diff --git a/presentation/main.tex b/presentation/main.tex index 3aad38d..60936b5 100644 --- a/presentation/main.tex +++ b/presentation/main.tex @@ -22,7 +22,7 @@ \setbeamercolor{background canvas}{bg=white!20} \title{An Efficient Quantum Computing Simulator using a Graphical Description for Many-Qbit Systems} -\subtitle{Simulation in the Stabilizer Formalism} +\subtitle{Simulation in the Graph Formalism} \author{Daniel Knüttel} \date{21.02.2020} \institute{Universität Regensburg} @@ -107,41 +107,45 @@ the bath statistically (\textit{Andreas Hackl}).} \end{itemize} } + \item{ + Long Term: Fault tolerant computing. + } \end{itemize} \end{frame} } -{ -\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 - \item Fault tolerant QC needs a way to correct for those errors. - %\pause - \item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}. - \item{Parts of the theoretical description of quantum errors can be used for physical - problems (see above).} - \end{itemize} -\end{frame} -} +%{ +%\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 +% \item Fault tolerant QC needs a way to correct for those errors. +% %\pause +% \item Several strategies exist; one important class of quantum error correction codes are \textbf{stabilizer codes}. +% \item{Parts of the theoretical description of quantum errors can be used for physical +% problems (see above).} +% \end{itemize} +%\end{frame} +%} \section{Binary Quantum Computing} -{ -\begin{frame}{Qbits and Gates} - \begin{itemize} - \item{A qbit is a two level quantum mechanical system. The Hilbert space - $\mathcal{H}$ is two dimensional and has the basis vectors $\ket{0}, \ket{1}$.} - \item{$n$ qbits have the Hilbert space $\mathcal{H}^{\otimes n}$.} - \item{Gates on a quantum computers are unitary operators acting on $\mathcal{H}^{\otimes n}$.} - \end{itemize} -\end{frame} -} +%{ +%\begin{frame}{Qbits and Gates} +% \begin{itemize} +% \item{A qbit is a two level quantum mechanical system. The Hilbert space +% $\mathcal{H}$ is two dimensional and has the basis vectors $\ket{0}, \ket{1}$.} +% \item{$n$ qbits have the Hilbert space $\mathcal{H}^{\otimes n}$.} +% \item{Gates on a quantum computers are unitary operators acting on $\mathcal{H}^{\otimes n}$.} +% \end{itemize} +%\end{frame} +%} { -\begin{frame}{Notable Gates on One Qbit} +\begin{frame}{One Qbit} \begin{itemize} + \item{One qbit has the Hilbert space $\mathcal{H}$ with basis vectors $\ket{0}$, $\ket{1}$.} \item{The Pauli matrices $X, Y, Z$ are gates commonly used. $X$ is also called the $NOT$ gate as it maps $\ket{0}$ to $\ket{1}$ and vice versa. @@ -157,22 +161,22 @@ \end{frame} } -{ -\begin{frame}{Notable gates on $n$ Qbits} - \begin{itemize} - \item{For a unitary $U$ acting on $\mathcal{H}$ - \begin{equation} - U_i := \left(\bigotimes\limits_{l < i} I\right) \otimes U \otimes \left(\bigotimes\limits_{l > i} I\right) - \end{equation} - is the $U$ gate acting on qbit $i$.} - - \item{For two qbits $i\neq j$ the controlled $X$ gate is - \begin{equation} - CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i. - \end{equation}} - \end{itemize} -\end{frame} -} +%{ +%\begin{frame}{Notable gates on $n$ Qbits} +% \begin{itemize} +% \item{For a unitary $U$ acting on $\mathcal{H}$ +% \begin{equation} +% U_i := \left(\bigotimes\limits_{l < i} I\right) \otimes U \otimes \left(\bigotimes\limits_{l > i} I\right) +% \end{equation} +% is the $U$ gate acting on qbit $i$.} +% +% \item{For two qbits $i\neq j$ the controlled $X$ gate is +% \begin{equation} +% CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i. +% \end{equation}} +% \end{itemize} +%\end{frame} +%} { \begin{frame}{Universal Gates} \begin{itemize} @@ -180,7 +184,12 @@ \item{Similarly to classical computers a universal set of operations is required.