Chapter 4: New Quantum

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\title{Lecture 1: Fundamentals of Quantum States and Measurements} \date{February 24, 2025}

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\section{Introduction} Quantum computing relies on quantum mechanics principles to manipulate information. This lecture introduces: \begin{itemize} \item Basis and Orthonormal Basis \item Bra-Ket Notation (Dirac Notation) \item Quantum Measurement \item Representing a state in different basis sets \end{itemize}

\section{Basis and Orthonormal Basis} A basis is a set of vectors that spans a vector space. Any vector in that space can be expressed as a linear combination of basis vectors. For example, in \(\mathbb{R}^2\), the standard basis is: $$ e_1 = \begin{bmatrix}1\0\end{bmatrix}, \quad e_2 = \begin{bmatrix}0\1\end{bmatrix} $$

A vector \(v \in \mathbb{R}^2\) can be written as: [ v = a e_1 + b e_2 ] A basis is orthonormal if: \begin{enumerate} \item The basis vectors are orthogonal: \(\langle e_i | e_j \rangle = 0\) for \(i \neq j\). \item The basis vectors are normalized: \(\langle e_i | e_i \rangle = 1\). \end{enumerate} The computational basis for qubits is: [ |0\rangle = \begin{bmatrix}1\0\end{bmatrix}, \quad |1\rangle = \begin{bmatrix}0\1\end{bmatrix} ]

\section{Bra-Ket Notation (Dirac Notation)} \begin{itemize} \item A ket \(|\psi\rangle\) represents a column vector in Hilbert space. \item A bra \(\langle \psi|\) is the conjugate transpose (row vector). \end{itemize} Example: [ |\psi\rangle = \begin{bmatrix}a\b\end{bmatrix}, \quad \langle \psi| = \begin{bmatrix}a^ & b^\end{bmatrix} ] Inner product: [ \langle \phi | \psi \rangle = \sum_i \phi^*_i \psi_i ] Outer product: [ |\psi\rangle \langle \phi| ]

\section{Quantum Measurement} A qubit state can be written as: [ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle ] where \(\alpha, \beta\) are complex numbers satisfying \(|\alpha|^2 + |\beta|^2 = 1\). When measured in the computational basis: \begin{itemize} \item \(|0\rangle\) occurs with probability \(|\alpha|^2\). \item \(|1\rangle\) occurs with probability \(|\beta|^2\). \end{itemize} Example: [ |\psi\rangle = \frac{1}{\sqrt{2}} |0\rangle + \frac{1}{\sqrt{2}} |1\rangle ] Measurement results in \(|0\rangle\) or \(|1\rangle\) with equal probability.

\section{Representing a State in Different Basis Sets} Besides the computational basis, we can use: \subsection{Hadamard Basis} [ |+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \quad |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}} ] \subsection{Eigenbasis of Pauli-X} Eigenvectors: [ X = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} ] [ |+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \quad |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}} ]

\section{Python and Qiskit Implementation} \begin{verbatim} from qiskit import QuantumCircuit, Aer, execute from qiskit.visualization import plot_bloch_vector import numpy as np

Define a quantum circuit with one qubit

qc = QuantumCircuit(1) qc.h(0) # Apply Hadamard gate qc.measure_all()

Simulate

simulator = Aer.get_backend('aer_simulator') result = execute(qc, simulator, shots=1000).result() counts = result.get_counts() print("Measurement Results:", counts)

def plot_state(): plot_bloch_vector([1/np.sqrt(2), 0, 1/np.sqrt(2)]) # |+> state plot_state() \end{verbatim}

\section{Summary} \begin{itemize} \item Basis and Orthonormal Basis: Computational and Hadamard basis. \item Bra-Ket Notation: Representation of quantum states. \item Measurement: Probabilistic collapse to basis states. \item Different Basis Representations: Transforming states across different bases. \end{itemize}

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