  We present a theoretical description for circuits consisting of weak
anharmonic qubits coupled to cavity multimodes. We obtain a unitary
transformation that diagonalizes harmonic sector of the circuit. Weak
anharmonicity does not alter the normal mode basis, however it can modify
energy levels. We study two examples of a transmon and two transmons coupled to
bus resonator, and we determine dressed frequencies and Kerr nonlinearities in
closed form formulas. Our results are valid for arbitrary frequency detuning
and coupling within and beyond dispersive regime.
