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In superconducting circuit quantum information technologies, classicalmicrowave pulses are applied to control and measure the qubit states.Currently, the design of these microwave pulses use simple theoretical ornumerical models that do not account for the self-consistent interactions ofhow the qubit state modifies the applied microwave pulse. In this work, wepresent the formulation and finite element time domain discretization of asemiclassical Maxwell-Schr\"{o}dinger method for describing theseself-consistent dynamics for the case of a superconducting qubit capacitivelycoupled to a general transmission line network. We validate the proposed methodby characterizing key effects related to common control and measurementapproaches for transmon and fluxonium qubits in systems that are amenable totheoretical analysis. Our numerical results also highlight scenarios whereincluding the self-consistent interactions are essential. By treating themicrowaves classically, our method is substantially more efficient thanfully-quantum methods for the many situations where the quantum statistics ofthe microwaves are not needed. Further, our approach does not require anyreformulations when the transmission line system is modified. In the future,our method can be used to rapidly explore broader design spaces to search formore effective control and measurement protocols for superconducting qubits.

Maxwell-Schr\"{o}dinger Modeling of Superconducting Qubits Coupled to Transmission Line Networks