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The Journal of Chemical Physics

Single-reference coupled-cluster theory based on the multi-purpose cluster operator

Karol Kowalski, Nicholas P. Bauman

Published Jul 2, 2026• 0 citations
Publisher

Abstract

In this paper, we develop a theoretical framework that extends single-reference (SR) coupled-cluster (CC) theory beyond its conventional role of describing a single electronic state-typically the lowest-energy state within the symmetry sector defined by the reference determinant. Rather than viewing the SR-CC cluster operator solely as a device for reproducing one target state, we consider more general constructions in which different components of the cluster operator play distinct roles, ranging from encoding states of different symmetry than the reference to enabling the SR-CC ansatz to describe multiple states simultaneously. These developments lead to a new class of SR-CC downfolding formalisms in which the resulting active-space effective Hamiltonians are capable of concurrently representing multiple correlated states non-orthogonal to the reference function. We establish three theorems that formalize this extension and demonstrate that standard CC downfolding emerges as a special case of the proposed framework. Finally, we introduce a Hermitian variant based on a unitary CC representation, which enables realistic simulations of ground and excited states while reducing the quantum resources required.

DOI
10.1063/5.0330044

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In this paper, we develop a theoretical framework that extends single-reference (SR) coupled-cluster (CC) theory beyond its conventional role of describing a single electronic state-typically the lowest-energy state within the symmetry sector defined by the reference determinant. Rather than viewing the SR-CC cluster operator solely as a device for reproducing one target state, we consider more general constructions in which different components of the cluster operator play distinct roles, ranging from encoding states of different symmetry than the reference to enabling the SR-CC ansatz to describe multiple states simultaneously. These developments lead to a new class of SR-CC downfolding formalisms in which the resulting active-space effective Hamiltonians are capable of concurrently representing multiple correlated states non-orthogonal to the reference function. We establish three theorems that formalize this extension and demonstrate that standard CC downfolding emerges as a special case of the proposed framework. Finally, we introduce a Hermitian variant based on a unitary CC representation, which enables realistic simulations of ground and excited states while reducing the quantum resources required.
In this paper, we develop a theoretical framework that extends single-reference (SR) coupled-cluster (CC) theory beyond its conventional role of describing a single electronic state-typically the lowest-energy state within the symmetry sector defined by the reference determinant. Rather than viewing the SR-CC cluster operator solely as a device for reproducing one target state, we consider more general constructions in which different components of the cluster operator play distinct roles, ranging from encoding states of different symmetry than the reference to enabling SR-CC Ansatz to describe multiple states simultaneously. These developments lead to a new class of SR-CC downfolding formalisms in which the resulting active-space effective Hamiltonians are capable of concurrently representing multiple correlated states nonorthogonal to the reference function. We establish three theorems that formalize this extension and demonstrate that standard CC downfolding emerges as a special case of the proposed framework. Finally, we introduce a Hermitian variant based on a unitary CC representation, which enables realistic simulations of ground and excited states while reducing the quantum resources required.
In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. 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 (DUCC) formalism, 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 CAS 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 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. Given growing interest in quantum computing, we provide energies for H2 and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.
In this manuscript, we provide an overview of the recent developments of the coupled cluster (CC) downfolding methods, where the ground-state problem of a quantum system is represented through effective/downfolded Hamiltonians defined using active spaces. All CC downfolding techniques discussed here are derived from a single-reference exponential ansatz for the ground-state problem. We discuss several extensions of the non-Hermitian and Hermitian downfolding approaches to the time domain and the so-called quantum flows. We emphasize the important role of downfolding formalisms in transitioning chemical applications from noisy quantum devices to scalable and error-corrected quantum computers.
Abstract The recently introduced coupled cluster (CC) downfolding techniques for reducing the dimensionality of quantum many-body problems recast the CC formalism in the form of the renormalization procedure allowing, for the construction of effective (or downfolded) Hamiltonians in small-dimensionality sub-space, usually identified with the so-called active space, of the entire Hilbert space. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space. This paper will discuss the extension of non-Hermitian (associated with standard CC formulations) and Hermitian (associated with the unitary CC approaches) downfolding formulations to composite quantum systems commonly encountered in materials science and chemistry. The non-Hermitian formulation can provide a platform for developing local CC approaches, while the Hermitian one can serve as an ideal foundation for developing various quantum computing applications based on the limited quantum resources. We also discuss the algorithm for extracting the semi-analytical form of the inter-electron interactions in the active spaces.
The recently introduced coupled cluster (CC) downfolding techniques for reducing the dimensionality of quantum many-body problems recast the CC formalism in the form of the renormalization procedure allowing, for the construction of effective (or downfolded) Hamiltonians in small-dimensionality sub-space, usually identified with the so-called active space, of the entire Hilbert space. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space. This paper will discuss the extension of non-Hermitian (associated with standard CC formulations) and Hermitian (associated with the unitary CC approaches) downfolding formulations to composite quantum systems. The non-Hermitian formulation can provide a platform for developing local CC approaches, while the Hermitian one can serve as an ideal foundation for developing various quantum computing applications based on the limited quantum resources. We also discuss the algorithm for extracting the semi-analytical form of the inter-electron interactions in the active spaces.
This paper evaluates the accuracy of the Hermitian form of the downfolding procedure using the double unitary coupled cluster (DUCC) ansatz on the benchmark systems of linear chains of hydrogen atoms, H6 and H8. The computational infrastructure employs the occupation-number-representation codes to construct the matrix representation of arbitrary second-quantized operators, allowing for the exact representation of exponentials of various operators. The tests demonstrate that external amplitudes from standard single-reference coupled cluster methods that sufficiently describe external (out-of-active-space) correlations reliably parameterize the Hermitian downfolded effective Hamiltonians in the DUCC formalism. The results show that this approach can overcome the problems associated with losing the variational character of corresponding energies in the corresponding SR-CC theories.