The Geometry of a Qubit
A classical bit is either 0 or 1. A qubit can be any point on the surface of a unit sphere — the Bloch sphere, named after physicist Felix Bloch. The north pole represents |0⟩, the south pole represents |1⟩, and every other point is a superposition of both. This geometric picture makes it intuitive to understand how quantum gates work: each gate is simply a rotation of the state vector on this sphere.
Angles and Amplitudes
The polar angle θ (theta) determines the measurement probabilities: cos²(θ/2) for |0⟩ and sin²(θ/2) for |1⟩. At θ=0 (north pole), you always measure |0⟩; at θ=π (south pole), always |1⟩; at θ=π/2 (equator), it's a perfect coin flip. The azimuthal angle φ (phi) controls the relative phase between the |0⟩ and |1⟩ components — invisible in Z measurements but critical for interference effects and X/Y measurements.
Decoherence and the Shrinking Sphere
Real qubits interact with their environment, causing decoherence. The T₂ time quantifies how long the phase relationship persists. As decoherence progresses, the Bloch vector shrinks from the surface toward the center, representing a mixed state. Superconducting qubits achieve T₂ times of 50–200 µs, while trapped ions can reach milliseconds. Every quantum algorithm must complete within this coherence window.
From Sphere to Quantum Gates
Each single-qubit quantum gate maps to a specific rotation on the Bloch sphere. The Pauli-X gate rotates 180° around the X-axis (equivalent to a classical NOT). The Hadamard gate takes |0⟩ to the equator, creating equal superposition. By composing rotations around different axes, any single-qubit unitary transformation can be achieved — this is the foundation of universal quantum computation.