  We theoretically study single and two-qubit dynamics in the circuit QED
architecture. We focus on the current experimental design [Wallraff et al.,
Nature 431, 162 (2004); Schuster et al., Nature 445, 515 (2007)] in which
superconducting charge qubits are capacitively coupled to a single high-Q
superconducting coplanar resonator. In this system, logical gates are realized
by driving the resonator with microwave fields. Advantages of this architecture
are that it allows for multi-qubit gates between non-nearest qubits and for the
realization of gates in parallel, opening the possibility of fault-tolerant
quantum computation with superconduting circuits. In this paper, we focus on
one and two-qubit gates that do not require moving away from the
charge-degeneracy `sweet spot'. This is advantageous as it helps to increase
the qubit dephasing time and does not require modification of the original
circuit QED. However these gates can, in some cases, be slower than those that
do not use this constraint. Five types of two-qubit gates are discussed, these
include gates based on virtual photons, real excitation of the resonator and a
gate based on the geometric phase. We also point out the importance of
selection rules when working at the charge degeneracy point.
