  We present a multi-level quantum theory of decoherence for a general circuit
realization of a superconducting qubit. Using electrical network graph theory,
we derive a Hamiltonian for the circuit. The dissipative circuit elements
(external impedances, shunt resistors) are described using the Caldeira-Leggett
model. The master equation for the superconducting phases in the Born-Markov
approximation is derived and brought into the Bloch-Redfield form in order to
describe multi-level dissipative quantum dynamics of the circuit. The model
takes into account leakage effects, i.e. transitions from the allowed qubit
states to higher excited states of the system. As a special case, we truncate
the Hilbert space and derive a two-level (Bloch) theory with characteristic
relaxation (T_1) and decoherence (T_2) times. We apply our theory to the class
of superconducting flux qubits; however, the formalism can be applied for both
superconducting flux and charge qubits.
