The Bloch Sphere
The Bloch sphere is a geometric picture where every pure single-qubit state is a point on a unit sphere and every gate is a rotation.
Mapping a qubit to a sphere
A general pure qubit state can be written, up to an irrelevant global phase, as |psi> = cos(theta/2)|0> + e^{i phi} sin(theta/2)|1>, with theta from 0 to pi and phi from 0 to 2pi. These two angles are exactly the polar and azimuthal angles of a point on a unit sphere, the Bloch sphere.
Landmarks
- North pole (theta=0): |0>
- South pole (theta=pi): |1>
- +x axis: (|0>+|1>)/sqrt(2)
- -x axis: (|0>-|1>)/sqrt(2)
- +y axis: (|0>+i|1>)/sqrt(2)
- -y axis: (|0>-i|1>)/sqrt(2)
Orthogonal quantum states sit at antipodal points, not at 90 degrees. |0> and |1> are perpendicular in Hilbert space but opposite poles on the sphere. The factor theta/2 is what produces this doubling.
Gates as rotations
Every single-qubit unitary is a rotation of the sphere about some axis by some angle. The Pauli-X gate is a 180-degree rotation about the x axis; the Hadamard gate is a 180-degree rotation about the diagonal x+z axis. Composing gates composes rotations, which makes the geometry a fast intuition tool for single-qubit reasoning.
Phase becomes longitude
Global phase does not move the point — it is unobservable — so the sphere naturally quotients it out. Relative phase phi is the longitude, which is why phase gates simply spin the state around the vertical axis and leave measurement probabilities in the computational basis unchanged.
Limits
The Bloch sphere describes one qubit only. Mixed states fill the interior of the ball, with the fully mixed state at the centre and pure states on the surface (see Bloch vector). There is no equally simple picture for two or more qubits, because entanglement has no low-dimensional visual analogue.