From 587af80d1409c80863dc742735df40f69a113552 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Wed, 19 Feb 2020 12:36:56 +0100 Subject: [PATCH] some more work --- thesis/chapters/quantum_computing.tex | 34 ++++++++++++++++++++-- thesis/chapters/stabilizer.tex | 42 +++++++++++++-------------- thesis/main.bib | 8 +++++ 3 files changed, 61 insertions(+), 23 deletions(-) diff --git a/thesis/chapters/quantum_computing.tex b/thesis/chapters/quantum_computing.tex index 8a09f2f..a3369b7 100644 --- a/thesis/chapters/quantum_computing.tex +++ b/thesis/chapters/quantum_computing.tex @@ -142,8 +142,7 @@ $CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1} One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough to generate an arbitrary $N$ qbit gate\cite[Chapter 4.3]{kaye_ea2007}\cite{barenco_ea_1995}. -The matrix representation of $CX$ and $CZ$ for two qbits is given by (this is quickly verified by applying the -matrices to the basis states) +The matrix representation of $CX$ and $CZ$ for two qbits is given by \begin{equation} CX_{1, 0} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{array}\right) @@ -216,6 +215,37 @@ the gate, for instance the circuit for $CZ_{2, 1}CX_{2,0}$ is } \] +Several qbits can be abbreviated by writing a slash on the qbit line: + +\[ +\Qcircuit @C=1em @R=.7em { + & {/} \qw & \qw & \gate{Z^{\otimes n}} & \qw \\ +} +\] + + \subsection{Quantum Algorithms} +The great hope behind quantum computing is that it will speed up some computations +exponentially using algorithms that utilize the laws of quantum mechanics. Current +algorithms are based upon quantum search and quantum fourier transform \cite{nielsen_chuang_2010}. +The latter is particular interesting for physical problems as it is used in the phase estimation +algorithm that can be used to analyze the spectrum of the transfer matrix + +\begin{equation} + T = \exp(itH) +\end{equation} + +The eigenvalue of $T$ can now be estimated by using the phase estimation circuit: + +\[ +\Qcircuit @C=1em @R=.7em { + & \lstick{\ket{0}} & {/^N} \qw & \gate{H^{\otimes n}} & \ctrl{1} & \gate{FT^\dagger} & \qw & \meter & \rstick{x}\\ + & \lstick{\ket{\varphi}} & {/} \qw & \qw & \gate{T} & \qw & {/} \qw & \qw& \rstick{\ket{\varphi}} \\ +} +\] + +Where $T\ket{\varphi} = \exp(2\pi i\varphi) \ket{\varphi}$ and the measurement result $\tilde{\varphi} = \frac{x}{2^n}$ is an +estimate for $\varphi$. If a success rate of $1-\epsilon$ and an accuracy of $| \varphi - \tilde{\varphi} | < 2^{-m}$ +is wanted $N = m + \log_2(2 + \frac{1}{2\epsilon})$ qbits are required\cite{nielsen_chuang_2010}\cite{lehner2019}. diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 36aec57..62a172f 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -31,7 +31,7 @@ the elements of $P$ either commute or anticommute. For $n$ qbits \begin{equation} - P_n := \left\{\bigotimes\limits_{i=0}^{n-1} p_i | p_i \in P\right\} + P_n := \left\{\bigotimes\limits_{i=0}^{n-1} p_i \middle| p_i \in P\right\} \end{equation} is called the multilocal Pauli group on $n$ qbits. @@ -68,12 +68,12 @@ via the tensor product as do the (anti-)commutator relationships. \begin{enumerate} \item{$(iI)^2 = (-iI)^2 = -I$. Which contradicts the definition of $S$.} \item{From the definition of $S$ ($G_n$ respectively) follows that any - $S^{(i)} \in S$ has the form $\pm i^l (\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$ where - $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$ + $S^{(i)} \in S$ has the form $\pm i^l \left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$ where + $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $\left(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j\right)$ is hermitian $(S^{(i)})^2$ is either $+I$ or $-I$. As $-I \notin S$ $(S^{(i)})^2 = I$ follows directly. } - \item{Following the argumentation above $(S^{(i)})^2 = -I \Leftrightarrow l=1$ - therefore $(S^{(i)})^2 = -I \Leftrightarrow (S^{(i)})^\dagger \neq (S^{(i)})$.} + \item{Following the argumentation above $\left(S^{(i)}\right)^2 = -I \Leftrightarrow l=1$ + therefore $\left(S^{(i)}\right)^2 = -I \Leftrightarrow \left(S^{(i)}\right)^\dagger \neq S^{(i)}$.