Quantum Computing: Qubits, Gates & Algorithms

Classical computing uses bits (0 or 1). Quantum computing uses QBs — quantum mechanical systems that exist in SUP of |0⟩ and |1⟩ simultaneously. A single QB state: |ψ⟩ = α|0⟩ + β|1⟩ where α,β are complex amplitudes satisfying |α|² + |β|² = 1. Measurement collapses SUP to definite state w/ probability |α|² for |0⟩ and |β|² for |1⟩. This probabilistic nature is fundamental, not a limitation — it's the source of quantum computational advantage.

Bloch sphere representation: any single QB pure state maps to a point on the unit sphere. North pole = |0⟩, south pole = |1⟩, equator = equal SUP states w/ varying phase. Polar angle θ determines measurement probabilities, azimuthal angle φ determines relative phase between amplitudes. Phase has no classical analogue but is critical for interference effects that power QAs.

Multi-QB systems: n QBs span a 2ⁿ-dimensional Hilbert space. Two QBs: |ψ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩. This exponential state space is why QCs potentially outperform CCs for certain problems. 50 QBs encode 2⁵⁰ ≈ 10¹⁵ amplitudes — more than petabyte of classical storage to fully describe.

ENT is the quintessential quantum resource. Bell state (|00⟩ + |11⟩)/√2 means measuring one QB instantly determines the other, regardless of distance. ENT is not faster-than-light communication (no-communication theorem) but enables quantum teleportation, superdense coding, & is essential for QA speedups. Creating ENT: apply Hadamard to first QB, then CNOT w/ second as target.

Quantum gates are unitary operations (preserve probability, reversible). Single-QB gates:
- Pauli-X (NOT gate): |0⟩↔|1⟩, bit flip
- Pauli-Y: rotation around Y-axis, combines bit flip + phase flip