Фізика квантової інформації та обчислень

⚛️ Кубити

Quantum Superposition

|ψ⟩ = α|0⟩ + β|1⟩ для single qubit, де |α|² + |β|² = 1. Bloch sphere representation: |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩. Measurement collapse до |0⟩ або |1⟩.

Entanglement

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

Decoherence

T₁ (energy relaxation): T₁ ~ мс для сверхпровідникових qubits. T₂ (dephasing): T₂ ≤ 2T₁. Phase relaxation через environment coupling. Error correction потрібна для fault tolerance.

Gate operations

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

🔬 Квантові коди

Shor Code

9 qubits: encode 1 logical qubit. Correct arbitrary single-qubit errors (bit flip, phase flip). Concatenation для 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 через stabilizers. Error threshold ~1% для depolarizing noise. Scalable для large qubit counts.

Fault-tolerant computing

Logical qubits з error correction. Threshold theorem: якщо physical error rate < threshold, то logical error rate → 0 з nesting.

📡 Quantum Teleportation

Protocol

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

Success probability

1 при ідеальній entangle; у реальності ~>0.9 через decoherence. Демонстровано на >143 км (Canary Islands, 2012).

Quantum repeaters

Квантові repeaters для long-distance entanglement distribution. Error correction, purification. QKD networks.

Quantum memory

Storage quantum states. Atomic ensembles, NV centers. Coherence time до секунд. Квантові мережі.

🔐 Протоколи

BB84 QKD

Quantum key distribution: Bennett-Brassard 1984. Random basis selection (Z or X). Security через no-cloning та 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 для precision beyond classical limit. Gravitational wave detection (LIGO). Quantum sensors.

Multi-party computation

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

🧮 Квантові алгоритми

Shor's algorithm

Факторизація 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. Барметр: feasible computation.

🔬 Hardware platforms

Superconducting qubits

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

Trapped ions

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

Photonic systems

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

Topological qubits

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

📊 Графіки та діаграми

Bloch sphere

Representation single qubit:

Quantum gates: rotations на 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 = entanglement.

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

🧪 Практичні приклади

Приклад 1: Google Sycamore

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

Приклад 2: IonQ system

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

Приклад 3: Chinese Micius satellite

Quantum teleportation 143 км (2012). QKD до 1,200 км ground stations. Quantum internet.

Приклад 4: BB84 implementation

Quantum key distribution: secure communication. Commercial systems: QKD networks у Європі, Азії.

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Квантова інформація: майбутнє обчислення

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