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arXiv:2003.09566 · The Journal of Chemical Physics

Sub-system quantum dynamics using coupled cluster downfolding techniques

Karol Kowalski, Nicholas P. Bauman

Published Jun 28, 2020• 37 citations
arXivHTMLPDF

Abstract

In this paper, we discuss extending the sub-system embedding sub-algebra coupled cluster formalism and the double unitary coupled cluster (DUCC) ansatz to the time domain. An important part of the analysis is associated with proving the exactness of the DUCC ansatz based on the general many-body form of anti-Hermitian cluster operators defining external and internal excitations. Using these formalisms, it is possible to calculate the energy of the entire system as an eigenvalue of downfolded/effective Hamiltonian in the active space, which is identifiable with the sub-system of the composite system. It can also be shown that downfolded Hamiltonians integrate out Fermionic degrees of freedom that do not correspond to the physics encapsulated by the active space. In this paper, we extend these results to the time-dependent Schrödinger equation, showing that a similar construct is possible to partition a system into a sub-system that varies slowly in time and a remaining sub-system that corresponds to fast oscillations. This time-dependent formalism allows coupled cluster quantum dynamics to be extended to larger systems and for the formulation of novel quantum algorithms based on the quantum Lanczos approach, which has recently been considered in the literature.

DOI
10.1063/5.0008436

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In this paper, we discuss extending the sub-system embedding sub-algebra coupled cluster formalism and the double unitary coupled cluster (DUCC) ansatz to the time domain. An important part of the analysis is associated with proving the exactness of the DUCC ansatz based on the general many-body form of anti-Hermitian cluster operators defining external and internal excitations. Using these formalisms, it is possible to calculate the energy of the entire system as an eigenvalue of downfolded/effective Hamiltonian in the active space, which is identifiable with the sub-system of the composite system. It can also be shown that downfolded Hamiltonians integrate out Fermionic degrees of freedom that do not correspond to the physics encapsulated by the active space. In this paper, we extend these results to the time-dependent Schrödinger equation, showing that a similar construct is possible to partition a system into a sub-system that varies slowly in time and a remaining sub-system that corresponds to fast oscillations. This time-dependent formalism allows coupled cluster quantum dynamics to be extended to larger systems and for the formulation of novel quantum algorithms based on the quantum Lanczos approach, which has recently been considered in the literature.
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.
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.
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.
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.
We discuss reduced-scaling strategies employing the recently introduced subsystem embedding subalgebra (SES) coupled-cluster (CC) formalism to describe quantum many-body systems. These strategies utilize properties of the SES CC formulations where the equations describing certain classes of subsystems can be integrated into computational flows composed of coupled eigenvalue problems of reduced dimensionality. Additionally, these flows can be determined at the level of the CC ansatz by the inclusion of selected classes of cluster amplitudes, which define the wave-function ``memory'' of possible partitioning of the many-body system into constituent subsystems. One of the possible ways of solving these coupled problems is through implementing procedures where the information is passed between the subsystems in a self-consistent manner. As a special case we consider local flow formulations where the local character of correlation effects can be closely related to the properties of subsystem embedding subalgebras employing a localized molecular basis. We also generalize flow equations to the time domain and to downfolding methods utilizing a double-exponential unitary CC ansatz, where the reduced dimensionality of constituent subproblems offers a possibility of efficient utilization of limited quantum resources in modeling realistic systems.
Many-body techniques based on the double unitary coupled cluster (DUCC) ansatz can be used to downfold electronic Hamiltonians into low-dimensional active spaces. It can be shown that the resulting dimensionality reduced Hamiltonians are amenable for quantum computing. Recent studies performed for several benchmark systems using phase estimation (PE) algorithms for quantum computers demonstrated that these formulations can recover a significant portion of ground-state dynamical correlation effects that stem from the electron excitations outside of the active space. These results have also been confirmed in studies of ground-state potential energy surfaces using quantum simulators. In this letter, we study the effectiveness of the DUCC formalism in describing excited states. We also emphasize the role of the PE formalism and its stochastic nature in discovering/identifying excited states or excited-state processes in situations when the knowledge about the true configurational structure of a sought after excited state is limited or postulated (due to the specific physics driving excited-state processes of interest). In this context, we can view PE algorithms as an engine for verifying various hypotheses for excited-state processes and providing statistically meaningful results that correspond to the electronic state(s) with the largest overlap with a postulated configurational structure. We illustrate these ideas on examples of strongly correlated molecular systems, characterized by small energy gaps and high density of quasidegenerate states around the Fermi level.