Variational

Data Reuploading

Re-encodes classical data multiple times across layers, achieving universal function approximation.

Qubits

4

Depth

12

Total Gates

21

Simulability

Not simulable

Mathematical Formulation

∣ψ(x)⟩=∏l=1L[Uent⋅∏iRY(xi mod n)]∣0⟩⊗n|\psi(\mathbf{x})\rangle = \prod_{l=1}^{L} \left[ U_{\text{ent}} \cdot \prod_{i} R_Y(x_{i \bmod n}) \right] |0\rangle^{\otimes n}

Description

Data reuploading is a powerful variational encoding strategy that re-encodes the classical input data at every layer of the quantum circuit, interleaved with entangling gates. This repeated injection of data, analogous to the universal approximation theorem for classical neural networks, provably enables approximation of any continuous function.

Each layer applies RY rotations with feature-dependent angles to encode data, followed by a CNOT ladder for entanglement. Features are cyclically mapped to qubits: feature x_i is encoded on qubit (i mod n_qubits), so all features are represented even when n_qubits < n_features. The Fourier expressivity of the circuit is f(x) = Σ_k c_k exp(ikx) for k ∈ {-L, ..., L}, where L is the number of layers.

The encoding excels at tasks requiring high expressibility and is particularly well-suited for time series data due to its sequential, recurrent-like structure. However, the deep circuits can suffer from barren plateaus and noise accumulation on current NISQ hardware.

Circuit Diagram

Property Radar

Properties

Qubits
4
Circuit Depth
12
Total Gates
21
Single-Qubit Gates
12
Two-Qubit Gates
9
Parameters
12
Entangling
Yes
Simulability
Not Simulable
Expressibility
—
Entanglement Capability
—
Trainability
0.75
Noise Resilience
—

Resource Scaling

How resource requirements grow with the number of input features.

FeaturesQubitsDepthGates2Q Gates
22693
4412219
88244521
1616489345

Code Examples

Data reuploading with PennyLane using 3 layers and 4 features.

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

enc = DataReuploading(n_features=4, n_layers=3)
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

  • Universal function approximation in quantum ML
  • Time series classification and forecasting
  • Tasks requiring high expressibility and rich Fourier spectra
  • Variational quantum eigensolvers with data-dependent ansätze
  • Quantum neural networks (QNNs) with data-driven layers

Pros & Cons

Advantages

  • Universal approximation capability (proven for single-qubit case)
  • Fourier expressivity grows linearly with number of layers
  • Cyclic feature mapping enables qubit-efficient encoding
  • Natural recurrent structure suits sequential data
  • Good trainability for moderate layer counts

Limitations

  • Deep circuits — depth grows with layers and qubit count
  • Barren plateau risk for many layers (>10 triggers warning)
  • Not NISQ-friendly for large feature counts
  • Feature count recommended ≤8 for practical use
  • Noise accumulation in deep circuits degrades fidelity

References

  1. [1]Pérez-Salinas, A., et al. (2020). Data re-uploading for a universal quantum classifier. Quantum, 4, 226.
  2. [2]Schuld, M., Sweke, R., & Meyer, J.K. (2021). Effect of data encoding on the expressive power of variational quantum machine learning models. Physical Review A, 103(3), 032430.
  3. [3]Vidal, G. & Dawson, C.M. (2004). Universal quantum circuit for two-qubit transformations. Physical Review A, 69(1), 010301.