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In this study, we introduce a novel approach to coupled-cluster Green'sfunction (CCGF) embedding by seamlessly integrating conventional CCGF theorywith the state-of-the-art sub-system embedding sub-algebras coupled cluster(SES-CC) formalism. This integration focuses primarily on delineating thecharacteristics of the sub-system and the corresponding segments of the Green'sfunction, defined explicitly by active orbitals. Crucially, our work involvesthe adaptation of the SES-CC paradigm, addressing the left eigenvalue problemthrough a distinct form of Hamiltonian similarity transformation. Thisadvancement not only facilitates a comprehensive representation of theinteraction between the embedded sub-system and its surrounding environment butalso paves the way for the quantum mechanical description of multiple embeddeddomains, particularly by employing the emergent quantum flow algorithms. Ourtheoretical underpinnings further set the stage for a generalization tomultiple embedded sub-systems. This expansion holds significant promise for theexploration and application of non-equilibrium quantum systems, enhancing theunderstanding of system-environment interactions. In doing so, the researchunderscores the potential of SES-CC embedding within the realm of quantumcomputations and multi-scale simulations, promising a good balance betweenaccuracy and computational efficiency.

Integrating Subsystem Embedding Subalgebras and Coupled Cluster Green's  Function: A Theoretical Foundation for Quantum Embedding in Excitation  Manifold