  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.
