  In this study, we introduce a novel approach to coupled-cluster Green's
function (CCGF) embedding by seamlessly integrating conventional CCGF theory
with the state-of-the-art sub-system embedding sub-algebras coupled cluster
(SES-CC) formalism. This integration focuses primarily on delineating the
characteristics of the sub-system and the corresponding segments of the Green's
function, defined explicitly by active orbitals. Crucially, our work involves
the adaptation of the SES-CC paradigm, addressing the left eigenvalue problem
through a distinct form of Hamiltonian similarity transformation. This
advancement not only facilitates a comprehensive representation of the
interaction between the embedded sub-system and its surrounding environment but
also paves the way for the quantum mechanical description of multiple embedded
domains, particularly by employing the emergent quantum flow algorithms. Our
theoretical underpinnings further set the stage for a generalization to
multiple embedded sub-systems. This expansion holds significant promise for the
exploration and application of non-equilibrium quantum systems, enhancing the
understanding of system-environment interactions. In doing so, the research
underscores the potential of SES-CC embedding within the realm of quantum
computations and multi-scale simulations, promising a good balance between
accuracy and computational efficiency.
