some work on the presentation
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@ -12,14 +12,10 @@
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\usepackage{listings}
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%\usepackage{struktex}
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\usepackage{qcircuit}
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\usepackage{adjustbox}
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\newtheorem{definition}{Definition}
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\newtheorem{postulate}{Postulate}
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\newtheorem{corrolary}{Corrolary}
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\newtheorem{lemma}{Lemma}
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\newtheorem{theorem}{Theorem}
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\usetheme[progressbar=frametitle]{metropolis}
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\usetheme{metropolis}
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\setbeamercolor{background canvas}{bg=white!20}
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@ -66,7 +62,9 @@
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\begin{itemize}
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\item Some (physical) problems are classically hard to solve.
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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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\pause
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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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@ -77,7 +75,9 @@
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\begin{frame}{Quantum Errors and Quantum Error Correction}
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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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\pause
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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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\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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@ -87,22 +87,387 @@
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\section{Binary Quantum Computing}
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{
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\begin{frame}{Qbits}
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\begin{definition}
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\textbf{Definition}
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{\itshape
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A qbit is a two-level quantum mechanical system $\ket{0}, \ket{1}$ with $\braket{0}{1} = 0$.
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In the following $Z = \sigma_Z, X = \sigma_X, Y = \sigma_Y, I = \id$ will be used.
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\end{definition}
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In the following $Z = \sigma_Z, X = \sigma_X, Y = \sigma_Y$ will be used. $I$ is the identity matrix.
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}
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Where $Z\ket{0} = +1\ket{0}$ and $Z\ket{1} = -1\ket{1}$.
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\begin{postulate}
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A $n$-qbit system is the tensor product of the single-qbit systems.
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\end{postulate}
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\end{frame}
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}
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{
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\begin{frame}{Qbits and Gates}
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\begin{itemize}
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\item{
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\textbf{Postulate}
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{\itshape
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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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\pause
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%\item{
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% For $n$ qbits define the integer state $\ket{j}$ as
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% \begin{equation}
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% \ket{j} := \ket{\mbox{0b}i_0i_1...i_{n-1}} := \ket{i_0}_s \otimes \ket{i_1}_s \otimes ... \otimes \ket{i_{n-1}}_s
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% \end{equation}
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%}
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\item{
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A transformation $U \in SU(2^n)$ is called \textit{gate} acting on the system.
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For $\bar{U} \in SU(2)$ the gate $U_i$ acting on qbit $i$ is given by
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\begin{equation}
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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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}
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\pause
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\item{
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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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CU_{i,j} := \ket{1}\bra{1}_j \otimes U_i + \ket{0}\bra{0}_j \otimes I
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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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}
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\pause
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Gates}
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Some notable gates are
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\begin{itemize}
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\item{
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the Hadamard gate
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\begin{equation}
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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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which transforms from the $Z$ to the $X$ basis}
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\pause
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\item{the rotation gate
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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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\end{equation}
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that performs a rotation around the $Z$ axis.}
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\pause
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Gates}
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\begin{itemize}
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\item{The $Z$ and $S$ gates are given by:
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\begin{equation}
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Z := \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right) = R_{\pi}
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\end{equation}
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\begin{equation}
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S := \left(\begin{array}{cc} 1 & 0 \\ 0 & i\end{array}\right) = R_{\frac{\pi}{2}}
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\end{equation}
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The $S$ gate transforms from $X$ to $Y$ basis.
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}
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\pause
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\item{
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\textbf{Theorem}
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{\itshape
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Any gate $U \in SU(2^n)$ can be approximated arbitrarely good using the $H$, $R_\phi$
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and $CZ_{i,j}$ gate.
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}
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Integer States}
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\begin{itemize}
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\item{
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The eigenstates of the $Z_i$ are called integer states. They have the form
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\begin{equation}
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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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}
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\pause
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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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\begin{equation}
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\ket{\psi} = \sum\limits_{i=0}^{n-1} \ket{i}\braket{i}{\psi} = \sum\limits_{i=0}^{n-1} c_i\ket{i}
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\end{equation}
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with the condition $\sum\limits_{i=0}^{n-1} c_i^2 = 1$.
