{itemize}
\item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation).
%\pause
\item Fault tolerant QC needs a way to correct for those errors.
%\pause
\item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}.
\item{Parts of the theoretical description of quantum errors can be used for physical
problems (see above).}
\end{itemize}
\end{frame}
}
%{
%\begin{frame}{Quantum Errors and Quantum Error Correction}
% \begin{itemize}
% \item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation).
% %\pause
% \item Fault tolerant QC needs a way to correct for those errors.
% %\pause
% \item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}.
% \item{Parts of the theoretical description of quantum errors can be used for physical
% problems (see above).}
% \end{itemize}
%\end{frame}
%}
\section{Binary Quantum Computing}
{
\begin{frame}{Qbits and Gates}
\begin{itemize}
\item{A qbit is a two level quantum mechanical system. The Hilbert space
$\mathcal{H}$ is two dimensional and has the basis vectors $\ket{0}, \ket{1}$.}
\item{$n$ qbits have the Hilbert space $\mathcal{H}^{\otimes n}$.}
\item{Gates on a quantum computers are unitary operators acting on $\mathcal{H}^{\otimes n}$.}
\end{itemize}
\end{frame}
}
%{
%\begin{frame}{Qbits and Gates}
% \begin{itemize}
% \item{A qbit is a two level quantum mechanical system. The Hilbert space
% $\mathcal{H}$ is two dimensional and has the basis vectors $\ket{0}, \ket{1}$.}
% \item{$n$ qbits have the Hilbert space $\mathcal{H}^{\otimes n}$.}
% \item{Gates on a quantum computers are unitary operators acting on $\mathcal{H}^{\otimes n}$.}
% \end{itemize}
%\end{frame}
%}
{
\begin{frame}{Notable Gates on One Qbit}
\begin{frame}{One Qbit}
\begin{itemize}
\item{One qbit has the Hilbert space $\mathcal{H}$ with basis vectors $\ket{0}$, $\ket{1}$.}
\item{The Pauli matrices $X, Y, Z$ are gates commonly used.
$X$ is also called the $NOT$ gate as it maps $\ket{0}$ to $\ket{1}$ and
vice versa.
@ -157,22 +161,22 @@
\end{frame}
}
{
\begin{frame}{Notable gates on $n$ Qbits}
\begin{itemize}
\item{For a unitary $U$ acting on $\mathcal{H}$
\begin{equation}
U_i := \left(\bigotimes\limits_{l < i} I\right) \otimes U \otimes \left(\bigotimes\limits_{l > i} I\right)
\end{equation}
is the $U$ gate acting on qbit $i$.}
\item{For two qbits $i\neq j$ the controlled $X$ gate is
\begin{equation}
CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i.
\end{equation}}
\end{itemize}
\end{frame}
}
%{
%\begin{frame}{Notable gates on $n$ Qbits}
% \begin{itemize}
% \item{For a unitary $U$ acting on $\mathcal{H}$
% \begin{equation}
% U_i := \left(\bigotimes\limits_{l < i} I\right) \otimes U \otimes \left(\bigotimes\limits_{l > i} I\right)
% \end{equation}
% is the $U$ gate acting on qbit $i$.}
%
% \item{For two qbits $i\neq j$ the controlled $X$ gate is
% \begin{equation}
% CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i.
% \end{equation}}
% \end{itemize}
%\end{frame}
%}
{
\begin{frame}{Universal Gates}
\begin{itemize}
@ -180,7 +184,12 @@
\item{Similarly to classical computers a universal set of operations is required.}
\item{One can show that any unitary acting on $\mathcal{H}^{\otimes n}$
can be generated using the $CX$ and universal gates
acting on $\mathcal{H}$.}
acting on $\mathcal{H}$ with
\begin{equation}
CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i.
\end{equation}
}
\item{The gates $\{H, R_\phi\}$ are universal on $\mathcal{H}$.}
\end{itemize}
\end{frame}
@ -190,14 +199,16 @@
\begin{frame}{Measurements and Computational Hardness}
\begin{itemize}
\item{When measuring a qbit $i$ the observable $Z_i$ is measured.}
\item{The Hilbert space $\mathcal{H}^{\otimes n}$ has the integer basis
\begin{equation}
\ket{j} = \ket{\mbox{0b}j_{n-1}...j_1j_0} = \bigotimes\limits_{l=0}^{n-1} \ket{j_l}.
\end{equation}
}
\item{A general state $\ket{\psi}$ has $2^n$ coefficients in this basis.}
\item{In general an operation on the state $\ket{\psi}$ will have to update $2^n$ coefficients.
Mapping a general state $\ket{\psi}$ to $\ket{\psi'}$ cannot be performed in $\mbox{poly}(n)$ time.}
\item{Operations on the $2^n$ dimensional state will have to update $2^n$ complex
coefficients. This cannot be performed in $\mbox{poly}(n)$ time.}
%\item{The Hilbert space $\mathcal{H}^{\otimes n}$ has the integer basis
% \begin{equation}
% \ket{j} = \ket{\mbox{0b}j_{n-1}...j_1j_0} = \bigotimes\limits_{l=0}^{n-1} \ket{j_l}.
% \end{equation}
%}
%\item{A general state $\ket{\psi}$ has $2^n$ coefficients in this basis.}
%\item{In general an operation on the state $\ket{\psi}$ will have to update $2^n$ coefficients.
% Mapping a general state $\ket{\psi}$ to $\ket{\psi'}$ cannot be performed in $\mbox{poly}(n)$ time.}
\end{itemize}
\end{frame}
@ -312,71 +323,71 @@
%\pause
\item{
$C_n$ is the normalizer of $P_n$, i.e. it maps $P_n$ to itself.\\
One can show that $C_n$ is generated by $H, R_\frac{\pi}{2}, CZ_{i,j}$.
