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