  We present detailed theory for Andreev level qubit, the system consisting of
a highly transmissive quantum point contact embedded in a superconducting loop.
The two-level Hamiltonian for Andreev levels interacting with quantum phase
fluctuations is derived by using the path integral method. We also derive
kinetic equation describing qubit decoherence due to interaction of the Andreev
levels with acoustic phonons. The collision terms are non-linear due to
fermionic nature of the Andreev states, leading to slow non-exponential
relaxation and dephasing of the qubit at temperature smaller than the qubit
level spacing.