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Measurement}
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\begin{itemize}
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\item{Measurements are performed in $Z$ basis, i.e. for a qbit $i$
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$Z_i$ is measured.
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}
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\item{
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The results of the measurements are associated with a classical result $s \in \{0, 1\}$
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using
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\begin{equation}
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Z_i\ket{\psi'} = (-1)^s \ket{\psi'}
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\end{equation}
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after the measurement.
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Quantum Circuits}
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\begin{itemize}
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\item{
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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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}
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\pause
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\item{
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Qbits are represented by horizontal lines.
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}
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\pause
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\item{
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Gates acting on a qbit are boxes on the lines.
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}
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\pause
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\item{
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Control-qbits are connected to the gate via a vertical line.
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}
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\pause
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\item{
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Circuits are read left to right.
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}
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\pause
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\item{
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Example:
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\Qcircuit @C=1em @R=.7em {
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& \gate{H} & \ctrl{1} & \qw &\qw \\
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& \gate{H} & \gate{Z} & \gate{H} &\qw \\
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}
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
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\begin{itemize}
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\item{
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For a set of $n$ spins in a magnetic field one can rescale the Hamiltonian
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of the system to
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\begin{equation}
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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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}
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\pause
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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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\begin{equation}
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T := \exp(-itH) \in SU(2^n)
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\end{equation}
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}
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\pause
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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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one should be able to simulate the behaviour of the spin chain using
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a quantum computer.
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
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\begin{itemize}
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\item{
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Trotterizing the matrix exponential
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\begin{equation}
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\exp(t(A + B)) = \left(\exp(\frac{t}{N}A)\exp(\frac{t}{N}B)\right)^N + \mathcal{O}\left(\frac{t^2}{N^2}\right)
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\end{equation}
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}
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\item{
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For $n=3$ spins one gets a circuit
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{\centering\adjustbox{max width=\textwidth}{
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\Qcircuit @C=1em @R=.7em {
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& \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw &\qw \\
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& \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \gate{X} & \gate{R_{-\frac{t}{2N}}} & \gate{X} & \gate{R_{\frac{t}{2N}}} & \gate{X} & \gate{X} &\qw \\
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& \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} & \gate{H} & \gate{R_{-2\frac{gt}{2N}}} & \gate{H} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \gate{R_{\frac{gt}{2N}}} & \gate{X} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrl{-1} & \qw & \qw & \qw & \qw & \ctrl{-1} &\qw \\
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}
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}}
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}
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\item{Applying this circuit $N$ times gives an approximation for the time evolution of a state.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Case Study: Spin Chain in a Magnetic Field}
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=\linewidth]{spin_chain/time_evo_6spin_g3.png}
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\end{center}
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\end{figure}
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\end{frame}
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}
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\section{Stabilizers}
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{
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\begin{frame}{The multilocal Pauli Group and the Clifford Group}
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\begin{itemize}
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\item{\textbf{Definition}
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{\itshape
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$P := \{\pm1, \pm i\} \cdot \{X, Y, Z, I\}$ 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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}}
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\pause
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\item{
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\textbf{Definition}
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{\itshape
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$C_n := \left\{U \in SU(2^n) \middle| \forall p \in P_n: U^\dagger p U \in P_n\right\}$
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is called the Clifford group
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}
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}
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\pause
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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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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Generators of a Group}
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\begin{itemize}
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\item{
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\textbf{Definition}
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{\itshape
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For a finite group $G$ one says $G$ is generated by $g_1, ..., g_N$ iff any $g \in G$ can be expressed
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as a product of $g_1, ..., g_N$. These generators are chosen to be the minimal set for which this
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condition holds.
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One also writes
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\begin{equation}
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G = \langle g_1,...,g_N \rangle \equiv \langle g_i\rangle_i
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\end{equation}
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}
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}
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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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}
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\item{
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The generators of a group have some kind of independence property.
