Quantum Embedding Theory Research Articles

Implementation of frozen density embedding in CP2K and OpenMolcas: CASSCF wavefunctions embedded in a Gaussian and plane wave DFT environment.

Most chemical processes happen at a local scale where only a subset of molecular orbitals is directly involved and only a subset of covalent bonds may be rearranged. To model such reactions, Density Functional Theory (DFT) is often inadequate, and the use of computationally more expensive correlated wavefunction (WF) methods is required for accurate results. Mixed-resolution approaches backed by quantum embedding theory have been used extensively to approach this imbalance. Based on the frozen density embedding freeze-and-thaw algorithm, we describe an approach to embed complete active space self-consistent field simulations run in the OpenMolcas code in a DFT environment calculated in CP2K without requiring any external tools. This makes it possible to study a local, active part of a chemical system in a larger and relatively static environment with a computational cost balanced between the accuracy of a WF method and the efficiency of DFT, which we test on environment-subsystem pairs. Finally, we apply the implementation to an oxygen molecule leaving an aluminum (111) surface and a ruthenium(IV) oxide (110) surface.

Accelerating Density Matrix Embedding with Stochastic Density Fitting Theory: An Application to Hydrogen Bonded Clusters.

In this work, we demonstrate how using semistochastic density fitting (ss-DF) can accelerate self-consistent density matrix embedding theory (DMET) calculations by reducing the number of auxiliary orbitals in the three-indexed DF integrals. This reduction results in significant time savings when building the Hartree-Fock (HF) Coulomb and Exchange Matrices and in transforming integrals from the atomic orbital (AO) basis to the embedding orbital (EO) basis. We apply ss-DF to a range of hydrogen-bonded clusters to showcase its effectiveness. First, we examine how the amount of deterministic space impacts the quality of the calculation in a (H2O)10 cluster. Next, we test the computational efficiency of ss-DF compared to deterministic DF (d-DF) in water clusters containing 6-30 water molecules using a triple-ζ basis set. Finally, we perform numerical structural optimizations on water and hydrogen fluoride clusters, revealing that DMET can recover weak interactions using a back-transformed energy formula. This work demonstrates the potential of using stochastic resolution of identity in quantum embedding theories and highlights its capability to recover weak interactions effectively.

Scalable semidefinite programming approach to variational embedding for quantum many-body problems

In quantum embedding theories, a quantum many-body system is divided into localized clusters of sites which are treated with an accurate ‘high-level’ theory and glued together self-consistently by a less accurate ‘low-level’ theory at the global scale. The recently introduced variational embedding approach for quantum many-body problems combines the insights of semidefinite relaxation and quantum embedding theory to provide a lower bound on the ground-state energy that improves as the cluster size is increased. The variational embedding method is formulated as a semidefinite program (SDP), which can suffer from poor computational scaling when treated with black-box solvers. We exploit the interpretation of this SDP as an embedding method to develop an algorithm which alternates parallelizable local updates of the high-level quantities with updates that enforce the low-level global constraints. Moreover, we show how translation invariance in lattice systems can be exploited to reduce the complexity of projecting a key matrix to the positive semidefinite cone.

Multi-scale QM/QM/MM molecular dynamics through quantum embedding.

Physical and chemical processes in bio-molecular systems typically span a wide range of time and space scales. These coupled processes at various scales must be taken into account concurrently in simulations depending on the problems of interest, yet they usually do not need to be depicted with the same accuracy. Large-scale protein conformational changes, for example, can be represented by carefully calibrated force fields, whereas chemical reactions at an enzyme's catalytic center necessitate a quantum mechanical approach. However, the number of atoms involved in the reaction center may still be quite large, limiting the applicability of some high-accuracy quantum chemistry methods, but the challenging part of the electronic structure may be even more local. A transition metal in enzymes, for example, may require some high-level multi-reference approaches, but a low-level mean-field description, such as the density functional theory, may be sufficient for metal ligands, serving as its environments. In this study, we used quantum embedding theory to rigorously partition a quantum system into multiple fragments, each of which could be solved using high-level quantum chemistry methods, allowing us to develop a multi-scale QM(high-level)/QM(low-level)/MM approach to achieve varying levels of accuracy in different scaled phenomenon.

