Data Re-uploading Classifier

Overview

Data re-uploading is a powerful technique that interleaves data encoding with variational layers. Remarkably, it achieves universal classification with just a single qubit.

The Key Insight

Traditional quantum classifiers encode data once, then apply variational layers. Data re-uploading encodes the same data multiple times, interspersed with trainable rotations:

CODE
|0⟩ → [Encode(x)][Variational] → [Encode(x)][Variational] → ... → Measure

The Circuit

CODE
┌─────────┐┌─────────┐┌─────────┐┌─────────┐┌─────────┐┌─────────┐
q_0: ┤ RX(x₀π) ├┤ RZ(x₁π) ├┤ RY(θ₀)  ├┤ RZ(θ₁)  ├┤ RX(x₀π) ├┤ RZ(x₁π) ├─...
     └─────────┘└─────────┘└─────────┘└─────────┘└─────────┘└─────────┘
         Data      Data      Weights    Weights      Data       Data
       Layer 1                                      Layer 2

Running the Circuit

PYTHON
from circuit import run_circuit

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

Universality Theorem

Theorem: A single-qubit data re-uploading classifier with L layers can approximate any Boolean function on n-dimensional input, given sufficient layers.

This means:

1 qubit is enough for any classification task
More layers = more complex decision boundaries
Universal approximation capability

Parameters

Layers	Parameters	Expressiveness
1	2	Linear boundaries
2	4	Simple curves
3	6	Complex boundaries
L	2L	Increasingly complex

Why It Works

Each layer applies a rotation that depends on both:

Input data x: Encoded as rotation angles
Trainable weights θ: Learned parameters

The composition creates rich, data-dependent transformations.

Advantages

Minimal qubits: Just 1 qubit needed
Noise-resilient: Single qubit = fewer errors
Universal: Can learn any decision boundary
Interpretable: Bloch sphere visualization

Comparison

Classifier	Qubits	Universality
Data Re-uploading	1	Yes
Standard VQC	N (# features)	Depends on ansatz
Kernel SVM	N	Depends on kernel

Applications

Binary classification: Any 2-class problem
NISQ devices: Minimal hardware requirements
Educational: Simple to understand and implement