and qubit-qubit crosstalk which show the need for better models/tools to understand and improve the operation of the superconducting quantum processors.

Several methods have been used to model and study the physics of superconducting qubit circuits. The Jaynes-Cummings model [@Jaynes-Cummings] originally introduced in quantum optics has routinely been applied to the study of the so-called circuit-QED architecture [@Blais-xQED; @Wallraff] in which superconducting qubits are coupled to readout resonators for their control and readout and two-qubit gate operations are mediated by the bus resonators. Readout and bus resonators are typically designed to be detuned away from the qubits to operate in the so-called dispersive regime. In that regime one can eliminate the resonators up to desired order in the bare qubit-resonator couplings and get an effective description of the system in the qubit subspace. However calculation of the dispersive quantities such as the exchange coupling or Purcell decay rates [@Controlling; @Spontaneous; @Emission; @-; @Houck] of the qubits with the single mode Jaynes-Cummings model showed significant discrepancy with the experimental measurements and attempts to include higher harmonics of the resonators with multi-mode extensions of the Jaynes-Cummings model failed due to divergence issues [@Bourassa-Multi-Mode-circuit-QED]. [@Gely-Adrian-Solano] showed the convergence of the Lamb shift in 