Properties of coupled-cluster equations originating in excitation sub-algebras

The accuracies of effective interactions in downfolding coupled-cluster approaches for small-dimensionality 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.

Projective quantum eigensolver via adiabatically decoupled subsystem evolution: A resource efficient approach to molecular energetics in noisy quantum computers

Quantum computers hold immense potential in the field of chemistry, ushering new frontiers to solve complex many-body problems that are beyond the reach of classical computers. However, noise in the current quantum hardware limits their applicability to large chemical systems. This work encompasses the development of a projective formalism that aims to compute ground-state energies of molecular systems accurately using noisy intermediate scale quantum (NISQ) hardware in a resource-efficient manner. Our approach is reliant upon the formulation of a bipartitely decoupled parameterized ansatz within the disentangled unitary coupled cluster framework based on the principles of nonlinear dynamics and synergetics. Such decoupling emulates total parameter optimization in a lower dimensional manifold, while a mutual synergistic relationship among the parameters is exploited to ensure characteristic accuracy via a non-iterative energy correction. Without any pre-circuit measurements, our method leads to a highly compact fixed-depth ansatz with shallower circuits and fewer expectation value evaluations. Through analytical and numerical demonstrations, we establish the method's superior performance under noise while concurrently ensuring requisite accuracy in future fault-tolerant systems. This approach enables rapid exploration of emerging chemical spaces by the efficient utilization of near-term quantum hardware resources.

Quantum Flow Algorithms for Simulating Many-Body Systems on Quantum Computers

We conducted quantum simulations of strongly correlated systems using the quantum flow (QFlow) approach, which enables sampling large subspaces of the Hilbert space through coupled variational problems in reduced dimensionality active spaces. Our QFlow algorithms significantly reduce circuit complexity and pave the way for scalable and constant-circuit-depth quantum computing. Our simulations show that QFlow can optimize the collective number of wave function parameters without increasing the required qubits using active spaces having an order of magnitude fewer number of parameters.

Exploring Parameter Redundancy in the Unitary Coupled-Cluster Ansätze for Hybrid Variational Quantum Computing

One of the commonly used chemically inspired approaches in variational quantum computing is the unitary coupled-cluster (UCC) ansätze. Despite being a systematic way of approaching the exact limit, the number of parameters in the standard UCC ansätze exhibits unfavorable scaling with respect to the system size, hindering its practical use on near-term quantum devices. Efforts have been taken to propose some variants of the UCC ansätze with better scaling. In this paper, we explore the parameter redundancy in the preparation of unitary coupled-cluster singles and doubles (UCCSD) ansätze employing spin-adapted formulation, small amplitude filtration, and entropy-based orbital selection approaches. Numerical results of using our approach on some small molecules have exhibited a significant cost reduction in the number of parameters to be optimized and in the time to convergence compared with conventional UCCSD-VQE simulations. We also discuss the potential application of some machine learning techniques in further exploring the parameter redundancy, providing a possible direction for future studies.

Sub-system self-consistency in coupled cluster theory

In this article, we provide numerical evidence indicating that the single-reference coupled-cluster (CC) energies can be calculated alternatively to their copybook definition. We demonstrate that the CC energy can be reconstructed by diagonalizing the effective Hamiltonians describing correlated sub-systems of the many-body system. In the extreme case, we provide numerical evidence that the CC energy can be reproduced through the diagonalization of the effective Hamiltonian describing sub-system composed of a single electron. These properties of the CC formalism can be exploited to design protocols to define effective interactions in sub-systems used as probes to calculate the energy of the entire system and introduce a new type of self-consistency for approximate CC approaches.

Projection-Based Density Matrix Renormalization Group in Density Functional Theory Embedding

The density matrix renormalization group (DMRG) method has already proved itself as a very efficient and accurate computational method, which can treat large active spaces and capture the major part of strong correlation. Its application on larger molecules is, however, limited by its own computational scaling as well as demands of methods for treatment of the missing dynamical electron correlation. In this work, we present the first step in the direction of combining DMRG with density functional theory (DFT), one of the most employed quantum chemical methods with favorable scaling, by means of the projection-based wave function (WF)-in-DFT embedding. On two proof-of-concept but important molecular examples, we demonstrate that the developed DMRG-in-DFT approach provides a very accurate description of molecules with a strongly correlated fragment.

