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We theoretically study single and two-qubit dynamics in the circuit QEDarchitecture. We focus on the current experimental design [Wallraff et al.,Nature 431, 162 (2004); Schuster et al., Nature 445, 515 (2007)] in whichsuperconducting charge qubits are capacitively coupled to a single high-Qsuperconducting coplanar resonator. In this system, logical gates are realizedby driving the resonator with microwave fields. Advantages of this architectureare that it allows for multi-qubit gates between non-nearest qubits and for therealization of gates in parallel, opening the possibility of fault-tolerantquantum computation with superconduting circuits. In this paper, we focus onone and two-qubit gates that do not require moving away from thecharge-degeneracy `sweet spot'. This is advantageous as it helps to increasethe qubit dephasing time and does not require modification of the originalcircuit QED. However these gates can, in some cases, be slower than those thatdo not use this constraint. Five types of two-qubit gates are discussed, theseinclude gates based on virtual photons, real excitation of the resonator and agate based on the geometric phase. We also point out the importance ofselection rules when working at the charge degeneracy point.Comment: 23 pages, 8 figure

Quantum-information processing with circuit quantum electrodynamics