Angle Encoding (Rotation Encoding)

Overview

Angle encoding is the simplest and most hardware-efficient quantum data encoding: each classical feature value becomes a rotation angle on a dedicated qubit. It is the default input method for variational quantum classifiers, quantum kernels, and most near-term QML algorithms.

The mapping is direct: a feature value x is encoded as a rotation R(x * pi) on the Bloch sphere. No normalization is required, and the circuit depth is just 1.

Bloch Sphere Mapping

For RY encoding, the feature value maps to a point on the Bloch sphere:

CODECopy
x = 0.0  -->  |0>   (north pole, P(0)=100%)
x = 0.5  -->  |+>   (equator, P(0)=50%, P(1)=50%)
x = 1.0  -->  |1>   (south pole, P(1)=100%)

The angle theta = x * pi determines the latitude on the Bloch sphere. Values in [-1, 1] cover the full sphere.

Encoding Variants

Standard Encoding (1 feature per qubit)

CODECopy
     +---------+
q_0: | RY(x0pi)|
     +---------+
q_1: | RY(x1pi)|
     +---------+
q_2: | RY(x2pi)|
     +---------+
q_3: | RY(x3pi)|
     +---------+

RZ Encoding (phase encoding)

CODECopy
     +---+ +---------+
q_0: | H |-| RZ(x0pi)|      Requires initial H to create
     +---+ +---------+      superposition for phase to matter
q_1: | H |-| RZ(x1pi)|
     +---+ +---------+

Dense Encoding (2 features per qubit)

CODECopy
     +---------+ +---------+
q_0: | RY(x0pi)|-| RZ(x1pi)|      RY for amplitude, RZ for phase
     +---------+ +---------+
q_1: | RY(x2pi)|-| RZ(x3pi)|      2x compression vs standard
     +---------+ +---------+

Dense encoding uses two orthogonal rotation axes (RY and RZ) on each qubit, packing 2 features per qubit. This halves the qubit requirement at the cost of slightly more complex readout.

Running the Circuit

PYTHONCopy
from circuit import (
    run_circuit,
    create_angle_encoding,
    create_dense_angle_encoding,
    verify_encoding,
)

# Standard RY encoding
result = run_circuit(data=[0.25, 0.5, 0.75, -0.5])
print(f"Qubits: {result['n_qubits']}")  # 4

# RZ encoding
result = run_circuit(data=[0.25, 0.5], encoding_type="rz")

# RX encoding
result = run_circuit(data=[0.25, 0.5], encoding_type="rx")

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

# Verify encoding properties
v = verify_encoding()
print(f"All checks passed: {v['passed']}")

Rotation Gate Comparison

For RY(theta)|0>:

Comparison with Other Encodings

Advantages

• Simple: Just rotation gates, no multi-qubit operations
• Hardware-efficient: Native gates on all quantum hardware
• Low depth: Single layer (depth 1) for standard encoding
• No normalization: Works with any feature scale
• Intuitive: Direct Bloch sphere geometry

Disadvantages

• Linear qubit scaling: N features need N qubits (or N/2 dense)
• No entanglement: Features are encoded independently
• Limited expressiveness: Less rich than amplitude encoding
• Feature interaction: Requires variational layers to create correlations

Applications

• VQC input: Standard input layer for variational quantum classifiers
• Quantum kernel: Feature encoding for kernel-based methods
• QAOA: Encoding problem parameters as rotation angles
• Hybrid models: Quantum layer input in classical-quantum networks

Learn More

• Robust Data Encodings for Quantum Classifiers (LaRose & Coyle, 2020)
• Data Encoding Patterns for Quantum Computing (Weigold et al., 2020)
• Expressibility of Data Encoding Strategies (Schuld et al., 2021)