Symmetry & Equivariant

Cyclic Equivariant Feature Map

Translationally invariant encoding preserving cyclic (Z_n) group symmetry via ring topology.

Qubits

4

Depth

6

Total Gates

24

Simulability

Not simulable

Mathematical Formulation

∣ψ(x)⟩=∏l=1reps[⨂iRX(π/6)⋅∏iRZZi,i+1(α)⋅⨂iRY(xi)]∣0⟩⊗n|\psi(\mathbf{x})\rangle = \prod_{l=1}^{\text{reps}} \left[ \bigotimes_i RX(\pi/6) \cdot \prod_{i} RZZ_{i,i+1}(\alpha) \cdot \bigotimes_i RY(x_i) \right] |0\rangle^{\otimes n}

Description

The Cyclic Equivariant Feature Map constructs a circuit that is equivariant under cyclic permutations of the input features. A cyclic shift σ of the input features x → (x_1, x_2, ..., x_n, x_0) results in the same cyclic shift of the quantum state, preserving the Z_n symmetry group.

Each layer applies: (1) RY rotations encoding features x_i on each qubit, (2) RZZ entangling gates in a ring topology connecting (0,1), (1,2), ..., (n-1,0) with a fixed coupling strength, and (3) RX rotations with a fixed angle (π/6) for basis mixing. The ring topology and uniform coupling ensure that the circuit commutes with cyclic permutation operators.

This encoding is ideal for data with periodic or cyclic structure, such as time series with seasonal patterns, molecular ring structures, or any problem where features have a natural circular ordering.

Circuit Diagram

Property Radar

Properties

Qubits
4
Circuit Depth
6
Total Gates
24
Single-Qubit Gates
16
Two-Qubit Gates
8
Parameters
0
Entangling
Yes
Simulability
Not Simulable
Expressibility
—
Entanglement Capability
—
Trainability
—
Noise Resilience
—

Resource Scaling

How resource requirements grow with the number of input features.

FeaturesQubitsDepthGates2Q Gates
226124
446248
8864816
161669632

Code Examples

Cyclic equivariant encoding with PennyLane using ring topology RZZ gates.

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

enc = CyclicEquivariantFeatureMap(n_features=4, 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

  • Periodic or seasonal time series data
  • Molecular ring structures (e.g., benzene, cyclopentane)
  • Data with natural circular ordering
  • Clock-like or angular measurements
  • Problems with discrete rotational symmetry (Z_n group)

Pros & Cons

Advantages

  • Rigorous Z_n equivariance — mathematically guaranteed
  • Ring topology provides efficient O(n) entanglement
  • Strong inductive bias for periodic/cyclic data
  • Constant depth per repetition (3 layers per rep)
  • Configurable coupling strength for interaction control

Limitations

  • Only captures cyclic symmetry — not general permutation equivariance
  • Ring topology limits to nearest-neighbor+wrap interactions
  • Requires features to have meaningful cyclic ordering
  • Fixed RX angle limits circuit flexibility
  • Performance degrades for non-periodic data

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]Meyer, J.J., et al. (2023). Exploiting symmetry in variational quantum machine learning. PRX Quantum, 4(1), 010328.