Classical Data → Quantum States

Angle Encoding maps classical numerical values into quantum rotation angles. Each classical feature rotates a qubit around the X, Y, or Z axis of the Bloch sphere.

📌 Concept

For vector \( x = (x_1, ..., x_n) \), each feature is encoded as:

⭐ Key Features

• Efficient depth: Constant circuit depth (1 gate per qubit).
• Use cases: QSVM, Variational Quantum Classifiers.
• Limitation: Requires \(N\) qubits for \(N\) features.

📘 Example

Input: x = 1.57 (approx π/2)
Gate: Ry(1.57)
State: |+⟩ (Superposition)
                        

Amplitude Encoding stores data in the amplitudes of a quantum state. It is exponentially efficient: \(N\) qubits can store \(2^N\) features.

📌 Concept

For normalized vector \( x \), the state is:

Constraint: \( \sum |x_i|^2 = 1 \) (Normalization required)

⭐ Key Features

• High Capacity: Encodes images/audio efficiently.
• Complexity: High circuit depth to prepare the state (requires many CNOTs).
• Applications: Quantum Machine Learning, QGANs.

📘 Example

Input: v = [0.6, 0.8]
State: 0.6|0⟩ + 0.8|1⟩
(Probabilities sum to 0.36 + 0.64 = 1)