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 squared N t) operations at fixed system dimension. In addition, we show that TCL 2n can be evaluated with O(N t n-1 N t) complexity. A fixed finite Matsubara expansion of the bath correlation function permits O(N t 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.