The Bloch sphere is the standard geometric visualization of a single qubit's state. Any pure state of a qubit can be represented as a point on the surface of a unit sphere, where the north pole corresponds to |0⟩, the south pole to |1⟩, and points on the equator represent equal superpositions with different relative phases. The state |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩ maps to the point (θ, φ) in spherical coordinates.
Single-qubit quantum gates correspond to rotations on the Bloch sphere. The X gate (quantum NOT) rotates 180° around the x-axis, flipping |0⟩ to |1⟩. The Z gate rotates 180° around the z-axis, changing the phase. The Hadamard gate rotates 180° around an axis between x and z, creating equal superpositions. This geometric picture makes it intuitive to understand gate compositions — two sequential rotations compose like physical rotations in 3D space.
Mixed states (partially decohered qubits) map to points inside the sphere rather than on its surface, with the center representing a completely mixed state carrying no quantum information. The radius from center to surface thus quantifies the "purity" of the qubit state. While the Bloch sphere is invaluable for single-qubit intuition, it does not generalize straightforwardly to multi-qubit systems — entangled states of two or more qubits cannot be represented as separate points on individual Bloch spheres, reflecting the fundamentally non-local nature of quantum correlations.