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