A Stabilizer Framework for the Contextual Subspace Variational Quantum Eigensolver and the Noncontextual Projection Ansatz

Quantum chemistry is a promising application for noisy intermediate-scale quantum (NISQ) devices. However, quantum computers have thus far not succeeded in providing solutions to problems of real scientific significance, with algorithmic advances being necessary to fully utilize even the modest NISQ machines available today. We discuss a method of ground state energy estimation predicated on a partitioning of the molecular Hamiltonian into two parts: one that is noncontextual and can be solved classically, supplemented by a contextual component that yields quantum corrections obtained via a Variational Quantum Eigensolver (VQE) routine. This approach has been termed Contextual Subspace VQE (CS-VQE); however, there are obstacles to overcome before it can be deployed on NISQ devices. The problem we address here is that of the ansatz, a parametrized quantum state over which we optimize during VQE; it is not initially clear how a splitting of the Hamiltonian should be reflected in the CS-VQE ansätze. We propose a "noncontextual projection" approach that is illuminated by a reformulation of CS-VQE in the stabilizer formalism. This defines an ansatz restriction from the full electronic structure problem to the contextual subspace and facilitates an implementation of CS-VQE that may be deployed on NISQ devices. We validate the noncontextual projection ansatz using a quantum simulator and demonstrate chemically precise ground state energy calculations for a suite of small molecules at a significant reduction in the required qubit count and circuit depth.

Efficient Adiabatic Connection Approach for Strongly Correlated Systems: Application to Singlet-Triplet Gaps of Biradicals

Strong electron correlation can be captured with multireference wave function methods, but an accurate description of the electronic structure requires accounting for the dynamic correlation, which they miss. In this work, a new approach for the correlation energy based on the adiabatic connection (AC) is proposed. The AC_n method accounts for terms up to order n in the coupling constant, and it is size-consistent and free from instabilities. It employs the multireference random phase approximation and the Cholesky decomposition technique, leading to a computational cost growing with the fifth power of the system size. Because of the dependence on only one- and two-electron reduced density matrices, AC_n is more efficient than existing ab initio multireference dynamic correlation methods. AC_n affords excellent results for singlet-triplet gaps of challenging organic biradicals. The development presented in this work opens new perspectives for accurate calculations of systems with dozens of strongly correlated electrons.

A quantum computing view on unitary coupled cluster theory

We present a review of the Unitary Coupled Cluster (UCC) ansatz and related ansätze which are used to variationally solve the electronic structure problem on quantum computers. A brief history of coupled cluster (CC) methods is provided, followed by a broad discussion of the formulation of CC theory. This includes touching on the merits and difficulties of the method and several variants, UCC among them, in the classical context, to motivate their applications on quantum computers. In the core of the text, the UCC ansatz and its implementation on a quantum computer are discussed at length, in addition to a discussion on several derived and related ansätze specific to quantum computing. The review concludes with a unified perspective on the discussed ansätze, attempting to bring them under a common framework, as well as with a reflection upon open problems within the field.

An approximate coupled cluster theory via nonlinear dynamics and synergetics: The adiabatic decoupling conditions

The coupled cluster iteration scheme is analyzed as a multivariate discrete time map using nonlinear dynamics and synergetics. The nonlinearly coupled set of equations to determine the cluster amplitudes are driven by a fraction of the entire set of cluster amplitudes. These driver amplitudes enslave all other amplitudes through a synergistic inter-relationship, where the latter class of amplitudes behave as the auxiliary variables. The driver and the auxiliary variables exhibit vastly different time scales of relaxation during the iteration process to reach the fixed points. The fast varying auxiliary amplitudes are small in magnitude, while the driver amplitudes are large, and they have a much longer time scale of relaxation. Exploiting their difference in relaxation time scale, we employ an adiabatic decoupling approximation, where each of the fast relaxing auxiliary modes is expressed as a unique function of the principal amplitudes. This results in a tremendous reduction in the independent degrees of freedom. On the other hand, only the driver amplitudes are determined accurately via exact coupled cluster equations. We will demonstrate that the iteration scheme has an order of magnitude reduction in computational scaling than the conventional scheme. With a few pilot numerical examples, we would demonstrate that this scheme can achieve very high accuracy with significant savings in computational time.

