  Two-qubit system is the foundation of constructing the universal quantum
gate. We have studied the two-qubit Rabi model for the general case and its
generalizations with dipole, XXX and XYZ Heisenberg qubit-qubit interactions,
which are commonly used in quantum computation. Their solutions are presented
analytically with eigenstates can be obtained in terms of extended coherent
states or Fock states and applied to the construction of the ultrafast
two-qubit quantum gate in circuit QED and quantum state storage and transfer.
Some novel kinds of quasi-exact solutions are found for specific sets of
parameters, causing level crossings within the same parity subspace, which do
not appear in the regular spectrum, indicating its non-integrability. These
eigenstates are very interesting for quantum computing and single photon
experiments because they are formed by just a few Fock states, and in some
cases, with at most one photon. They are also easy to prepare, since they exist
for all qubit-photon coupling values with constant eigenenergy and the qubit
energy splittings can be fine tuned in the experiment in contrast to the
coupling.
