Bloch Sphere

The Bloch sphere is a unit sphere used to visualize the state of a single qubit. Every pure qubit state |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩ corresponds to a unique point on the sphere surface using polar angle θ and azimuthal angle φ. The north pole represents |0⟩ and the south pole represents |1⟩. The equator points represent equal superpositions: |+⟩, |−⟩, |i⟩, |−i⟩. Single-qubit gates correspond to rotations of the Bloch sphere: the X gate is a 180° rotation about the X axis, Z gate about the Z axis, and H gate about the axis between X and Z. Mixed states (from decoherence) are represented by points inside the sphere. The Bloch sphere is purely a visualization tool for single qubits — multi-qubit states cannot be represented this way.