Benchmarking TCL4: Assessing the usability and reliability of fourth-order approximations

The non-Markovian dynamics of an open quantum system can be rigorously derived using the Feynman–Vernon influence functional approach. Although this formalism is exact, practical numerical implementations often require compromises. The time-convolutionless (TCL) master equation offers an exact framework, yet its application typically relies on a perturbative expansion of both the time forward and time backward state propagators. Due to the significant computational effort involved—and the scarcity of analytical solutions for most open quantum systems—the fourth-order perturbative TCL generator (TCL4) has only been benchmarked on a limited range of systems and parameter spaces. Recent advancements, however, have made the computation of TCL4 faster and more accessible. In this paper, we benchmark the TCL4 master equation against numerically exact methods for the biased spin-boson model. We focus on the regime near critical bath coupling where perturbative master equations are expected to become inaccurate. Our findings reveal that the TCL4 approach is most reliable at low temperatures and more efficient than the numerical exact methods. This study aims to delineate the conditions under which the TCL4 perturbative master equation enhances the accuracy of the TCL2.