Qubits: Understanding Quantum States and Operations

This lecture introduces the concept of qubits, the fundamental units of quantum information. It begins by defining qubits as systems with two states, often represented as spin-up and spin-down or horizontal and vertical polarizations. The instructor explains the significance of superposition, where a qubit can exist in a combination of both states, and discusses the mathematical representation of these states using complex numbers. The lecture covers the Bloch sphere, a geometrical representation of qubit states, and the role of phase in determining the state of a qubit. The instructor also explains the properties of Pauli matrices and their relevance in quantum mechanics. The discussion extends to Hamiltonians and propagators, illustrating how they govern the evolution of qubit states over time. The lecture concludes with examples of Rabi oscillations and the importance of resonance in controlling qubit transitions, emphasizing the practical applications of qubits in quantum computing and information processing.