Dynamics of Josephson junction systems in the computational subspace

The quantum dynamics of the Josephson junction system in the computational subspace is investigated. A scheme for the controlled not operation is given for two capasitively coupled SQUIDs. In this system, there is no systematic error for the two qubit operation. For the inductively coupled SQUIDs, the effective Hamiltonian causes systematic errors in the computational subspace for the two qubit operation. Using the purterbation theory, we construct a more precise effective Hamiltonian. This new effective Hamiltonian reduces the systematic error to the level much lower than the threshold of the fault resilent quantum computation.