Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)

A two-sphere (``Bloch'' or ``Poincare'') is familiar for describing the dynamics of a spin-1/2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2). We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field.