  The hierarchical equations of motion (HEOM), derived from the exact
Feynman-Vernon path integral, is one of the most powerful numerical methods to
simulate the dynamics of open quantum systems that are embedded in thermal
environments. However, its applicability is restricted to specific forms of
spectral reservoir distributions and relatively elevated temperatures. Here we
solve this problem and introduce an effective treatment of quantum noise in
frequency space by systematically clustering higher order Matsubara poles
equivalent to an optimized rational decomposition. This leads to an elegant
extension of the HEOM to arbitrary temperatures and very general reservoirs in
combination with efficiency, high accuracy and long-time stability. Moreover,
the technique can directly be implemented in alternative approaches such as
Green's function, stochastic, and pseudo-mode formulations. As one highly
non-trivial application, for the sub-ohmic spin-boson model at vanishing
temperature the Shiba relation is quantitatively verified which predicts the
long-time decay of correlation functions.