} \end{enumerate} \end{proof} @@ -108,7 +108,7 @@ can be diagonalized simultaneously. This motivates and justifies the following d For a set of stabilizers $S$ the vector space \begin{equation} - V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = +1\ket{\psi} \forall S^{(i)} \in S\} + V_S := \left\{\ket{\psi} \middle| S^{(i)}\ket{\psi} = +1\ket{\psi} \forall S^{(i)} \in S\right\} \end{equation} is called the space of stabilizer states associated with $S$ and one says @@ -124,7 +124,7 @@ result: \begin{theorem} \label{thm:unique_s_state} - For a $n$ qbit system and a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer + For a $n$ qbit system and stabilizers $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer space $V_S$ has $dim V_S = 1$, in particular there exists an up to a trivial phase unique state $\ket{\psi}$ that is stabilized by $S$. @@ -157,14 +157,14 @@ however some statements that can still be made: \end{aligned} \end{equation} -Note that in \ref{def:stabilizer} it has been demanded that stabilizers are a +Note that in Definition \ref{def:stabilizer} it has been demanded that stabilizers are a subgroup of the multilocal Pauli operators. This does not hold true for an arbitrary $U$ but there exists a group for which $S'$ will be a set of stabilizers. \begin{definition} For $n$ qbits \begin{equation} - C_n := \left\{U \in SU(n) | UpU^\dagger \in P_n \forall p \in P_n\right\} + C_n := \left\{U \in SU(n) \middle| UpU^\dagger \in P_n \forall p \in P_n\right\} \end{equation} is called the Clifford group. $C_1 =: C_L$ is called the local Clifford group. @@ -235,12 +235,12 @@ the result of the measurement. \begin{lemma} \label{lemma:stab_measurement} - Let $J := \{ S^{(i)} | [g_a, S^{(i)}] \neq 0\} \neq \{\}$. When measuring + Let $J := \left\{ S^{(i)} \middle| [g_a, S^{(i)}] \neq 0\right\} \neq \{\}$. When measuring $\frac{I + (-1)^s g_a}{2} $ $1$ and $0$ are obtained with probability $\frac{1}{2}$ and after choosing a $j \in J$ the new state $\ket{\psi'}$ is stabilized by \begin{equation} - \langle \{(-1)^s g_a\} \cup \{S^{(i)} S^{(j)} | S^{(i)} \in J \setminus \{S^{(j)}\} \} \cup J^c\rangle + \langle \{(-1)^s g_a\} \cup \left\{S^{(i)} S^{(j)} \middle| S^{(i)} \in J \setminus \{S^{(j)}\} \right\} \cup J^c\rangle \end{equation} \end{lemma} @@ -287,9 +287,9 @@ vertex operator-free will be clear in the following section about graph states. \label{def:graph} The tuple $(V, E)$ is called a graph iff $V$ is a set of vertices with $|V| = n \in \mathbb{N}$. In the following $V = \{0, ..., n-1\}$ will be used. - $E$ is the set of edges $E = \left\{\{i, j\} | i,i \in V, i \neq j\right\}$. + $E$ is the set of edges $E = \left\{\{i, j\} \middle| i,i \in V, i \neq j\right\}$. - For a vertex $i$ $n_i := \left\{j \in V | \{i, j\} \in E\right\}$ is called the neighbourhood + For a vertex $i$ $n_i := \left\{j \in V \middle| \{i, j\} \in E\right\}$ is called the neighbourhood of $i$. \end{definition} @@ -436,11 +436,11 @@ that will be used later. = K_{G'}^{(j)}K_{G}^{(a)}\ket{\bar{G}'} = K_{G'}^{(j)}\ket{\bar{G}'} \end{equation} - Because $\{K_G^{(i)} | i \notin n_a\} \cup \{S^{(i)} | i\in n_a\}$ and - $\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}$ are both $n$ commuting + Because $\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{S^{(i)} \middle| i\in n_a\right\}$ and + $\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}$ are both $n$ commuting multi-local Pauli operators where the $S^{(i)}$ can be generated from the $K_{G'}^{(i)}$ and $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$ - $\langle\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}\rangle$ + $\langle\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}\rangle$ are the stabilizers of $\ket{\bar{G}'}$ and the associated graph is changed as given in the third equation. \end{proof} @@ -665,8 +665,8 @@ so it is easier to list the operators that anticommute: \begin{equation} \begin{aligned} - A_{\pm X_a} &= \left\{j | \{j, a\} \in E\right\}\\ - A_{\pm Y_a} &= \left\{j | \{j, a\} \in E\right\} \cup \{a\} \\ + A_{\pm X_a} &= \left\{j \middle| \{j, a\} \in E\right\}\\ + A_{\pm Y_a} &= \left\{j \middle| \{j, a\} \in E\right\} \cup \{a\} \\ A_{\pm Z_a} &= \{a\}\\ \end{aligned} \end{equation} @@ -695,8 +695,8 @@ $K_G^{(a)}$ is chosen. The graph is changed according to \begin{equation} \begin{aligned} - E'_{Z} &= E \setminus \left\{\{i,a\} | i \in V\right\}\\ - E'_{Y} &= E \Delta (n_a \otimes n_a) \setminus \left\{\{i,a\} | i \in V\right\}\\ + E'_{Z} &= E \setminus \left\{\{i,a\} \middle| i \in V\right\}\\ + E'_{Y} &= E \Delta (n_a \otimes n_a) \setminus \left\{\{i,a\} \middle| i \in V\right\}\\ \end{aligned} \end{equation} @@ -714,7 +714,7 @@ For $g_a = X_a$ one has to choose a $b \in n_a$ and the transformations are E'_{X} = E &\Delta (n_b \otimes n_a) \\ & \Delta ((n_b \cap n_a) \otimes (n_b \cap n_a)) \\ & \Delta (\{b\} \otimes (n_a \setminus \{b\})) \\ - & \setminus \left\{\{i,a\} | i \in V\right\}\\ + & \setminus \left\{\{i,a\} \middle| i \in V\right\}\\ \end{aligned} \end{equation} diff --git a/thesis/main.bib b/thesis/main.bib index afacd94..7d417ed 100644 --- a/thesis/main.bib +++ b/thesis/main.bib @@ -147,3 +147,11 @@ author={M. Hein, J. Eisert, H.J. Briegel}, note={https://arxiv.org/abs/quant-ph/0307130v7} } +@article{ + lehner2019, + title={Lecture Notes on Quantum Computing SS 2019}, + year=2019, + author={Christoph Lehner}, + note={Unpublished Work} +} +