  Modelling the electrical response of multi-level quantum systems at finite
frequency has been typically performed in the context of two incomplete
paradigms: (i) input-output theory, which is valid at any frequency but
neglects dynamic losses, and (ii) semiclassical theory, which captures well
dynamic dissipation effects but is only accurate at low frequencies. Here, we
develop a unifying theory, valid for arbitrary frequencies, that captures both
the quantum behaviour and the non-unitary effects introduced by relaxation and
dephasing. The theory allows a multi-level system to be described by a
universal small-signal equivalent circuit model, a resonant RLC circuit, whose
topology only depends on the number of energy levels. We apply our model to a
double quantum-dot charge qubit and a Majorana qubit, showing the capability to
continuously describe the systems from adiabatic to resonant and from coherent
to incoherent, suggesting new and realistic experiments for improved quantum
state readout. Our model will facilitate the design of hybrid quantum-classical
circuits and the simulation of qubit control and quantum state readout.
