Interactive Simulator
Every qubit state lives on this sphere. Gates are rotations around specific axes. Watch X rotate 180° around the X axis, flipping |0⟩ to |1⟩. Watch H rotate 180° around the diagonal (X+Z)/√2 axis, landing on the equator. Drag to orbit · scroll to zoom.
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|0⟩
Circuit so far
A qubit state |ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩ can be represented as a point on a unit sphere using two angles: θ (polar, 0 to π) and φ (azimuthal, 0 to 2π). The north pole is |0⟩, the south pole is |1⟩, and the equator is pure superposition. Every quantum gate is a rotation of this sphere.
The Pauli matrices X, Y, Z each generate rotations via e−iθP/2.
At θ=π (180°) this gives the quantum gate.
The Rodrigues formula implements exactly this rotation without matrix exponentiation:
v′ = v·cos(θ) + (k×v)·sin(θ) + k(k·v)(1−cos(θ))
where k is the gate's rotation axis.