The Bloch sphere is a geometric representation of single-qubit quantum states.

Mapping States to Points

Any pure single-qubit state |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩ maps to a point on the unit sphere with polar angle θ and azimuthal angle φ.

Special States

• ▸|0⟩: north pole
• ▸|1⟩: south pole
• ▸|+⟩ = (|0⟩ + |1⟩)/√2: +X axis
• ▸|-⟩ = (|0⟩ - |1⟩)/√2: -X axis
• ▸|+i⟩ = (|0⟩ + i|1⟩)/√2: +Y axis

Rotations

Rotation around the X, Y, or Z axis by angle θ is performed by the gates R_x(θ), R_y(θ), R_z(θ). Any single-qubit unitary is a rotation on the Bloch sphere.