Fast Evaluation of the Sixth-Order Time-Convolutionless Master-Equation Generator and Beyond

Direct quadrature of the Hadamard-reduced sixth-order time-convolutionless (TCL6) generator at all $N_t$ sampled times requires $O(N_t^3)$ operations at fixed system dimension. We derive an exact reduction for a finite-dimensional system coupled through one Hermitian operator to a stationary centered Gaussian bath. By separating fixed operator coefficients from scalar bath kernels, the complete TCL6 time series is reduced to cumulative sums and first-order recurrences, one-dimensional causal convolutions, and an exact dyadic recursion for interlocked histories. With the algorithm, evaluating the complete TCL6 time series requires $O(N_t{\log }^2{N}_t)$ operations at fixed system dimension. In addition, we show that TCL$2n$ can be evaluated with $O(N_t{\log }^{n-1}{N}_t)$ complexity. A fixed finite Matsubara expansion of the bath correlation function permits $O(N_t\log N_t)$ evaluation at any fixed TCL order. These reductions enable fast long-time simulations of non-Markovian open quantum systems within the regime of validity of the TCL expansion.