Quantum Spin — Bloch Sphere

A qubit |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩ lives on the Bloch sphere surface

|ψ⟩ = |0⟩ | P(|0⟩) = 1.000 | P(|1⟩) = 0.000

θ: 0.00 φ: 0.00

A qubit state |ψ⟩ = α|0⟩ + β|1⟩ with |α|²+|β|²=1 corresponds to a point on the Bloch sphere via θ=2arccos|α|, φ=arg(β/α). Quantum gates are rotations on this sphere — the Hadamard gate is a 180° rotation about (X+Z)/√2.