Variational Quantum Classifier

Dataset

Training

Learning rate η 0.40

Live Metrics

Loss (MSE)

0.000

Accuracy

Epoch

|∇θ|

0.000

θ₁=0.00 θ₂=0.00 θ₃=0.00

Feature Space

class +1 class −1 misclassified

How it works

A single qubit is a full binary classifier. Each 2D point (x₁,x₂) is encoded into the qubit with fixed rotations, then a trainable circuit with parameters θ₁,θ₂,θ₃ rotates the state further:

|ψ(x;θ)⟩ = R_z(θ₃) R_y(θ₂) R_z(θ₁) · R_y(πx₂) R_z(πx₁) |0⟩
prediction  ẑ(x;θ) = ⟨ψ|Z|ψ⟩ ∈ [-1, 1]
loss        L(θ)   = (1/N) Σᵢ (ẑ(xᵢ;θ) − yᵢ)²

The gradient of a quantum expectation value with respect to a rotation-gate angle is computed exactly on real quantum hardware with the parameter-shift rule — no backprop through the circuit is needed:

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

Each "Train Step" evaluates the circuit at θₖ±π/2 for all three parameters over the whole dataset, assembles ∂L/∂θ, and applies one gradient-descent update θ ← θ − η∇L. The Bloch sphere shows every data point's output state; the translucent equator plane is the decision boundary ⟨Z⟩=0 — points above it are classified +1, below are −1. The Circles dataset is not linearly separable in the raw plane, so watch the model plateau below 100% accuracy — a real limitation of a one-qubit model, not a bug.

class +1 (world) class −1 (world) misclassified

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