# Downfolding of many-body Hamiltonians using active-space models: extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms

## TL;DR

This paper extends the SES-CC formalism to unitary CC methods, enabling efficient downfolding of many-electron Hamiltonians into low-energy models with Hermitian effective Hamiltonians suitable for quantum algorithms.

## Contribution

It introduces a unitary CC extension of the SES-CC formalism, providing a Hermitian effective Hamiltonian for active space models and separating dynamical from internal correlations.

## Key findings

- Hermitian effective Hamiltonians suitable for FCI solvers
- Separation of external and internal cluster amplitudes
- Potential for hybrid classical-quantum algorithms

## Abstract

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.