Qubits are the building blocks of quantum computing, existing in superposition and holding more information than classical bits. They're visualized using the Bloch sphere and mathematically represented as linear combinations of basis states.

Quantum gates manipulate qubits, performing operations like creating superposition or entanglement. Single-qubit gates like Hadamard and Pauli gates, and multi-qubit gates like CNOT, form universal sets for implementing complex quantum algorithms.

Qubits and Superposition

Fundamental Concepts of Qubits

• Qubit represents the basic unit of quantum information
• Qubits exist in superposition of states, allowing for multiple values simultaneously
• Superposition enables qubits to hold more information than classical bits
• Quantum measurement collapses superposition, forcing qubit into definite state
• Measurement outcome probabilistic, based on qubit's state before measurement

Visualizing Qubit States

• Bloch sphere provides 3D representation of qubit state
• Sphere's surface represents all possible pure states of a qubit
• North and south poles correspond to classical bit states (|0⟩ and |1⟩)
• Points on sphere's equator represent equal superposition of |0⟩ and |1⟩
• Any point on sphere's surface represents unique qubit state

Mathematical Representation of Qubits

• Qubit state expressed as linear combination of basis states: $|\psi⟩ = α|0⟩ + β|1⟩$
• α and β complex numbers, representing probability amplitudes
• Normalization condition: $|α|^2 + |β|^2 = 1$
• Measurement probabilities: $P(|0⟩) = |α|^2$ and $P(|1⟩) = |β|^2$
• Phase difference between α and β determines qubit's position on Bloch sphere

Quantum Gates

Single-Qubit Gates

• Quantum gates perform unitary operations on qubits
• Hadamard gate creates superposition from basis states
• Hadamard gate representation: $H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$
• Pauli gates (X, Y, Z) rotate qubit state around Bloch sphere axes
• X gate (NOT gate) flips qubit state: $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
• Z gate changes qubit's phase: $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Multi-Qubit Gates

• CNOT (Controlled-NOT) gate operates on two qubits
• CNOT flips target qubit if control qubit |1⟩, does nothing if control qubit |0⟩
• CNOT gate crucial for creating entanglement between qubits
• CNOT gate representation: $CNOT = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$
• Toffoli gate (CCNOT) extends CNOT to three qubits, flips target if both controls |1⟩

Universal Gate Sets

• Combination of gates forming universal set can implement any quantum operation
• Common universal set includes Hadamard, CNOT, and T gate
• T gate introduces phase shift: $T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$
• Universal sets enable construction of complex quantum algorithms

Quantum Circuits and Entanglement

Quantum Circuit Fundamentals

• Quantum circuits represent sequences of quantum operations
• Circuits composed of quantum gates applied to qubits
• Horizontal lines represent qubits, progressing from left to right
• Vertical lines with controls represent multi-qubit gates (CNOT)
• Measurement operations depicted by meter symbols at end of qubit lines
• Circuit diagrams crucial for designing and analyzing quantum algorithms

Quantum Entanglement and Its Applications

• Quantum entanglement describes correlated states between multiple qubits
• Entangled qubits exhibit stronger-than-classical correlations
• Creating entanglement involves applying gates to initially separable qubits
• Bell states represent maximally entangled two-qubit states
• Bell state example: $|\Phi^+⟩ = \frac{1}{\sqrt{2}}(|00⟩ + |11⟩)$
• Entanglement enables quantum teleportation and superdense coding
• Quantum key distribution protocols utilize entanglement for secure communication

Measuring Entanglement and Quantum State Tomography

• Entanglement quantified using measures like concurrence or entanglement entropy
• Quantum state tomography reconstructs full quantum state from measurements
• Tomography requires multiple measurements in different bases
• Process involves preparing identical copies of quantum state and performing various measurements
• Reconstructed density matrix provides complete description of quantum system's state
• Challenges in tomography include exponential scaling with number of qubits