Data Re-uploading Classifier - Pérez-Salinas et al. 2020

Overview

Reproduction of the data re-uploading paper - demonstrating that a single qubit can be a universal classifier when data is re-uploaded through multiple layers.

Property	Value
Category	Research
Difficulty	Advanced
Framework	PennyLane
Qubits	1
Paper	Quantum 4, 226 (2020)

Key Insight

A single qubit with L layers of data re-uploading can:

Approximate any Boolean function on 2L bits
Achieve universal classification
Match deep neural network expressivity

This challenges the assumption that more qubits = more power.

The Data Re-uploading Idea

Traditional quantum classifiers encode data once, then apply parametrized gates:

CODE
|0⟩ → S(x) → U(θ) → Measure

Data re-uploading interleaves data and parameters:

CODE
|0⟩ → U(θ₁,x) → U(θ₂,x) → ... → U(θₗ,x) → Measure

Circuit Structure

Each layer applies:

CODE
U(θ,x) = RZ(θ₁ + x₁w₁) RY(θ₂ + x₂w₂) RZ(θ₃)

For L layers on a single qubit:

CODE
┌──────────────┐┌──────────────┐     ┌──────────────┐
q_0: ┤ U(θ₁,x)     ├┤ U(θ₂,x)     ├ ... ┤ U(θₗ,x)     ├ → Z
     └──────────────┘└──────────────┘     └──────────────┘

Running the Circuit

PYTHON
from circuit import run_circuit

result = run_circuit(
    n_layers_list=[1, 2, 3, 4],
    n_train=50,
    n_test=20,
    n_epochs=30
)

for n_layers, data in result['layer_analysis'].items():
    print(f"{n_layers} layers: {data['test_accuracy']:.1%} accuracy")

Expected Results

Circle Dataset (2D Classification)

Layers	Parameters	Train Acc	Test Acc
1	6	~60%	~55%
2	12	~75%	~70%
3	18	~85%	~80%
4	24	~90%	~85%

Universality Theorem

From the paper:

"A single-qubit quantum classifier with L layers can approximate any Boolean function f: {0,1}^{2L} → {0,1}"

Implementation Details

Feature Encoding

PYTHON
angle = θ + x * w

Where:

θ: Learnable bias
x: Input feature (scaled to [0, 2π])
w: Learnable weight

Training

Gradient descent on cross-entropy loss
Numerical gradients (or PennyLane autodiff)
Typical learning rate: 0.1

Why It Works

Non-linearity: Repeated rotations create highly non-linear decision boundaries
Data mixing: Each layer mixes data with parameters differently
Interference: Quantum interference creates complex classification patterns

Comparison with Classical

Model	Parameters	Qubits	Expressivity
Single perceptron	3	0	Linear
Data re-upload (L=3)	18	1	Universal
Deep NN (3 layers)	100+	0	Universal

Paper Citation

BIBTEX
@article{perez2020data,
  title={Data re-uploading for a universal quantum classifier},
  author={P{\'e}rez-Salinas, Adri{\'a}n and others},
  journal={Quantum},
  volume={4},
  pages={226},
  year={2020}
}

Data Re-uploading (Pérez-Salinas 2020)