Four-level and two-qubit systems, subalgebras, and unitary integration

Four-level systems in quantum optics, and for representing two qubits in quantum computing, are difficult to solve for general time-dependent Hamiltonians. A systematic procedure is presented which combines analytical handling of the algebraic operator aspects with simple solutions of classical, first-order differential equations. In particular, by exploiting su(2)+su(2) and su(2)+su(2)+u(1) subalgebras of the full SU(4) dynamical group of the system, the nontrivial part of the final calculation is reduced to a single Riccati (first-order, quadratically nonlinear) equation, itself simply solved. Examples are provided of two-qubit problems from the recent literature, including implementation of two-qubit gates with Josephson junctions.