Markovian master equations beyond the Lindblad master equation

The Lindblad master equation is the standard description of dissipation in the presence of Markovian noise. However, for simulations of quantum gates, phenomenological master equations are frequently used for numerical and analytical simplicity, despite debates about their validity. The requirement for increasingly accurate gate simulations necessitates a better understanding of these equations. We present a framework to derive phenomenological master equations that describe dissipation in three key types of systems: weakly-anharmonic systems (e.g., transmons), dispersively coupled systems (e.g., cavity-qubit setups), and transversely coupled systems (e.g., the Rabi model). Through this framework, we assess the approximations involved in deriving these equations and compute higher-order corrections to enhance the accuracy of these equations, showing agreement with previous studies while extending their scope.