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Stabilizers and Stabilizer Spaces}
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\begin{itemize}
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\item{
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\textbf{Definition}
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{\itshape
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A finite abelian subgroup $S$ of $P_n$ is called a set of stabilizers iff $-I \notin S$ and
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all elements of $S$ commute.
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}
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}
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\item{
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From $-I \notin S$ follows that all elements of $S$ are hermitian.
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}
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\item{
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From the definition as tensor products of Pauli matrices follows that the
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elements of $S$ have eigenvalues $\pm1$.
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}
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\item{
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These properties together yield that all elements of $S$ can be diagonalized simultaneously.
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Further there exists a vector space $V_S$ with $s \ket{\psi} = +1 \ket{\psi}$ for all $s \in S$
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and all $\ket{\psi} \in V_S$. This space is called the stabilizer space of $S$
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and all $\ket{\psi}$ are called stabilizer states.
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Stabilizer States}
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\begin{itemize}
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\item{
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One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$ the stabilizer space $V_S$
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has dimension $1$.
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}
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\item{
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Therefore the state $\ket{\psi}$ that is the +1 eigenstate of all stabilizers is (up to a
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global phase) unique.
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}
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\item{Notable stabilizer states are:
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\begin{itemize}
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\item{ $\ket{0b0..0} = \ket{0}\otimes ... \otimes \ket{0}$ and $\ket{0b1..1} = \ket{1}\otimes ... \otimes \ket{1}$
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which are stabilized by $\langle Z_i \rangle_i$ and $\langle -Z_i \rangle_i$ respectively.
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}
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\item{
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$\ket{+}$ stabilized by $\langle X_i \rangle_i$ (and $\ket{-}$ analogously).
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}
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\item{
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$\ket{0b00} + \ket{0b11}$ the Bell/EPR state which is stabilized by $\langle X_1X_2, Z_1Z_2\rangle$.
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}
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\end{itemize}
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}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Dynamics of Stabilizer States}
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\begin{itemize}
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\item{
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Under a transformation $U \in SU(2^n)$ the state changes to
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\begin{equation}
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\ket{\psi'} = U \ket{\psi}
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\end{equation}
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}
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\item{
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The stabilizers of $\ket{\psi}$ change to
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\begin{equation}
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\ket{\psi'} = U\ket{\psi} = US^{(i)}\ket{\psi} = US^{(i)}U^\dagger U\ket{\psi} = US^{(i)}U^\dagger\ket{\psi'}
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\end{equation}
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this gives that if $U \in C_n$ $\ket{\psi'}$ is a stabilizer state again with the stabilizers
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$S' = \langle US^{(i)}U^\dagger\rangle_{i=1,...,n}$
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}
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\item{
