removed the pauses for now

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Daniel Knüttel 2020-03-09 14:40:49 +01:00
parent d6d300a1b2
commit c9cc7b914c

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@ -63,9 +63,9 @@
\begin{itemize} \begin{itemize}
\item Some (physical) problems are classically hard to solve. \item Some (physical) problems are classically hard to solve.
\pause %\pause
\item The quantum simulator: Mapping a hard problem to quantum hardware that can simulate this specific problem. \item The quantum simulator: Mapping a hard problem to quantum hardware that can simulate this specific problem.
\pause %\pause
\item The (universal) quantum computer: able to simulate any unitary transformation on the system. \item The (universal) quantum computer: able to simulate any unitary transformation on the system.
\end{itemize} \end{itemize}
@ -76,9 +76,9 @@
\begin{frame}{Quantum Errors and Quantum Error Correction} \begin{frame}{Quantum Errors and Quantum Error Correction}
\begin{itemize} \begin{itemize}
\item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation). \item Quantum systems at non-zero temperature often have dephasing effects and a finite population lifetime (relaxation).
\pause %\pause
\item Fault tolerant QC needs a way to correct for those errors. \item Fault tolerant QC needs a way to correct for those errors.
\pause %\pause
\item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}. \item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}.
\end{itemize} \end{itemize}
@ -111,7 +111,7 @@
A $n$-qbit system is the tensor product of the single-qbit systems. A $n$-qbit system is the tensor product of the single-qbit systems.
} }
} }
\pause %\pause
%\item{ %\item{
% For $n$ qbits define the integer state $\ket{j}$ as % For $n$ qbits define the integer state $\ket{j}$ as
@ -126,7 +126,7 @@
U_i := \left(\bigotimes\limits_{k < i} I\right) \otimes \bar{U} \otimes \left(\bigotimes\limits_{k > i} I\right) U_i := \left(\bigotimes\limits_{k < i} I\right) \otimes \bar{U} \otimes \left(\bigotimes\limits_{k > i} I\right)
\end{equation} \end{equation}
} }
\pause %\pause
\item{ \item{
For $\bar{U} \in SU(2)$ and qbits $i \neq j$ For $\bar{U} \in SU(2)$ and qbits $i \neq j$
\begin{equation} \begin{equation}
@ -134,7 +134,7 @@
\end{equation} \end{equation}
is the controlled $\ket{U}$ gate acting on $i$ with control-qbit $j$. is the controlled $\ket{U}$ gate acting on $i$ with control-qbit $j$.
} }
\pause %\pause
\end{itemize} \end{itemize}
\end{frame} \end{frame}
} }
@ -150,13 +150,13 @@
H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right)
\end{equation} \end{equation}
which transforms from the $Z$ to the $X$ basis} which transforms from the $Z$ to the $X$ basis}
\pause %\pause
\item{the rotation gate \item{the rotation gate
\begin{equation} \begin{equation}
R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right) R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right)
\end{equation} \end{equation}
that performs a rotation around the $Z$ axis.} that performs a rotation around the $Z$ axis.}
\pause %\pause
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -175,7 +175,7 @@
The $S$ gate transforms from $X$ to $Y$ basis. The $S$ gate transforms from $X$ to $Y$ basis.
} }
\pause %\pause
\item{ \item{
\textbf{Theorem} \textbf{Theorem}
{\itshape {\itshape
@ -196,7 +196,7 @@
\ket{j} = \ket{\mbox{0b}l_1...l_n} = \ket{l_1}_s \otimes ... \otimes \ket{l_n}s \ket{j} = \ket{\mbox{0b}l_1...l_n} = \ket{l_1}_s \otimes ... \otimes \ket{l_n}s
\end{equation} \end{equation}
} }
\pause %\pause
\item{ \item{
For $n$ qbits there exist $2^n$ such states and they form a basis For $n$ qbits there exist $2^n$ such states and they form a basis
@ -235,23 +235,23 @@
Writing a unitary transformation as a product of the generator gates is unreadable. Writing a unitary transformation as a product of the generator gates is unreadable.
To fix this problem quantum circuits have been introduced. To fix this problem quantum circuits have been introduced.
} }
\pause %\pause
\item{ \item{
Qbits are represented by horizontal lines. Qbits are represented by horizontal lines.
} }
\pause %\pause
\item{ \item{
Gates acting on a qbit are boxes on the lines. Gates acting on a qbit are boxes on the lines.
} }
\pause %\pause
\item{ \item{
Control-qbits are connected to the gate via a vertical line. Control-qbits are connected to the gate via a vertical line.
} }
\pause %\pause
\item{ \item{
Circuits are read left to right. Circuits are read left to right.
} }
\pause %\pause
\item{ \item{
Example: Example:
\Qcircuit @C=1em @R=.7em { \Qcircuit @C=1em @R=.7em {
@ -273,14 +273,14 @@
H = -\sum\limits_{i=1}^{n-1} Z_i Z_{i-1} + g\sum\limits_{i=0}^{n-1} X_i H = -\sum\limits_{i=1}^{n-1} Z_i Z_{i-1} + g\sum\limits_{i=0}^{n-1} X_i
\end{equation} \end{equation}
} }
\pause %\pause
\item{ \item{
The time evolution of such a system is given by the transfer matrix The time evolution of such a system is given by the transfer matrix
\begin{equation} \begin{equation}
T := \exp(-itH) \in SU(2^n) T := \exp(-itH) \in SU(2^n)
\end{equation} \end{equation}
} }
\pause %\pause
\item{ \item{
By associating every qbit with one spin (both are two-level systems) By associating every qbit with one spin (both are two-level systems)
one should be able to simulate the behaviour of the spin chain using one should be able to simulate the behaviour of the spin chain using
@ -334,7 +334,7 @@
is called the Pauli group.\\ is called the Pauli group.\\
$P_n := \left\{p_1 \otimes ... \otimes p_n \middle| p_i \in P \right\}$ is called the multilocal Pauli group. $P_n := \left\{p_1 \otimes ... \otimes p_n \middle| p_i \in P \right\}$ is called the multilocal Pauli group.
}} }}
\pause %\pause
\item{ \item{
\textbf{Definition} \textbf{Definition}
{\itshape {\itshape
@ -342,7 +342,7 @@
is called the Clifford group is called the Clifford group
} }
} }
\pause %\pause
\item{ \item{
One can show that $C_n$ is generated by $H, S, CZ_{i,j}$. One can show that $C_n$ is generated by $H, S, CZ_{i,j}$.
} }
@ -370,7 +370,7 @@
\item{ \item{
The generators are not unique. For instance $C_1$ can be generated using $H, S$ or $\sqrt{-iX}, \sqrt{iZ}$. The generators are not unique. For instance $C_1$ can be generated using $H, S$ or $\sqrt{-iX}, \sqrt{iZ}$.
} }
\pause %\pause
\item{ \item{
The generators of a group have some kind of independence property. The generators of a group have some kind of independence property.
} }