  In this paper, we discuss extending the sub-system embedding sub-algebra
coupled cluster (SESCC) formalism and the double unitary coupled cluster (DUCC)
Ansatz to the time domain. An important part of the analysis is associated with
proving the exactness of the DUCC Ansatz based on the general many-body form of
anti-Hermitian cluster operators defining external and internal excitations.
Using these formalisms, it is possible to calculate the energy of the entire
system as an eigenvalue of downfolded/effective Hamiltonian in the active
space, that is identifiable with the sub-system of the composite system. It can
also be shown that downfolded Hamiltonians integrate out Fermionic degrees of
freedom that do not correspond to the physics encapsulated by the active space.
In this paper, we extend these results to the time-dependent Schroedinger
equation, showing that a similar construct is possible to partition a system
into a sub-system that varies slowly in time and a remaining sub-system that
corresponds to fast oscillations. This time-dependent formalism allows coupled
cluster quantum dynamics to be extended to larger systems and for the
formulation of novel quantum algorithms based on the quantum Lanczos approach,
which has recently been considered in the literature.