} \item{One can show that any unitary acting on $\mathcal{H}^{\otimes n}$ can be generated using the $CX$ and universal gates - acting on $\mathcal{H}$.} + acting on $\mathcal{H}$ with + + \begin{equation} + CX_{i,j} = \ket{0}\bra{0}_j \otimes I_i + \ket{1}\bra{1}_j \otimes X_i. + \end{equation} + } \item{The gates $\{H, R_\phi\}$ are universal on $\mathcal{H}$.} \end{itemize} \end{frame} @@ -190,14 +199,16 @@ \begin{frame}{Measurements and Computational Hardness} \begin{itemize} \item{When measuring a qbit $i$ the observable $Z_i$ is measured.} - \item{The Hilbert space $\mathcal{H}^{\otimes n}$ has the integer basis - \begin{equation} - \ket{j} = \ket{\mbox{0b}j_{n-1}...j_1j_0} = \bigotimes\limits_{l=0}^{n-1} \ket{j_l}. - \end{equation} - } - \item{A general state $\ket{\psi}$ has $2^n$ coefficients in this basis.} - \item{In general an operation on the state $\ket{\psi}$ will have to update $2^n$ coefficients. - Mapping a general state $\ket{\psi}$ to $\ket{\psi'}$ cannot be performed in $\mbox{poly}(n)$ time.} + \item{Operations on the $2^n$ dimensional state will have to update $2^n$ complex + coefficients. This cannot be performed in $\mbox{poly}(n)$ time.} + %\item{The Hilbert space $\mathcal{H}^{\otimes n}$ has the integer basis + % \begin{equation} + % \ket{j} = \ket{\mbox{0b}j_{n-1}...j_1j_0} = \bigotimes\limits_{l=0}^{n-1} \ket{j_l}. + % \end{equation} + %} + %\item{A general state $\ket{\psi}$ has $2^n$ coefficients in this basis.} + %\item{In general an operation on the state $\ket{\psi}$ will have to update $2^n$ coefficients. + % Mapping a general state $\ket{\psi}$ to $\ket{\psi'}$ cannot be performed in $\mbox{poly}(n)$ time.} \end{itemize} \end{frame} @@ -312,71 +323,71 @@ %\pause \item{ $C_n$ is the normalizer of $P_n$, i.e. it maps $P_n$ to itself.\\ - One can show that $C_n$ is generated by $H, R_\frac{\pi}{2}, CZ_{i,j}$. + $C_n$ is generated by $H, R_\frac{\pi}{2}, CZ_{i,j}$. } + \item{One can show that the normalizers of a group such as the Pauli group can be simulated efficiently.} + \item{The simulation using $P_n$ is called stabilizer formalism.} \end{itemize} \end{frame} } -{ -\begin{frame}{Stabilizers and Stabilizer Spaces} - \begin{itemize} - \item{ - Choose a finite commuting Abelian subgroup $S$ of $P_n$ with $-I \notin S$. - One can show that all elements of $S$ are hermitian. - } - \item{ - One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$ - if every element of $S$ can be expressed as a product of the $S^{(i)}$ and the - $S^{(i)}$ are the minimal amount of matrices required for this property. - } - \item{ - One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$ the stabilizer space - $V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = \ket{\psi} \forall i\}$ - has dimension $1$. $\ket{\psi}$ is therefore up to a trivial phase unique. - } - \end{itemize} - -\end{frame} -} - -{ -\begin{frame}{Some Notable Stabilizer States} - \begin{itemize} - \item{The state $\ket{\mbox{0b}00}$ is stabilized by - $\langle Z_0, Z_1\rangle$.} - \item{Applying the Hadamard gate to the first qbit changes the state to - $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}01}\right)$. - This state is stabilized by - $\langle H_0 Z_0 H_0^\dagger, Z_1 \rangle = \langle X_0, Z_1 \rangle$. - } - \item{Applying a $CX_{1, 0}$ gate yields - $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}11}\right)$ - the famous EPR/Bell state which is stabilized by - $\langle CX_{1, 0} X_0 CX_{1, 0}^\dagger, CX_{1, 0} Z_1 CX_{1, 0}^\dagger \rangle = \langle X_0 X_1, Z_0 Z_1 \rangle$. - } - \item{When measuring qbit $0$ the resulting state is either $\ket{\mbox{0b}00}$ or $\ket{\mbox{0b}11}$ - and the stabilizers are either $\langle Z_0, Z_1\rangle$ or $\langle -Z_0, -Z_1\rangle$.} - \end{itemize} -\end{frame} -} - -{ -\begin{frame}{Dynamics and Measurements} - \begin{itemize} - \item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state. - The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.