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

Abstract: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.

Quantum Physics (quant-ph); Computational Physics (physics.comp-ph)

arXiv:2609.18806 [quant-ph]

(or arXiv:2609.18806v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.18806 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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