in a minimally extended state space. Starting from the Feynman–Vernon path integral formalism for open quantum systems with the system degrees of freedom being the nonlinear elements of the circuit, we eliminate the temporally non-local influence functional of all linear elements by introducing auxiliary harmonic modes with complex-valued frequencies coupled to the non-linear degrees of freedom of the circuit. In our work, we propose a concept of time-averaged observables, inspired by experiment, and provide an explicit formula for producing their quasiprobability distribution. Furthermore, we systematically derive a weak-coupling approximation in the presence of a drive, and demonstrate the applicability of our formalism through a study on the dispersive readout of a superconducting qubit. The developed framework enables a comprehensive fully quantum-mechanical treatment of nonlinear quantum circuits coupled to their environment, without the limitations of typical approaches to weak dissipation, high temperature, and weak drive. Furthermore, we discuss the implications of our findings to the quantum measurement theory.