} - \item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.} - \item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are - possible.} - \end{itemize} -\end{frame} -} +%{ +%\begin{frame}{Stabilizers and Stabilizer Spaces} +% \begin{itemize} +% \item{ +% Choose a finite commuting Abelian subgroup $S$ of $P_n$ with $-I \notin S$. +% One can show that all elements of $S$ are hermitian. +% } +% \item{ +% One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$ +% if every element of $S$ can be expressed as a product of the $S^{(i)}$ and the +% $S^{(i)}$ are the minimal amount of matrices required for this property. +% } +% \item{ +% One can show that for $S = \langle S^{(i)} \rangle_{i=1,...,n}$ the stabilizer space +% $V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = \ket{\psi} \forall i\}$ +% has dimension $1$. $\ket{\psi}$ is therefore up to a trivial phase unique. +% } +% \end{itemize} +% +%\end{frame} +%} +% +%{ +%\begin{frame}{Some Notable Stabilizer States} +% \begin{itemize} +% \item{The state $\ket{\mbox{0b}00}$ is stabilized by +% $\langle Z_0, Z_1\rangle$.} +% \item{Applying the Hadamard gate to the first qbit changes the state to +% $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}01}\right)$. +% This state is stabilized by +% $\langle H_0 Z_0 H_0^\dagger, Z_1 \rangle = \langle X_0, Z_1 \rangle$. +% } +% \item{Applying a $CX_{1, 0}$ gate yields +% $\frac{1}{\sqrt{2}}\left(\ket{\mbox{0b}00} + \ket{\mbox{0b}11}\right)$ +% the famous EPR/Bell state which is stabilized by +% $\langle CX_{1, 0} X_0 CX_{1, 0}^\dagger, CX_{1, 0} Z_1 CX_{1, 0}^\dagger \rangle = \langle X_0 X_1, Z_0 Z_1 \rangle$. +% } +% \item{When measuring qbit $0$ the resulting state is either $\ket{\mbox{0b}00}$ or $\ket{\mbox{0b}11}$ +% and the stabilizers are either $\langle Z_0, Z_1\rangle$ or $\langle -Z_0, -Z_1\rangle$.} +% \end{itemize} +%\end{frame} +%} +% +%{ +%\begin{frame}{Dynamics and Measurements} +% \begin{itemize} +% \item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state. +% The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.} +% \item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.} +% \item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are +% possible.} +% \end{itemize} +%\end{frame} +%} { \begin{frame}{Graphical States} \begin{itemize} - \item{The graphical representation is an optimized way to write - the stabilizers.} \item{ For a set of vertices $V = \{0, ..., n-1\}$, some edges $E \subset V \otimes V$, and local Clifford operators @@ -390,7 +401,8 @@ where $\ket{+} = H\ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$. } \item{One can show that all stabilizer states can be brought into this form.} - \item{The stabilizers of this state are $K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j$.} + \item{For the derivation of the stabilizer and graphical states see my bachelor's thesis.} + %\item{The stabilizers of this state are $K_G^{(i)} = X_i \prod\limits_{\{i,j\} \in E} Z_j$.} %\item{The stabilizers associated with $(V, E, O)$ are % \begin{equation} % \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 @@ -405,12 +417,17 @@ \end{frame} } -%{ -%\begin{frame}{Graphical States} -% \begin{itemize} -% \end{itemize} -%\end{frame} -%} +{ +\begin{frame}{Counting Graphical States} + \begin{itemize} + \item{Graphical states are finite and do not span a vector space.} + \item{There are $24$ local Clifford operators.} + \item{The amount of edges in a $n$ qbit state is $n^2 - n$.} + \item{For $n$ qbits there exist $24^n(n^2 - n)$ tuples $(V, E, O)$. But many of those states are equivalent.} + \item{There are fewer parameters that have to be updated under a quantum operation (see below).} + \end{itemize} +\end{frame} +} { \begin{frame}{Dynamics of Graphical States}