The Mathematics
Any single-qubit state can be written:
|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩
where 0 ≤ θ ≤ π and 0 ≤ φ < 2π. The Bloch vector is:
r = (sin θ cos φ, sin θ sin φ, cos θ)
Pure states lie on the surface (|r| = 1); mixed states (classical probabilistic mixtures) live inside the sphere. Measurement in the computational basis collapses the state to the north pole (with probability cos²(θ/2)) or the south pole (probability sin²(θ/2)).
Gates as rotations
Quantum gates are rotations on the Bloch sphere:
- X gate (NOT): 180° rotation around the x-axis. Flips |0⟩ to |1⟩.
- Hadamard (H): rotates the north pole to the equator — creates equal superposition.
- T gate: 45° rotation around the z-axis — creates the phase needed for quantum speedups.
Any single-qubit gate is a rotation R_n(α) by angle α around axis n. This geometric picture — quantum computation as choreographed rotations — is the foundation of quantum circuit design.
Further reading
- Nielsen & Chuang, Quantum Computation and Quantum Information — the standard textbook, Chapter 1.
- Bloch sphere (Wikipedia)
- 3Blue1Brown & MinutePhysics, Quantum Computing — excellent visual series.