Angle Encoding (Rotation Encoding)

Overview

Angle encoding is the simplest quantum data encoding method: each classical feature value becomes a rotation angle. It's hardware-efficient and widely used in variational quantum algorithms.

The Concept

Classical data: x = [x₀, x₁, x₂, x₃]

Quantum encoding: RY(x₀π)|0⟩ ⊗ RY(x₁π)|0⟩ ⊗ RY(x₂π)|0⟩ ⊗ RY(x₃π)|0⟩

Encoding Types

Standard (1 feature per qubit)

CODE
┌─────────┐
q_0: ┤ RY(x₀π) ├
     ├─────────┤
q_1: ┤ RY(x₁π) ├
     ├─────────┤
q_2: ┤ RY(x₂π) ├
     └─────────┘

Dense (2 features per qubit)

CODE
┌─────────┐┌─────────┐
q_0: ┤ RY(x₀π) ├┤ RZ(x₁π) ├
     ├─────────┤├─────────┤
q_1: ┤ RY(x₂π) ├┤ RZ(x₃π) ├
     └─────────┘└─────────┘

Running the Circuit

PYTHON
from circuit import run_circuit

# Standard encoding
result = run_circuit(data=[0.25, 0.5, 0.75, -0.5])
print(f"Qubits used: {result['n_qubits']}")

# Dense encoding
result = run_circuit(data=[0.25, 0.5, 0.75, 1.0], encoding_type='dense')
print(f"Qubits used: {result['n_qubits']}")  # 2 instead of 4

Rotation Effects

For RY(θ)|0⟩:

θ	State	P(0)	P(1)
0	\|0⟩	100%	0%
π/2	\|+⟩	50%	50%
π	\|1⟩	0%	100%

Comparison with Other Encodings

Encoding	Qubits for N features	Depth
Angle	N	1
Dense Angle	N/2	1
Amplitude	log₂(N)	O(N)
Basis	N	1

Advantages

Simple: Just rotation gates
Hardware-efficient: Native gates on most hardware
Low depth: Single layer of rotations
Intuitive: Direct mapping of features

Disadvantages

Linear scaling: One qubit per feature (or 2 with dense)
Limited expressiveness: Compared to amplitude encoding
No entanglement: Features encoded independently

Applications

VQC input: Standard input for variational classifiers
Quantum kernel: Feature map for kernel methods
QAOA: Encoding problem parameters