Comparative study of adaptive variational quantum eigensolvers for multi-orbital impurity models

Hybrid quantum-classical embedding methods for correlated materials simulations provide a path towards potential quantum advantage. However, the required quantum resources arising from the multi-band nature of d and f electron materials remain largely unexplored. Here we compare the performance of different variational quantum eigensolvers in ground state preparation for interacting multi-orbital embedding impurity models, which is the computationally most demanding step in quantum embedding theories. Focusing on adaptive algorithms and models with 8 spin-orbitals, we show that state preparation with fidelities better than 99.9% can be achieved using about 214 shots per measurement circuit. When including gate noise, we observe that parameter optimizations can still be performed if the two-qubit gate error lies below 10−3, which is slightly smaller than current hardware levels. Finally, we measure the ground state energy on IBM and Quantinuum hardware using a converged adaptive ansatz and obtain a relative error of 0.7%.

Quantum embedding theories to simulate condensed systems on quantum computers.

Quantum computers hold promise to improve the efficiency of quantum simulations of materials and to enable the investigation of systems and properties that are more complex than tractable at present on classical architectures. Here, we discuss computational frameworks to carry out electronic structure calculations of solids on noisy intermediate-scale quantum computers using embedding theories, and we give examples for a specific class of materials, that is, solid materials hosting spin defects. These are promising systems to build future quantum technologies, such as quantum computers, quantum sensors and quantum communication devices. Although quantum simulations on quantum architectures are in their infancy, promising results for realistic systems appear to be within reach.

Simulating the Electronic Structure of Spin Defects on Quantum Computers

We present calculations of both the ground- and excited-state energies of spin defects in solids carried out on a quantum computer, using a hybrid classical-quantum protocol. We focus on the negatively charged nitrogen-vacancy center in diamond and on the double vacancy in 4H SiC, which are of interest for the realization of quantum technologies. We employ a recently developed first-principles quantum embedding theory to describe point defects embedded in a periodic crystal and to derive an effective Hamiltonian, which is then transformed to a qubit Hamiltonian by means of a parity transformation. We use the variational quantum eigensolver (VQE) and quantum subspace expansion methods to obtain the ground and excited states of spin qubits, respectively, and we propose a promising strategy for noise mitigation. We show that by combining zero-noise extrapolation techniques and constraints on electron occupation to overcome the unphysical-state problem of the VQE algorithm, one can obtain reasonably accurate results on near-term-noisy architectures for ground- and excited-state properties of spin defects.1 MoreReceived 7 December 2021Revised 27 January 2022Accepted 8 February 2022DOI:https://doi.org/10.1103/PRXQuantum.3.010339Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum computationQuantum simulationPhysical SystemsNitrogen vacancy centers in diamondQuantum InformationCondensed Matter, Materials & Applied Physics

Systematic Improvability in Quantum Embedding for Real Materials

Quantum embedding methods have become a powerful tool to overcome deficiencies of traditional quantum modelling in materials science. However, while these are systematically improvable in principle, in practice it is rarely possible to achieve rigorous convergence and often necessary to employ empirical parameters. Here, we formulate a quantum embedding theory, building on the methods of density-matrix embedding theory combined with local correlation approaches from quantum chemistry, to ensure the ability to systematically converge properties of real materials with accurate correlated wave~function methods, controlled by a single, rapidly convergent parameter. By expanding supercell size, basis set, and the resolution of the fluctuation space of an embedded fragment, we show that the systematic improvability of the approach yields accurate structural and electronic properties of realistic solids without empirical parameters, even across changes in geometry. Results are presented in insulating, semi-metallic, and more strongly correlated regimes, finding state of the art agreement to experimental data.

Extending density matrix embedding: A static two-particle theory

We introduce Extended Density Matrix Embedding Theory (EDMET), a static quantum embedding theory explicitly self-consistent with respect to local two-body physics. This overcomes the biggest practical and conceptual limitation of more traditional one-body embedding methods, namely the lack of screening and treatment of longer-range interactions. This algebraic zero-temperature embedding augments a local interacting cluster model with a minimal number of bosons from a description of the full system correlations via the random phase approximation, and admits an analytic approach to build a self-consistent Coulomb-exchange-correlation kernel. For extended Hubbard models with non-local interactions, this leads to the accurate description of phase transitions, static quantities and dynamics. We also move towards ab initio systems via the Parriser--Parr--Pople model of conjugated coronene derivatives, finding good agreement with experimental optical gaps.

