Symmetry & Equivariant

SO(2) Equivariant Feature Map

Rigorously equivariant encoding for 2D rotational symmetry using angular momentum eigenstates.

Qubits

2

Depth

6

Total Gates

5

Simulability

Not simulable

Mathematical Formulation

∣ψ(r,θ)⟩=∑m=−MMcm(r)⋅eimθ∣m⟩|\psi(r, \theta)\rangle = \sum_{m=-M}^{M} c_m(r) \cdot e^{im\theta} |m\rangle

Description

The SO(2) Equivariant Feature Map encodes 2D data points (r, θ) into quantum states that transform correctly under rotations: applying a rotation by angle φ to the input is equivalent to a unitary rotation of the quantum state. This is achieved by encoding data into angular momentum eigenstates |m⟩ with amplitudes modulated by a radial function.

The encoding converts Cartesian (x, y) coordinates to polar (r, θ), then prepares a superposition of angular momentum eigenstates m ∈ {-max_m, ..., +max_m}. The radial component c_m(r) can be Gaussian (centered at |m|) or uniform, while the angular component e^{imθ} ensures exact equivariance under SO(2) rotations.

This encoding requires exactly 2 input features (x, y coordinates) and uses ⌈log₂(2·max_m + 1)⌉ qubits. It provides the strongest theoretical guarantees among the equivariant encodings but is restricted to 2D rotation problems.

Circuit Diagram

Property Radar

Properties

Qubits
2
Circuit Depth
6
Total Gates
5
Single-Qubit Gates
5
Two-Qubit Gates
0
Parameters
0
Entangling
No
Simulability
Not Simulable
Expressibility
—
Entanglement Capability
—
Trainability
—
Noise Resilience
—

Resource Scaling

How resource requirements grow with the number of input features.

FeaturesQubitsDepthGates2Q Gates
22650

Code Examples

SO(2) equivariant encoding with PennyLane for 2D point classification.

python
from encoding_atlas import SO2EquivariantFeatureMap
import pennylane as qml
import numpy as np

enc = SO2EquivariantFeatureMap(n_features=2, max_angular_momentum=1)
dev = qml.device("default.qubit", wires=enc.n_qubits)

@qml.qnode(dev)
def circuit(x):
    enc.get_circuit(x, backend="pennylane")
    return qml.state()

x = np.array([1.0, 0.5])  # (x, y) coordinates
state = circuit(x)

When to Use This Encoding

  • 2D point cloud classification with rotational symmetry
  • Image classification on rotationally symmetric data
  • Molecular property prediction for 2D molecules
  • Signal processing with circular symmetry
  • Research into equivariant quantum ML

Pros & Cons

Advantages

  • Rigorous SO(2) equivariance — mathematically guaranteed
  • Compact encoding — few qubits for angular momentum states
  • Strong inductive bias reduces sample complexity
  • Configurable angular momentum resolution
  • Physically meaningful angular momentum basis

Limitations

  • Restricted to exactly 2 features (2D data only)
  • Limited applicability to strictly rotational problems
  • State preparation depth grows with angular momentum
  • Gaussian radial function may not suit all data distributions
  • Cannot capture non-rotational data patterns

References

  1. [1]Nguyen, Q.T., et al. (2022). Theory for equivariant quantum neural networks. PRX Quantum, 3(3), 030322.
  2. [2]Larocca, M., et al. (2022). Group-invariant quantum machine learning. PRX Quantum, 3(3), 030341.
  3. [3]Schatzki, L., et al. (2022). Theoretical guarantees for permutation-equivariant quantum neural networks. npj Quantum Information, 8, 130.