Modeling 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 dynamic dissipation effects well but is only accurate at low frequencies. Here, we develop a unifying theory, valid for arbitrary frequencies, that captures both the small-signal quantum behavior 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.

在有限频率下对多能级量子系统的电响应进行建模通常在两个不完整范式的背景下进行：（i） 输入-输出理论，在任何频率下都有效，但忽略了动态损耗，以及 （ii） 半经典理论，它很好地捕捉了动态耗散效应，但仅在低频下准确。在这里，我们开发了一个对任意频率有效的统一理论，它同时捕获了小信号量子行为以及弛豫和移相引入的非幺正效应。该理论允许用通用的小信号等效电路模型（谐振 RLC 电路）来描述多电平系统，其拓扑结构仅取决于能级的数量。我们将模型应用于双量子点电荷量子比特和 Majorana 量子比特，展示了从绝热到共振、从相干到非相干连续描述系统的能力，为改进量子态读出提出了新的和现实的实验。我们的模型将有助于混合量子-经典电路的设计以及量子比特控制和量子状态读出的仿真。