Tailored coupled cluster theory in varying correlation regimes

The tailored coupled cluster (TCC) approach is a promising ansatz that preserves the simplicity of single-reference coupled cluster theory while incorporating a multi-reference wave function through amplitudes obtained from a preceding multi-configurational calculation. Here, we present a detailed analysis of the TCC wave function based on model systems, which require an accurate description of both static and dynamic correlation. We investigate the reliability of the TCC approach with respect to the exact wave function. In addition to the error in the electronic energy and standard coupled cluster diagnostics, we exploit the overlap of TCC and full configuration interaction wave functions as a quality measure. We critically review issues, such as the required size of the active space, size-consistency, symmetry breaking in the wave function, and the dependence of TCC on the reference wave function. We observe that possible errors caused by symmetry breaking can be mitigated by employing the determinant with the largest weight in the active space as reference for the TCC calculation. We find the TCC model to be promising in calculations with active orbital spaces which include all orbitals with a large single-orbital entropy, even if the active spaces become very large and then may require modern active-space approaches that are not restricted to comparatively small numbers of orbitals. Furthermore, utilizing large active spaces can improve on the TCC wave function approximation and reduce the size-consistency error because the presence of highly excited determinants affects the accuracy of the coefficients of low-excited determinants in the active space.

Resource-Efficient Chemistry on Quantum Computers with the Variational Quantum Eigensolver and the Double Unitary Coupled-Cluster Approach

Applications of quantum simulation algorithms to obtain electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices require careful consideration of resources describing the complex electron correlation effects. In modeling second-quantized problems, the biggest challenge confronted is that the number of qubits scales linearly with the size of the molecular basis. This poses a significant limitation on the size of the basis sets and the number of correlated electrons included in quantum simulations of chemical processes. To address this issue and enable more realistic simulations on NISQ computers, we employ the double unitary coupled-cluster (DUCC) method to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space. Using downfolding techniques, we demonstrate that properly constructed effective Hamiltonians can capture the effect of the whole orbital space in small-size active spaces. Combining the downfolding preprocessing technique with the variational quantum eigensolver, we solve for the ground-state energy of H₂, Li₂, and BeH₂ in the cc-pVTZ basis using the DUCC-reduced active spaces. We compare these results to full configuration-interaction and high-level coupled-cluster reference calculations.

Near-Linear Scaling in DMRG-Based Tailored Coupled Clusters: An Implementation of DLPNO-TCCSD and DLPNO-TCCSD(T)

We present a new implementation of density matrix renormalization group based tailored coupled clusters method (TCCSD), which employs the domain-based local pair natural orbital approach (DLPNO). Compared to the previous local pair natural orbital (LPNO) version of the method, the new implementation is more accurate, offers more favorable scaling, and provides more consistent behavior across the variety of systems. On top of the singles and doubles, we include the perturbative triples correction (T), which is able to retrieve even more dynamic correlation. The methods were tested on three systems: tetramethyleneethane, oxo-Mn(Salen), and iron(II)-porphyrin model. The first two were revisited to assess the performance with respect to LPNO-TCCSD. For oxo-Mn(Salen), we retrieved between 99.8 and 99.9% of the total canonical correlation energy which is an improvement of 0.2% over the LPNO version in less than 63% of the total LPNO runtime. Similar results were obtained for iron(II)-porphyrin. When the perturbative triples correction was employed, irrespective of the active space size or system, the obtained energy differences between two spin states were within the chemical accuracy of 1 kcal/mol using the default DLPNO settings.

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

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.