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⚛️ Qubits

Quantum Superposition

|ψ⟩ = α|0⟩ + β|1⟩ for a single qubit, where |α|² + |β|² = 1. Bloch sphere representation: |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩. Measurement collapse to |0⟩ or |1⟩.

Entanglement

|Ψ⁻⟩ = (|01⟩ - |10⟩)/√2 for a Bell state. Nonlocal correlations: violation of the Bell inequality. EPR paradox, quantum nonlocality.

Decoherence

T₁ (energy relaxation): T₁ ~ ms for superconductive qubits. T₂ (dephasing): T₂ ≤ 2T₁. Phase relaxation due to environment coupling. Error correction is needed for fault tolerance.

Gate operations

Pauli gates (X, Y, Z), Hadamard (H), CNOT. Single-qubit rotation: R_z(θ) = exp(-iθZ/2). Universal gate set. Error rates: ~10⁻³ for NISQ-era.

🔬 Quantum Codes

Shor Code

9 qubits: encode 1 logical qubit. Correct arbitrary single-qubit errors (bit flip, phase flip). Concatenation for better performance. Error threshold.

Steane Code

7 qubits: CSS code (Calderbank-Shor-Steane). Correct single errors. Lower overhead vs Shor. Fault-tolerant gates.

Surface Codes

2D lattice, syndrome measurement via stabilizers. Error threshold ~1% for depolarizing noise. Scalable for large qubit counts.

Fault-tolerant computing

Logical qubits with error correction. Threshold theorem: if physical error rate < threshold, then logical error rate → 0 with nesting.

📡 Quantum Teleportation

Protocol

Alice: unknown state |ψ⟩, entangled Bell pair. Bell measurement (4 outcomes). Classical communication to Bob. Bob applies correction Pauli gate.

Success probability

1 for ideal entanglement; in reality ~0.9 due to decoherence. Demonstrated over >143 km (Canary Islands, 2012).

Quantum repeaters

Quantum repeaters for long-distance entanglement distribution. Error correction, purification. QKD networks.

Quantum memory

Storage of quantum states. Atomic ensembles, NV centers. Coherence time up to seconds. Quantum networks.

🔐 Protocols

BB84 QKD

Quantum key distribution: Bennett-Brassard 1984. Random basis selection (Z or X). Security due to no-cloning and measurement disturbance.

Bell tests

CHSH inequality: S = ⟨AB⟩ - ⟨AB'⟩ + ⟨A'B⟩ + ⟨A'B'⟩. Quantum: S ≤ 2√2, classical: S ≤ 2. Loophole-free tests (2015).

Quantum metrology

Squeezed states for precision beyond the classical limit. Gravitational wave detection (LIGO). Quantum sensors.

Multi-party computation

Secure computation among parties. Quantum secret sharing. Blind quantum computation.

🧮 Quantum Algorithms

Shor's algorithm

Factorization of N: exponential speedup O((log N)³) vs classical O(exp((log N)^(1/3))). Quantum Fourier transform, period finding. RSA security.

Grover's algorithm

Unstructured search: quadratic speedup O(√N) vs classical O(N). Amplitude amplification. Oracle queries.

QAOA

Quantum Approximate Optimization Algorithm. Combinatorial optimization. Variational approach. NISQ-friendly.

Quantum supremacy

Google Sycamore: 53 qubits, random circuit sampling (2019). Classical simulation time ~10,000 years. Barmerter: feasible computation.

🔬 Hardware platforms

Superconducting qubits

Transmon and flux qubits. Coherence: T₁ ~ ms, T₂ ≤ ms. High connectivity. Google, IBM, Rigetti.

Trapped ions

Yb⁺ and Ca⁺ ions. High fidelity: >99.9%. Long coherence: seconds. IonQ, Quantinuum.

Photonic systems

Linear optics and quantum photonics. Boson sampling for supremacy. Photonic QKD. PsiQuantum, Xanadu.

Topological qubits

Anyons for fault tolerance. Microsoft Station Q. Error correction built-in.

📊 Graphs and Diagrams

Bloch sphere

Representation of a single qubit:

Quantum gates: rotations on the Bloch sphere. Z-axis: phase. X, Y: bit flips.

Bell states

4 entangled states:

CHSH inequality: S_max = 2√2 ≈ 2.83 vs classical 2. Violation indicates entanglement.

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩ Where θ ∈ [0,π], φ ∈ [0,2π] North pole: |0⟩, South: |1⟩ Equator: superposition states

🧪 Practical Examples

Example 1: Google Sycamore

53 superconducting qubits, random circuit sampling (2019). Quantum supremacy. Classical simulation ~10,000 years.

Example 2: IonQ System

Trapped ion qubits: >99.9% fidelity. Long coherence. Quantum chemistry, optimization.

Example 3: Chinese Micius Satellite

Quantum teleportation 143 km (2012). QKD to 1,200 km ground stations. Quantum internet.

Example 4: BB84 Implementation

Quantum key distribution: secure communication. Commercial systems: QKD networks in Europe, Asia.

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