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Under the transformations $C_n$ one can describe the dynamics of the stabilizer states by their
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stabilizers.
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}
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\item{Because the stabilizers are given by $n$ matrices which are the tensor product of $n$ Pauli matrices
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this can be simulated in $n^2$ time instead of $2^n$.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Measurements on Stabilizer States}
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\begin{itemize}
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\item{
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}
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\end{itemize}
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\end{frame}
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}
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\end{document}
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@ -1,64 +1,87 @@
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import numpy as np
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import matplotlib.pyplot as plt
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from pyqcs import State, sample
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from transfer_matrix import T_time_slice
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from hamiltonian import H
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import matplotlib
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from scipy.linalg import expm
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from pyqcs import State, sample
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nqbits = 2
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g = 0.20
|
||||
N = 50
|
||||
from transfer_matrix import T_time_slice
|
||||
from hamiltonian import H
|
||||
from bootstrap import bootstrap
|
||||
|
||||
|
||||
np.random.seed(0xdeadbeef)
|
||||
|
||||
matplotlib.rcParams.update(
|
||||
{'errorbar.capsize': 2
|
||||
, 'figure.figsize': (16, 9)}
|
||||
)
|
||||
|
||||
nqbits = 6
|
||||
g = 3
|
||||
N_trot = 80
|
||||
t_stop = 9
|
||||
delta_t = 0.05
|
||||
delta_t = 0.09
|
||||
qbits = list(range(nqbits))
|
||||
|
||||
n_sample = 400
|
||||
n_sample = 2200
|
||||
measure = 0b10
|
||||
|
||||
measure_coefficient_mask = [False if (i & measure) else True for i in range(2**nqbits)]
|
||||
|
||||
|
||||
results_qc = []
|
||||
results_np = []
|
||||
errors_sampling = []
|
||||
|
||||
print()
|
||||
for t in np.arange(0, t_stop, delta_t):
|
||||
# QC simulation
|
||||
state = State.new_zero_state(nqbits)
|
||||
|
||||
for _ in range(N):
|
||||
state = T_time_slice(qbits, t, g, N) * state
|
||||
T_dt = T_time_slice(qbits, t, g, N_trot)
|
||||
for _ in range(N_trot):
|
||||
state = T_dt * state
|
||||
|
||||
#result = sample(state, measure, n_sample)
|
||||
result = sample(state, measure, n_sample)
|
||||
results_qc.append(result[0] / n_sample)
|
||||
|
||||
#results_qc.append(result[0] / n_sample)
|
||||
errors_sampling.append(bootstrap(result[0], n_sample, n_sample // 4, n_sample // 10, np.average))
|
||||
|
||||
amplitude = np.sqrt(np.sum(np.abs(state._qm_state[[False if (i & measure) else True for i in range(2**nqbits)]])**2))
|
||||
results_qc.append(amplitude)
|
||||
#amplitude = np.sqrt(np.sum(np.abs(state._qm_state[measure_coefficient_mask])**2))
|
||||
#results_qc.append(amplitude)
|
||||
|
||||
# Simulation using matrices
|
||||
np_zero_state = np.zeros(2**nqbits)
|
||||
np_zero_state[0] = 1
|
||||
|
||||
itH = np.matrix(-1j * t * H(nqbits, g))
|
||||
itH = np.matrix(-0.5j * t * H(nqbits, g))
|
||||
T = expm(itH)
|
||||
|
||||
np_state = T.dot(np_zero_state)
|
||||
amplitude = np.sqrt(np.sum(np.abs(np_state[[False if (i & measure) else True for i in range(2**nqbits)]])**2))
|
||||
amplitude = (np.sum(np.abs(np_state[measure_coefficient_mask])**2))
|
||||
results_np.append(amplitude)
|
||||
|
||||
print(f"simulating... {int(t/t_stop*100)} % ", end="\r")
|
||||
print()
|
||||
print("done.")
|
||||
|
||||
errors_trotter = np.arange(0, t_stop, delta_t)**2 / N**2
|
||||
results_qc = np.array(results_qc)
|
||||
|
||||
errors_trotter = (np.arange(0, t_stop, delta_t) * g)**2 / N_trot**2
|
||||
errors_sampling = np.array(errors_sampling)
|
||||
|
||||
|
||||
h0 = plt.errorbar(np.arange(0, t_stop, delta_t), results_qc, yerr=errors_trotter, label=f"Quantum computing ({n_sample} samples, {N} trotterization steps)")
|
||||
h0 = plt.errorbar(np.arange(0, t_stop, delta_t)
|
||||
, results_qc
|
||||
, yerr=(errors_trotter + errors_sampling)
|
||||
, label=f"Quantum computing ({n_sample} samples, {N_trot} trotterization steps)"
|
||||
, )
|
||||
h1, = plt.plot(np.arange(0, t_stop, delta_t), results_np, label="Classical simulation using explicit transfer matrix")
|
||||
plt.xlabel("t")
|
||||
plt.ylabel(r"$|0\rangle$ probability amplitude for second spin")
|
||||
plt.title(f"{nqbits} site spin chain with g={g} coupling to external field")
|
||||
plt.legend(handles=[h0, h1])
|
||||
|
||||
plt.savefig("time_evo_6spin_g3.png", dpi=400)
|
||||
plt.show()
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user