Real Space Quantum Cluster Formulation for the Typical Medium Theory of Anderson Localization

We develop a real space cluster extension of the typical medium theory (cluster-TMT) to study Anderson localization. By construction, the cluster-TMT approach is formally equivalent to the real space cluster extension of the dynamical mean field theory. Applying the developed method to the 3D Anderson model with a box disorder distribution, we demonstrate that cluster-TMT successfully captures the localization phenomena in all disorder regimes. As a function of the cluster size, our method obtains the correct critical disorder strength for the Anderson localization in 3D, and systematically recovers the re-entrance behavior of the mobility edge. From a general perspective, our developed methodology offers the potential to study Anderson localization at surfaces within quantum embedding theory. This opens the door to studying the interplay between topology and Anderson localization from first principles.

First-Principles Predictions of Out-of-Plane Group IV and V Dimers as High-Symmetry, High-Spin Defects in Hexagonal Boron Nitride.

Hexagonal boron nitride (h-BN) has been recently found to host a variety of quantum point defects, which are promising candidates as single-photon sources for solid-state quantum nanophotonic applications. Most recently, optically addressable spin qubits in h-BN have been the focus of intensive research due to their unique potential in quantum computation, communication, and sensing. However, the number of high-symmetry, high-spin defects that are desirable for developing spin qubits in h-BN is highly limited. Here, we combine density functional theory (DFT) and quantum embedding theories to show that out-of-plane XNYi dimer defects (X, Y = C, N, P, and Si) form a new class of stable C3v spin-triplet defects in h-BN. We find that the dimer defects have a robust 3A2 ground state and 3E excited state, both of which are isolated from the h-BN bulk states. We show that 1E and 1A shelving states exist and they are positioned between the 3E and 3A2 states for all the dimer defects considered in this study. To support future experimental identification of the XNYi dimer defects, we provide extensive characterization of the defects in terms of their spin and optical properties. We predict that the zero-phonon line of the spin-triplet XNYi defects lies in the visible range (800 nm to 500 nm). We compute the zero-field splitting of the dimers' spin to range from 1.79 GHz (SiNPi0) to 29.5 GHz (CNNi0). Our results broaden the scope of high-spin defect candidates that would be useful for the development of spin-based solid-state quantum technologies in two-dimensional hexagonal boron nitride.

Quantum Embedding Theory for Strongly Correlated States in Materials.

Quantum embedding theories are promising approaches to investigate strongly correlated electronic states of active regions of large-scale molecular or condensed systems. Notable examples are spin defects in semiconductors and insulators. We present a detailed derivation of a quantum embedding theory recently introduced, which is based on the definition of effective Hamiltonians. The effect of the environment on a chosen active space is accounted for through screened Coulomb interactions evaluated using density functional theory. Importantly, the random phase approximation is not required, and the evaluation of virtual electronic orbitals is circumvented with algorithms previously developed in the context of calculations based on many-body perturbation theory. In addition, we generalize the quantum embedding theory to active spaces composed of orbitals that are not eigenstates of Kohn-Sham Hamiltonians. Finally, we report results for spin defects in semiconductors.

Variational Embedding for Quantum Many‐Body Problems

AbstractQuantum embedding theories are powerful tools for approximately solving large‐scale, strongly correlated quantum many‐body problems. The main idea of quantum embedding is to glue together a highly accurate quantum theory at the local scale and a less accurate quantum theory at the global scale. We introduce the first quantum embedding theory that is also variational, in that it is guaranteed to provide a one‐sided bound for the exact ground‐state energy. Our method, which we call the variational embedding method, provides a lower bound for this quantity. The method relaxes the representability conditions for quantum marginals to a set of linear and semidefinite constraints that operate at both local and global scales, resulting in a semidefinite program (SDP) to be solved numerically. The accuracy of the method can be systematically improved. The method is versatile and can be applied, in particular, to quantum many‐body problems for both quantum spin systems and fermionic systems, such as those arising from electronic structure calculations. We describe how the proper notion of quantum marginal, sufficiently general to accommodate both of these settings, should be phrased in terms of certain algebras of operators. We also investigate the duality theory for our SDPs, which offers valuable perspective on our method as an embedding theory. As a byproduct of this investigation, we describe a formulation for efficiently implementing the variational embedding method via a partial dualization procedure and the solution of quantum analogues of the Kantorovich problem from optimal transport theory. © 2021 Wiley Periodicals LLC.

