Chapter 11: LatexA

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\section*{Lecture 11: Quantum Gates and Operators}

\subsection*{Pauli-X Gate} The Pauli-X gate can be represented in terms of the Hadamard ($H$) and Pauli-Z ($Z$) gates as:

\[\begin{equation} X = H Z H \end{equation}\]

Matrix representation of the Pauli-X gate:

\[\begin{equation} X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \end{equation}\]

\subsection*{Hadamard Gate} The Hadamard gate transforms the basis states as follows:

\[\begin{equation} H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle), \quad H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) \end{equation}\]

Matrix representation:

\[\begin{equation} H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \end{equation}\]

\subsection*{Bloch Sphere Representation} A qubit state can be visualized on the Bloch Sphere and is given by:

\[\begin{equation} |\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi} \sin\frac{\theta}{2}|1\rangle \end{equation}\]

where: \begin{itemize} \item $ \theta $ represents the polar angle, determining the probability distribution between $ |0\rangle $ and $ |1\rangle $. \item $ \phi $ represents the phase difference between $ |0\rangle $ and $ |1\rangle $. \end{itemize}

\subsection*{Eigenstates and Operators}

\textbf{Pauli-Z Operator:}

\[\begin{equation} Z|0\rangle = +|0\rangle, \quad Z|1\rangle = -|1\rangle \end{equation}\]

Matrix representation:

\[\begin{equation} Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{equation}\]

\subsection*{Projection Operators} Projection operators for measuring in the computational basis:

\[\begin{equation} \mathbf{P}_0 = |0\rangle\langle0| = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad \mathbf{P}_1 = |1\rangle\langle1| = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}\]

Measurement probabilities for a general qubit state $ \psi = \alpha|0\rangle + \beta|1\rangle $:

\[\begin{equation} P(|0\rangle) = |\alpha|^2, \quad P(|1\rangle) = |\beta|^2 \end{equation}\]

\subsection*{CNOT Gate (Controlled-NOT)} Matrix representation of the Controlled-NOT (CNOT) gate:

\[\begin{equation} \text{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \end{equation}\]

\section*{Lecture 11: Quantum Gates and Operators}

\subsection*{Pauli-X Gate} The Pauli-X gate can be represented in terms of the Hadamard ($H$) and Pauli-Z ($Z$) gates as:

\[\begin{equation} X = H Z H \end{equation}\]