Both panels see the same 2D points. The classical panel fits a straight-line (logistic-regression) boundary directly in the original 2D plane — on rings or spirals no single line can separate the classes, so it settles for the least-bad line and misclassifies a cluster of points on each side.
The quantum panel never draws a line in 2D at all. Each point (x, y) is encoded onto 3 qubits via per-qubit RY/RZ rotations plus an entangling ring of CNOTs, producing a genuine state vector in a 8-dimensional complex Hilbert space (dimension = 2qubits, growing exponentially with each extra qubit). The quantum kernel k(a,b) = |⟨ψ(a)|ψ(b)⟩|² measures how much two points' encoded states overlap in that space. A kernel perceptron then classifies using only these overlaps — never the raw coordinates — and because the encoding folds the plane into a much richer space, its boundary can wrap around rings and spirals that defeat the straight line.
|ψ(x,y)⟩ = CNOT-ring · ⊗ᵢ RZ(φᵢ)RY(θᵢ) |0…0⟩
k(a,b) = |⟨ψ(a)|ψ(b)⟩|²
f(x) = sign(Σⱼ αⱼ yⱼ k(xⱼ,x))
- Noise — how much random jitter is added to each point; higher noise makes both classifiers' job harder.
- Encoding qubits — more qubits ⇒ exponentially larger Hilbert space (2, 3 or 4 qubits ⇒ dimension 4, 8 or 16) and a richer, more flexible quantum boundary.
- Show quantum encoding — animates a few sample points settling onto a Bloch sphere, a simplified single-qubit picture of what "encoding a classical point onto a quantum state" means.