$C_n$ is generated by $H, R_\frac{\pi}{2}, CZ_{i,j}$.
}
\item{One can show that the normalizers of a group such as the Pauli group can be simulated efficiently.}
\item{The simulation using $P_n$ is called stabilizer formalism.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Stabilizers and Stabilizer Spaces}
\begin{itemize}
\item{
Choose a finite commuting Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
One can show that all elements of $S$ are hermitian.
}
\item{
One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$
if every element of $S$ can be expressed as a product of the $S^{(i)}$ and the
$S^{(i)}$ are the minimal amount of matrices required for this property.
}
\item{
One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$ the stabilizer space
$V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = \ket{\psi} \forall i\}$
has dimension $1$. $\ket{\psi}$ is therefore up to a trivial phase unique.
}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Some Notable Stabilizer States}
\begin{itemize}
\item{The state $\ket{\mbox{0b}00}$ is stabilized by
$\langle Z_0, Z_1\rangle$.}
\item{Applying the Hadamard gate to the first qbit changes the state to
$\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}01}\right)$.
This state is stabilized by
$\langle H_0 Z_0 H_0^\dagger, Z_1 \rangle = \langle X_0, Z_1 \rangle$.
}
\item{Applying a $CX_{1, 0}$ gate yields
$\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}11}\right)$
the famous EPR/Bell state which is stabilized by
$\langle CX_{1, 0} X_0 CX_{1, 0}^\dagger, CX_{1, 0} Z_1 CX_{1, 0}^\dagger \rangle = \langle X_0 X_1, Z_0 Z_1 \rangle$.
}
\item{When measuring qbit $0$ the resulting state is either $\ket{\mbox{0b}00}$ or $\ket{\mbox{0b}11}$
and the stabilizers are either $\langle Z_0, Z_1\rangle$ or $\langle -Z_0, -Z_1\rangle$.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Dynamics and Measurements}
\begin{itemize}
\item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state.
The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.}
\item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.}
\item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are
possible.}
\end{itemize}
\end{frame}
}
%{
%\begin{frame}{Stabilizers and Stabilizer Spaces}
% \begin{itemize}
% \item{
% Choose a finite commuting Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
% One can show that all elements of $S$ are hermitian.
% }
% \item{
% One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$
% if every element of $S$ can be expressed as a product of the $S^{(i)}$ and the
% $S^{(i)}$ are the minimal amount of matrices required for this property.
% }
% \item{
% One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$ the stabilizer space
% $V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = \ket{\psi} \forall i\}$
% has dimension $1$. $\ket{\psi}$ is therefore up to a trivial phase unique.
% }
% \end{itemize}
%
%\end{frame}
%}
%
%{
%\begin{frame}{Some Notable Stabilizer States}
% \begin{itemize}
% \item{The state $\ket{\mbox{0b}00}$ is stabilized by
% $\langle Z_0, Z_1\rangle$.}
% \item{Applying the Hadamard gate to the first qbit changes the state to
% $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}01}\right)$.
% This state is stabilized by
% $\langle H_0 Z_0 H_0^\dagger, Z_1 \rangle = \langle X_0, Z_1 \rangle$.
% }
% \item{Applying a $CX_{1, 0}$ gate yields
% $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}11}\right)$
% the famous EPR/Bell state which is stabilized by
% $\langle CX_{1, 0} X_0 CX_{1, 0}^\dagger, CX_{1, 0} Z_1 CX_{1, 0}^\dagger \rangle = \langle X_0 X_1, Z_0 Z_1 \rangle$.
% }
% \item{When measuring qbit $0$ the resulting state is either $\ket{\mbox{0b}00}$ or $\ket{\mbox{0b}11}$
% and the stabilizers are either $\langle Z_0, Z_1\rangle$ or $\langle -Z_0, -Z_1\rangle$.}
% \end{itemize}
%\end{frame}
%}
%
%{
%\begin{frame}{Dynamics and Measurements}
% \begin{itemize}
% \item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state.
% The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.}
% \item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.}
% \item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are
% possible.}
% \end{itemize}
%\end{frame}
%}
{
\begin{frame}{Graphical States}
\begin{itemize}
\item{The graphical representation is an optimized way to write
the stabilizers.}
\item{
For a set of vertices $V = \{0, ..., n-1\}$, some edges
$E \subset V \otimes V$, and local Clifford operators
@ -390,7 +401,8 @@
where $\ket{+} = H\ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$.
}
\item{One can show that all stabilizer states can be brought into this form.}
\item{The stabilizers of this state are $K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j$.}
\item{For the derivation of the stabilizer and graphical states see my bachelor's thesis.}
%\item{The stabilizers of this state are $K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j$.}
%\item{The stabilizers associated with $(V, E, O)$ are
% \begin{equation}
% \left\langle\left(\bigotimes\limits_{j=0}^{n-1} o_j \right) K_G^{(i)} \left(\bigotimes\limits_{j=0}^{n-1} o_j \right)^\dagger\right\rangle_i
@ -405,12 +417,17 @@
\end{frame}
}
%{
%\begin{frame}{Graphical States}
% \begin{itemize}
% \end{itemize}
%\end{frame}
%}
{
\begin{frame}{Counting Graphical States}
\begin{itemize}
\item{Graphical states are finite and do not span a vector space.}
\item{There are $24$ local Clifford operators.}
\item{The amount of edges in a $n$ qbit state is $n^2 - n$.}
\item{For $n$ qbits there exist $24^n(n^2 - n)$ tuples $(V, E, O)$. But many of those states are equivalent.}
\item{There are fewer parameters that have to be updated under a quantum operation (see below).}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Dynamics of Graphical States}