  We present an efficient, accurate, and comprehensive analysis framework for
generic, multi-port nonlinear parametric circuits, in the presence of
dissipation from lossy circuit components, based on "quantum-adapted"
X-parameters. We apply this method to Josephson traveling-wave parametric
amplifiers (JTWPAs) - a key component in superconducting and spin qubit quantum
computing architectures - which are challenging to model accurately due to
their thousands of linear and nonlinear circuit components. X-parameters are
generated from a harmonic balance solution of the classical nonlinear circuit
and then mapped to the field ladder operator basis, so that the energy
associated with each of the multiple interacting modes corresponds to photon
occupancy, rather than classical power waves. Explicit relations for the
quantum efficiency of a generic, multi-port, multi-frequency parametric circuit
are presented and evaluated for two distinct JTWPA designs. The gain and
quantum efficiency are consistent with those obtained from Fourier analysis of
time-domain solutions, but with enhanced accuracy, speed, and the ability to
include real-world impairments, statistical variations, parasitic effects, and
impedance mismatches (in- and out-of-band) seamlessly. The unified flow is
implemented in Keysight's PathWave Advanced Design System (ADS) and
independently in an open-source simulation code, JosephsonCircuits.jl, from the
MIT authors.
