Geometric phase for N -level systems through unitary integration

Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin 1a•2, the Bloch sphere S2, together with a U(1) phase, provides a complete SU(2) description. We generalize to N -level systems and SU (N) in terms of a 2 (Na 1) -dimensional base space and reduction to a (Na 1) -level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an N -level system and gives (Na 1) geometric phases explicitly. A complete analytical construction of an S4 Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4). © 2006 The American Physical Society