Quantum Data Re-uploading Classifier

Circuit

Re-uploading layers L 3

Learning rate η 0.30

Dataset pattern

Training

Live Metrics

Step

Accuracy

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Loss (MSE)

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⟨Z⟩ at cursor

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How it works

A single qubit, re-used many times, can act as a universal binary classifier (Pérez-Salinas et al., "Data re-uploading for a universal quantum classifier"). Each of the L layers re-encodes the same 2D input point and adds one trainable rotation:

layer l:  R_x(π·x₁) · R_z(π·x₂) · R_y(θ_l)
output:   ⟨Z⟩ = ⟨ψ_L| Z |ψ_L⟩ ∈ [-1, 1]
predict:  ŷ = sign(⟨Z⟩)

Because the classical input is re-injected before every trainable rotation, a single qubit can carve out a decision boundary of arbitrary complexity as L grows — the Bloch-sphere path on the left visibly folds more with each extra layer.

Training minimises MSE loss L = mean[(⟨Z⟩ − y)²] over the dataset. Each θ_l is a Pauli-Y rotation angle, so its exact gradient comes from the parameter-shift rule — no backprop through the quantum state is needed, only two extra circuit evaluations:

∂⟨Z⟩/∂θ_l = [ ⟨Z⟩(θ_l+π/2) − ⟨Z⟩(θ_l−π/2) ] / 2

L slider — more re-uploading layers ⇒ more expressive boundary, but slower/harder to train.
Hover the right-hand grid — traces that exact point's Bloch trajectory live, with the current ⟨Z⟩ readout.
Train — runs one parameter-shift gradient-descent step roughly every 150 ms; the boundary tiles and point heights update as θ improves.

This is the same trick behind NISQ-era hybrid quantum-classical models: keep the trainable circuit shallow (few qubits, few gates) and let classical gradient descent do the optimisation.

Bloch trajectory (hovered point) Predicted class +1 Predicted class −1

Hover the grid →

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