Qubit Gates on the Bloch Sphere

Single-Qubit Gates

Phase rotation R_z(φ) 0°

Quantum State

|ψ⟩ = 1.00|0⟩

P(|0⟩) = 100%P(|1⟩) = 0%

Polar angle θ

0.0°

Azimuth φ

0.0°

How it works

Any pure single-qubit state can be written |ψ⟩ = α|0⟩ + β|1⟩ with |α|² + |β|² = 1. Up to a global phase it is fully described by two real angles on the Bloch sphere:

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩
Bloch vector: n = (sinθ cosφ, sinθ sinφ, cosθ)
Born rule:    P(0) = |α|²,  P(1) = |β|²

A gate is a 2×2 unitary matrix acting on (α, β). Every single-qubit gate corresponds to a rotation of this vector on the sphere:

H = 1/√2 [[1, 1],[1,-1]]   → rotates X-axis ↔ Z-axis
X = [[0,1],[1,0]]           → π rotation about X (bit flip)
Y = [[0,-i],[i,0]]          → π rotation about Y
Z = [[1,0],[0,-1]]          → π rotation about Z (phase flip)
S = [[1,0],[0,i]]           → π/2 rotation about Z
T = [[1,0],[0,e^(iπ/4)]]    → π/4 rotation about Z

H, X, Y, Z, S, T — apply the matrix to the current state; the arrow rotates to the new Bloch vector.
R_z(φ) slider — a continuous phase rotation about the Z-axis (multiplies β by e^(iφ)), showing that gates are just points on a continuous family of rotations.
Measure — samples an outcome from the Born-rule probabilities |α|² and |β|²; the state collapses to the measured pole, exactly as a real qubit readout would.

Real-world relevance: this is the exact single-qubit model used inside every gate-based quantum computer (IBM, Google, Rigetti) — multi-qubit circuits are built by composing gates like these across entangled qubits.

State vector |ψ⟩ Bloch sphere axes

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