  The combination of chain-mapping and tensor-network techniques provides a
powerful tool for the numerically exact simulation of open quantum systems
interacting with structured environments. However, these methods suffer from a
quadratic scaling with the physical simulation time, and therefore they become
challenging in the presence of multiple environments. This is particularly true
when fermionic environments, well-known to be highly correlated, are
considered. In this work we first illustrate how a thermo-chemical modulation
of the spectral density allows replacing the original fermionic environments
with equivalent, but simpler, ones. Moreover, we show how this procedure
reduces the number of chains needed to model multiple environments. We then
provide a derivation of the fermionic Markovian closure construction,
consisting of a small collection of damped fermionic modes undergoing a
Lindblad-type dynamics and mimicking a continuum of bath modes. We describe, in
particular, how the use of the Markovian closure allows for a polynomial
reduction of the time complexity of chain-mapping based algorithms when
long-time dynamics are needed.
