A qubit state (the bright dot) precessing around an external field direction on the Bloch sphere.
The Core Idea
A qubit is not a "bit that can be 0 and 1 at the same time" — it's more nuanced than that. The state of a single qubit is a point on the Bloch sphere, a unit sphere where the north pole is |0⟩, the south pole is |1⟩, and every other point represents a quantum superposition cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩. The two angles θ and φ specify the state completely.
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)).
Stronger coupling — the precession is faster. The tilted field axis determines the cone the state traces.
Larmor precession
When a qubit couples to an external field (like a magnetic field on a spin-½ particle), the state vector precesses around the field axis:
d|ψ⟩/dt = −iγ(H/ℏ)|ψ⟩
where γ is the gyromagnetic ratio and H is the Hamiltonian. On the Bloch sphere, this looks like the state vector tracing a cone around the field direction — exactly like a gyroscope precessing in a gravitational field.
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.
Field along the equator — the state precesses around the x-axis, cycling between superposition states.
Why a sphere and not a circle?
A classical bit is 0 or 1 — a point on a line. A probabilistic bit is a probability p ∈ [0, 1] — a point on a line segment. A qubit needs two angles because of the phase φ. Two states that differ only in phase (e.g., (|0⟩ + |1⟩)/√2 vs (|0⟩ − |1⟩)/√2) produce identical measurement probabilities but behave differently under further operations. The phase is invisible to measurement but critical to interference — and interference is what makes quantum algorithms work.
Further reading
- Nielsen & Chuang, Quantum Computation and Quantum Information — the standard textbook, Chapter 1.
- Bloch sphere (Wikipedia)
- 3Blue1Brown & MinutePhysics, Quantum Computing — excellent visual series.