diff --git a/presentation/main.tex b/presentation/main.tex index 0e41ccb..7a437e4 100644 --- a/presentation/main.tex +++ b/presentation/main.tex @@ -63,9 +63,9 @@ \begin{itemize} \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. - \pause + %\pause \item The (universal) quantum computer: able to simulate any unitary transformation on the system. \end{itemize} @@ -76,9 +76,9 @@ \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 + %\pause \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}. \end{itemize} @@ -111,7 +111,7 @@ A $n$-qbit system is the tensor product of the single-qbit systems. } } - \pause + %\pause %\item{ % 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) \end{equation} } - \pause + %\pause \item{ For $\bar{U} \in SU(2)$ and qbits $i \neq j$ \begin{equation} @@ -134,7 +134,7 @@ \end{equation} is the controlled $\ket{U}$ gate acting on $i$ with control-qbit $j$. } - \pause + %\pause \end{itemize} \end{frame} } @@ -150,13 +150,13 @@ H := \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1\end{array}\right) \end{equation} which transforms from the $Z$ to the $X$ basis} - \pause + %\pause \item{the rotation gate \begin{equation} R_{\phi} := \left(\begin{array}{cc} 1 & 0 \\ 0 & \exp(i\phi)\end{array}\right) \end{equation} that performs a rotation around the $Z$ axis.} - \pause + %\pause \end{itemize} \end{frame} @@ -175,7 +175,7 @@ The $S$ gate transforms from $X$ to $Y$ basis. } - \pause + %\pause \item{ \textbf{Theorem} {\itshape @@ -196,7 +196,7 @@ \ket{j} = \ket{\mbox{0b}l_1...l_n} = \ket{l_1}_s \otimes ... \otimes \ket{l_n}s \end{equation} } - \pause + %\pause \item{ 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. To fix this problem quantum circuits have been introduced. } - \pause + %\pause \item{ Qbits are represented by horizontal lines. } - \pause + %\pause \item{ Gates acting on a qbit are boxes on the lines. } - \pause + %\pause \item{ Control-qbits are connected to the gate via a vertical line. } - \pause + %\pause \item{ Circuits are read left to right. } - \pause + %\pause \item{ Example: \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 \end{equation} } - \pause + %\pause \item{ The time evolution of such a system is given by the transfer matrix \begin{equation} T := \exp(-itH) \in SU(2^n) \end{equation} } - \pause + %\pause \item{ 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 @@ -334,7 +334,7 @@ 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. }} - \pause + %\pause \item{ \textbf{Definition} {\itshape @@ -342,7 +342,7 @@ is called the Clifford group } } - \pause + %\pause \item{ One can show that $C_n$ is generated by $H, S, CZ_{i,j}$. } @@ -370,7 +370,7 @@ \item{ The generators are not unique. For instance $C_1$ can be generated using $H, S$ or $\sqrt{-iX}, \sqrt{iZ}$. } - \pause + %\pause \item{ The generators of a group have some kind of independence property. }