Symmetry & Equivariant

Symmetry-Inspired Feature Map

Heuristic encoding incorporating symmetry-aware gates for data with known group structure.

Qubits

4

Depth

12

Total Gates

48

Simulability

Not simulable

Mathematical Formulation

∣ψ(x)⟩=∏l=1reps[Usym⋅Ueq⋅Uenc(x)⋅H⊗n]∣0⟩⊗n|\psi(\mathbf{x})\rangle = \prod_{l=1}^{\text{reps}} \left[ U_{\text{sym}} \cdot U_{\text{eq}} \cdot U_{\text{enc}}(\mathbf{x}) \cdot H^{\otimes n} \right] |0\rangle^{\otimes n}

Description

The Symmetry-Inspired Feature Map incorporates symmetry information into the encoding circuit through symmetry-aware gate sequences. Unlike rigorously equivariant encodings, this approach uses heuristic circuit designs that respect the symmetry structure without formally guaranteeing equivariance.

Each layer applies: (1) Hadamard gates for superposition, (2) RY encoding gates with feature-dependent angles, (3) RZ equivariant rotation gates, and (4) symmetry-dependent entangling gates. The entangling gates vary by symmetry type: rotation symmetry uses controlled-RZ (CRZ) gates on coordinate pairs, cyclic symmetry uses CNOT-RZ-CNOT chains, reflection symmetry uses CZ gates with RZ rotations, and full symmetry uses a richer CNOT-RY-CNOT-RY-CNOT decomposition.

This encoding serves as a general-purpose symmetry-aware feature map when the specific equivariant encodings (SO2, Cyclic, Swap) do not match the problem's symmetry group. It provides an inductive bias toward symmetry-preserving representations while maintaining flexibility.

Circuit Diagram

Property Radar

Properties

Qubits
4
Circuit Depth
12
Total Gates
48
Single-Qubit Gates
36
Two-Qubit Gates
12
Parameters
0
Entangling
Yes
Simulability
Not Simulable
Expressibility
—
Entanglement Capability
—
Trainability
0.43
Noise Resilience
—

Resource Scaling

How resource requirements grow with the number of input features.

FeaturesQubitsDepthGates2Q Gates
228204
44124812
882010428
16163621660

Code Examples

Symmetry-Inspired Feature Map with PennyLane using rotation symmetry.

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

enc = SymmetryInspiredFeatureMap(n_features=4, symmetry="rotation", reps=2)
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([0.1, 0.5, 1.2, 2.3])
state = circuit(x)

When to Use This Encoding

  • Data with known but complex symmetry structure
  • General-purpose symmetry-aware encoding
  • Inductive bias for symmetry-preserving quantum ML models
  • Problems where rigorous equivariance is desirable but not required
  • Research into symmetry-informed quantum feature maps

Pros & Cons

Advantages

  • Incorporates symmetry information as inductive bias
  • Supports four symmetry types (rotation, cyclic, reflection, full)
  • More flexible than rigorously equivariant encodings
  • Multiple feature preprocessing options (angle, fourier, polynomial)
  • Configurable entanglement topology

Limitations

  • Heuristic — does not formally guarantee equivariance
  • More complex circuit than non-symmetry encodings
  • Requires knowing the data's symmetry type a priori
  • Full entanglement scales O(n²) for large feature counts
  • Lower trainability with deep circuits and many entangling pairs

References

  1. [1]Meyer, J.J., et al. (2023). Exploiting symmetry in variational quantum machine learning. PRX Quantum, 4(1), 010328.
  2. [2]Larocca, M., et al. (2022). Group-invariant quantum machine learning. PRX Quantum, 3(3), 030341.
  3. [3]Nguyen, Q.T., et al. (2022). Theory for equivariant quantum neural networks. PRX Quantum, 3(3), 030322.