Quantum Computing

Quantum Computing – Structured Overview

1. What Is Quantum Computing?

Key Idea: Use quantum‑mechanical phenomena (superposition, entanglement, interference) to process information in ways that classical bits cannot.

2. Fundamental Concepts

2.1 Qubit

• Definition: Two‑level quantum system (e.g., electron spin, photon polarization).
• State representation:
  [ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle,\qquad |\alpha|^{2}+|\beta|^{2}=1 ]
• Bloch sphere: Visual tool; any pure qubit state maps to a point on the sphere’s surface.

2.2 Superposition

• A qubit simultaneously occupies multiple basis states.
• Enables parallel evaluation of many possibilities.

2.3 Entanglement

• Correlation between qubits such that the state of one instantly determines the state of another, regardless of distance.
• Example (Bell state):
  [ |\Phi^{+}\rangle = \frac{1}{\sqrt{2}}\bigl(|00\rangle + |11\rangle\bigr) ]

2.4 Quantum Interference

• Probability amplitudes add constructively or destructively.
• Algorithmic step: amplify correct answers, cancel wrong ones.

2.5 Measurement

• Collapses a superposed state to a classical outcome (|0⟩ or |1⟩) with probabilities given by the squared amplitudes.
• Measurement is destructive for the quantum information.

3. Quantum Gates & Circuits

Circuit model:

1. Initialize qubits (usually to |0⟩).
2. Apply a sequence of quantum gates (unitary operations).
3. Measure selected qubits to obtain classical results.

4. Major Quantum Algorithms

| Algorithm | Problem Addressed | Speed