  We introduce a numerically exact and yet computationally feasible nonlinear
response theory developed for lossy superconducting quantum circuits based on a
framework of quantum dissipation 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.
