QUANTUM ENTANGLEMENT (2-Qubit System)

Entanglement is represented using Bell States.

Circuits:

Perfect Correlation: 6 (phi +), 7 (phi -)

Perfect Anti-correlation: 8 (psi +), 9 (psi -)

1. Φ⁺ (Phi-plus)

Formula:

Nature:
Perfect correlation → measurements are always the same (00 or 11).

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2. Φ⁻ (Phi-minus)

Formula:

Nature:
Perfect correlation + phase shift → same outcomes (00, 11), but negative phase changes interference behavior.

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3. Ψ⁺ (Psi-plus)

Formula:

Nature:
Perfect anti-correlation → measurements are always opposite (01 or 10).

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4. Ψ⁻ (Psi-minus)

Formula:

Nature:
Perfect anti-correlation + phase shift → opposite outcomes (01, 10) with negative phase affecting interference.

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QUANTUM ENTANGLEMENT (3-Qubit System)

Circuit: 10 (GHZ state for 3 qubits)

GHZ States- (Greenberger-Horne-Zeilinger states) represent type of multi-quvit entangled state, ususally 3 or more qubits, hence show entanglement shared across all qubits together. These states are more fraglie since measuring or loosing 1 qubit can destory entanglement of whole system. They demonstrate Global Correlations, showing full anti-correlation is impossible.

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QUANTUM TELEPORTATION (3-Qubit System)

Circuit: 11 (teleportation using 3 qubits)

Transfers a quantum state using entanglement + classical bits.

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1. Prepare Message State

Apply:

H(q₀)

State:

Nature: Creates state ψ to teleport.

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2. Create Entangled Pair

Apply:

H(q₁) → CNOT(q₁→q₂)

Nature: Forms Bell pair between q₁ and q₂.

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3. Encode Message

Apply:

CNOT(q₀→q₁)
H(q₀)

Nature: Mixes message with entangled system.

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4. Measure

Measure:

q₀, q₁ → 00 / 01 / 10 / 11

Nature: Produces 2 classical bits.

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5. Send & Correct

Possible corrections on q₂:

00 → I
01 → X
10 → Z
11 → XZ

Nature: Recovers original state ψ.

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Final

Initial:
q₀ = ψ
q₂ = |0⟩

Final:
q₀ → measured
q₂ = ψ

Nature: State transferred; original destroyed (No-Cloning preserved).

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DEUTSCH ALGORITHM (2-Qubit System)

Circuit: 12 First quantum algorithm showing quantum speedup.

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Purpose

Determines whether a hidden function is:

Constant  → same outputs
Balanced  → different outputs

using only 1 oracle query through superposition + interference.

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Classical vs Quantum

Classical

Need f(0) and f(1)
→ 2 evaluations

Quantum

Processes both inputs simultaneously using superposition.

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1. Initial State

q₀ = |0⟩
q₁ = |1⟩

Apply:

H(q₀)
X(q₁)
H(q₁)

State:

q₀ → (|0⟩ + |1⟩)/√2
q₁ → (|0⟩ - |1⟩)/√2

Nature: Creates superposition of both inputs.

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2. Oracle (Uf)

Oracle:

Uf|x,y⟩ = |x, y⊕f(x)⟩

For balanced function:

f(x)=x

Oracle becomes:

CNOT(q₀ → q₁)

Nature: Encodes function using phase kickback.

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3. Interference

Apply:

H(q₀)

Nature: Interference removes wrong possibilities and keeps correct result.

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4. Measurement

Measure:

q₀

Result:

0 → Constant
1 → Balanced

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Final

Classical → 2 evaluations
Quantum   → 1 oracle query

Nature: Shows how quantum computers use superposition + interference for faster computation.

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QUANTUM RANDOM NUMBER GENERATOR (QRNG)

Circuit 13 (1 Qubit: generates random numbers between 0 to 1)

Circuit 14 (2 Qubit: generates random numbers between 0 to 3 i.e. 00, 01, 10, 11)

Classical computers generate pseudo-random numbers since algorithm's internal state is unknown.

But quantum measurement gives true randomness using superposition and measurement as the randomness comes from nature of quantum physics.