Data Re-uploading Classifier

Overview

Data re-uploading is a quantum classification technique that achieves universal approximation with just a single qubit. Instead of encoding data once and then applying variational layers (as in a standard VQC), it re-encodes the same input data multiple times, interspersed with trainable rotations.

This remarkable result, proved by Perez-Salinas et al. (2020), means that a single qubit with enough layers can learn any classification boundary -- no entanglement, no multi-qubit operations required.

The Universality Theorem

Theorem (Perez-Salinas et al., 2020): A single-qubit data re-uploading classifier with L re-uploading layers can approximate any function from R^n to {0, 1} to arbitrary precision, given sufficient layers.

The mathematical foundation: each re-uploading layer contributes a term to a Fourier series on the Bloch sphere. With L layers, the circuit computes a Fourier series of degree L, and increasing L increases the complexity of learnable decision boundaries.

Layers	Parameters	Fourier Degree	Expressiveness
1	2	1	Linear boundaries
2	4	2	Simple curves
3	6	3	Complex boundaries
L	2L	L	Arbitrary (L -> infinity)

The Circuit

CODE
     +---------+ +---------+ +--------+ +--------+     +---------+ +---------+ +--------+ +--------+ +-+
q_0: | RX(x0pi)|-| RZ(x1pi)|-| RY(th0)|-| RZ(th1)|--*--| RX(x0pi)|-| RZ(x1pi)|-| RY(th2)|-| RZ(th3)|-|M|
     +---------+ +---------+ +--------+ +--------+     +---------+ +---------+ +--------+ +--------+ +-+
      Data         Data       Weights    Weights         Data         Data       Weights    Weights
     <--------- Layer 1 --------->                      <--------- Layer 2 --------->

Each layer has the same structure:

Data encoding: RX(x0 * pi), RZ(x1 * pi) -- the same data, every layer
Variational rotations: RY(theta_j), RZ(theta_j+1) -- trainable parameters

The key is that data is uploaded L times, not once. This repeated encoding creates an increasingly rich data-dependent transformation on the Bloch sphere.

Why It Works

Consider the effect of L layers on the Bloch sphere:

CODE
Layer 1: Rotate by data -> Rotate by weights -> Point p1 on sphere
Layer 2: Rotate by data -> Rotate by weights -> Point p2 on sphere
  ...
Layer L: Rotate by data -> Rotate by weights -> Final point pL

The composition of these rotations creates a complex, data-dependent trajectory on the Bloch sphere. The measurement probability P(|1>) at the end depends on where the final point lands -- above or below the equator.

With enough layers, this trajectory can carve out any decision boundary in the input space.

Running the Circuit

PYTHON
from circuit import (
    run_circuit,
    predict,
    train_classifier,
    verify_classifier,
)

# Single prediction (random parameters)
result = run_circuit(x=[0.5, -0.3], n_layers=3)
print(f"Qubits: {result['n_qubits']}")  # Just 1!
print(f"Prediction: class {result['prediction']}")

# Train on XOR-like dataset
trained = train_classifier(max_iterations=120, seed=42)
print(f"Accuracy: {trained['accuracy']:.1%}")

# Predict with trained model
theta_opt = trained["optimal_theta"]
pred = predict([0.5, 0.5], theta_opt, n_layers=3)
print(f"Class: {pred['prediction']}, Confidence: {pred['confidence']:.1%}")

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

Default Dataset: XOR Pattern

The default dataset tests non-linear classification (XOR is not linearly separable):

CODE
         x1
    1 |  0  |  1
      |     |
   ---|-----|-----
      |     |
   -1 |  1  |  0
      -1    0    1
             x0

Input	Label	Pattern
[0.5, 0.5]	0	Same sign
[-0.5, -0.5]	0	Same sign
[0.5, -0.5]	1	Opposite sign
[-0.5, 0.5]	1	Opposite sign

A single layer (linear boundary) cannot solve XOR. Multiple re-uploading layers can.

Comparison with Other Classifiers

Classifier	Qubits	Entanglement	Universality	Min Layers
Data Re-uploading	1	None	Yes	Depends on task
Standard VQC	N (features)	Required	Depends on ansatz	1-3
Quantum Kernel SVM	N	Required	Depends on kernel	N/A
Classical Perceptron	N/A	N/A	No (linear only)	1
Classical Neural Net	N/A	N/A	Yes	Depends on task

Advantages

Minimal hardware: Just 1 qubit, no entanglement gates
Noise-resilient: Single qubit = fewest error sources possible
Universal: Can learn any decision boundary with enough layers
Interpretable: Bloch sphere visualization of decision boundary
NISQ-friendly: Runs on any quantum hardware, even the noisiest

Limitations

Training cost: COBYLA optimization can require many iterations
Shot noise: Single-qubit measurement has high variance
Scaling: For high-dimensional data, need more rotations per layer
Circuit depth: Deep circuits (many layers) still suffer from noise

Applications

Binary classification: Any 2-class problem on 2D input
NISQ benchmarking: Minimal hardware requirements for testing
Educational: Clear demonstration of quantum ML universality
Hybrid models: Single-qubit quantum layer in classical networks

Learn More

Data Re-uploading for a Universal Quantum Classifier (Perez-Salinas et al., 2020)
One Qubit as a Universal Approximant (Perez-Salinas et al., 2021)
Effect of Data Encoding on Expressive Power (Schuld et al., 2021)