additions

Author: mhjensen

doc/src/week6/Latexexamples/add2.tex

@@ -1,26 +1,3 @@
-\chapter{Quantum Computing}
-\label{ch:qc}
-In this chapter, we provide details of quantum computing and quantum algorithms. Starting by introducing the basic concepts of quantum computing, we then discuss the quantum algorithms used in this thesis. Unlike the previous chapter which provides details of the physics problem we are trying to solve, this chapter focuses on the algorithms and their applications of them. Thus all physical objects such as the Hamiltonian or the Schr{\"o}dinger equation will be treated as pure mathematical objects. 
-
-\section{Qubits}
-In analogy to the classical bit, the basic unit of information in classical computing, the basic unit of information in quantum computing is the quantum bit or qubit. A qubit lives in a two-dimensional Hilbert space which can be spanned by two basis vectors, theoretically of choice. Unlike the classical bit, the quantum nature of qubits allows them to be in a superposition of the two basis states. Conventionally, we denote the two basis state as $ \ket{0} $ and $ \ket{1} $, chosen to be the eigenstates of the Pauli-$Z$ operator, such that:
-\begin{equation}
-	\ket{0} = \begin{pmatrix}
-		1 \\
-		0
-	\end{pmatrix}, \quad
-	\ket{1} = \begin{pmatrix}
-		0 \\
-		1
-	\end{pmatrix}.
-\end{equation}
-
-The state of a qubit can then be written as a linear combination of the two basis states,
-\begin{equation}
-	\ket{\psi} = \alpha \ket{0} + \beta \ket{1},
-\end{equation}
-where $\alpha$ and $\beta$ are complex numbers, with normalisation condition $\alpha^2 + \beta^2 = 1$. 
-
 \subsection{Bloch Sphere}
 The state of a qubit can also be represented by a point on the surface Bloch sphere, as shown in Figure~\ref{fig:bloch_sphere}. The Bloch sphere is a unit sphere, where the $ z $ direction the represents the $\ket{0}$ state and the $-z$ direction represents the state $\ket{1}$. Note that although the $x,y$ and $z$ axes are perpendicular in this representation on the Bloch sphere, they are not orthonormal. Vectors pointing at opposite directions are orthonormal, such as the $\ket{0}$ and $\ket{1}$ states.