Quantum Data Re-Uploading Classifier

Dataset

Model

Re-uploading layers L 2

Learning rate η 0.30

Training Stats

Epoch

Loss (MSE)

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Accuracy

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Probe score f(x)

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Decision Boundary

click to move the probe point ✛

How it works

A single qubit is not just re-encoded once: the data-re-uploading classifier repeats an encoding gate and a trainable rotation across L layers, so the qubit "sees" the input several times:

|ψ(x,θ)⟩ = U(θ_L)·S(x)·⋯·U(θ_1)·S(x)·|0⟩
S(x)    = R_y(x₂π)·R_z(x₁π)              (data encoding, fixed)
U(θ_l)  = R_z(θ_l,3)·R_y(θ_l,2)·R_z(θ_l,1)  (trainable, per layer)
f(x,θ)  = ⟨ψ|Z|ψ⟩ = |α|² − |β|²           (classifier output)

Because each rotation is repeated with fresh trainable parameters between encodings, a single qubit can approximate decision boundaries that one encoding alone cannot — the same trick that gives classical feature maps their expressivity, done entirely inside one 2-level quantum system.

Training uses the exact parameter-shift rule — no numerical differentiation needed, because shifting a rotation angle by ±π/2 gives the true analytic gradient of a quantum expectation value:

∂f/∂θ = [f(θ+π/2) − f(θ−π/2)] / 2
Loss   = mean( (f(xᵢ,θ) − yᵢ)² )
θ ← θ − η·∇Loss

Circle / XOR — switch the target concept the qubit must learn.
Layers L — more re-uploads let the single qubit fit sharper, less linear boundaries.
Train / Step — run gradient descent; the Bloch sphere shows the probe point's full re-upload trajectory update live.
Probe point — click the boundary panel to test any (x,y) and watch its exact rotation path.

Final state |ψ⟩ Encoding step S(x) Trainable step U(θ)

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