  In this paper we outline the extension of recently introduced the sub-system
embedding sub-algebras coupled cluster (SES-CC) formalism to the unitary CC
formalism. In analogy to the standard single-reference SES-CC formalism, its
unitary CC extension allows one to include the dynamical (outside the active
space) correlation effects in an SES induced complete active space (CAS)
effective Hamiltonian. In contrast to the standard single-reference SES-CC
theory, the unitary CC approach results in a Hermitian form of the effective
Hamiltonian. Additionally, for the double unitary CC formalis (DUCC) the
corresponding CAS eigenvalue problem provides a rigorous separation of external
cluster amplitudes that describe dynamical correlation effects - used to define
the effective Hamiltonian - from those corresponding to the internal (inside
the active space) excitations that define the components of eigenvectors
associated with the energy of the entire system. The proposed formalism can be
viewed as an efficient way of downfolding many-electron Hamiltonian to the
low-energy model represented by a particular choice of CAS. In principle, this
technique can be extended to any type of complete active space representing an
arbitrary energy window of a quantum system. The Hermitian character of
low-dimensional effective Hamiltonians makes them an ideal target for several
types of full configuration interaction (FCI) type eigensolvers. As an example,
we also discuss the algebraic form of the perturbative expansions of the
effective DUCC Hamiltonians corresponding to composite unitary CC theories and
discuss possible algorithms for hybrid classical and quantum computing.