Bypassing the computational bottleneck of quantum-embedding theories for strong electron correlations with machine learning

A cardinal obstacle to performing quantum-mechanical simulations of strongly-correlated matter is that, with the theoretical tools presently available, sufficiently-accurate computations are often too expensive to be ever feasible. Here we design a computational framework combining quantum-embedding (QE) methods with machine learning. This allows us to bypass altogether the most computationally-expensive components of QE algorithms, making their overall cost comparable to bare Density Functional Theory (DFT). We perform benchmark calculations of a series of actinide systems, where our method describes accurately the correlation effects, reducing by orders of magnitude the computational cost. We argue that, by producing a larger-scale set of training data, it will be possible to apply our method to systems with arbitrary stoichiometries and crystal structures, paving the way to virtually infinite applications in condensed matter physics, chemistry and materials science.

Sparse modeling of large-scale quantum impurity models with low symmetries

Quantum embedding theories can be used for obtaining quantitative descriptions of correlated materials. However, a critical challenge is solving an effective impurity model of correlated orbitals embedded in an electron bath. Many advanced impurity solvers require the approximation of a bath continuum using a finite number of bath levels, producing a highly nonconvex, ill-conditioned inverse problem. To address this drawback, this study proposes an efficient fitting algorithm for matrix-valued hybridization functions based on a data-science approach, sparse modeling, and a compact representation of Matsubara Green's functions. The efficiency of the proposed method is demonstrated by fitting random hybridization functions with large off-diagonal elements and those of a 20-orbital impurity model for a high-${T}_{\mathrm{c}}$ compound, LaAsFeO, at low temperatures $(T)$. The results set quantitative goals for the future development of impurity solvers toward quantum embedding simulations of complex correlated materials.

Quantum simulations of materials on near-term quantum computers

Quantum computers hold promise to enable efficient simulations of the properties of molecules and materials; however, at present they only permit ab initio calculations of a few atoms, due to a limited number of qubits. In order to harness the power of near-term quantum computers for simulations of larger systems, it is desirable to develop hybrid quantum-classical methods where the quantum computation is restricted to a small portion of the system. This is of particular relevance for molecules and solids where an active region requires a higher level of theoretical accuracy than its environment. Here, we present a quantum embedding theory for the calculation of strongly-correlated electronic states of active regions, with the rest of the system described within density functional theory. We demonstrate the accuracy and effectiveness of the approach by investigating several defect quantum bits in semiconductors that are of great interest for quantum information technologies. We perform calculations on quantum computers and show that they yield results in agreement with those obtained with exact diagonalization on classical architectures, paving the way to simulations of realistic materials on near-term quantum computers.

Bootstrap Embedding For Large Molecular Systems.

Recent developments in quantum embedding theories have provided attractive approaches to correlated calculations for large systems. In this work, we extend our previous work [J. Chem. Theory Comput. 2019, 15, 4497-4506; J. Phys. Chem. Lett. 2019, 10, 6368-6374] on bootstrap embedding (BE) to enable correlated ab initio calculations at the coupled cluster with singles and doubles (CCSD) level for large molecules. We introduce several new algorithmic developments that significantly reduce the computational cost of BE, while maintaining its accuracy. The resulting implementation scales as O(N3) for the integral transform and O(N) for the CCSD calculation. Numerical results on a series of conjugated molecules suggest that BE with reasonably sized fragments can recover more than 99.5% of the total correlation energy of a full CCSD calculation, while the required computational resources (time and storage) compare favorably to one popular local correlation scheme: domain localized pair natural orbital (DLPNO). The largest BE calculation in this work involves ∼2900 basis functions and can be performed on a single node with 16 CPU cores and 64 GB of memory in a few days. We anticipate that these developments represent an important step toward the application of BE to solve practical problems.

First-principles studies of strongly correlated states in defect spin qubits in diamond.

Using a recently developed quantum embedding theory, we present first-principles calculations of strongly correlated states of spin defects in diamond. Using this theory, effective Hamiltonians are constructed, which can be solved by classical and quantum computers; the latter promise a much more favorable scaling as a function of system size than the former. In particular, we report a study on the neutral group-IV vacancy complexes in diamond, and we discuss their strongly correlated spin-singlet and spin-triplet excited states. Our results provide valuable predictions for experiments aimed at optical manipulation of these defects for quantum information technology applications.

Projected density matrix embedding theory with applications to the two-dimensional Hubbard model

Density matrix embedding theory (DMET) is a quantum embedding theory for strongly correlated systems. From a computational perspective, one bottleneck in DMET is the optimization of the correlation potential to achieve self-consistency, especially for heterogeneous systems of large size. We propose a new method, called projected DMET (p-DMET), which achieves self-consistency without needing to optimize the correlation potential. We demonstrate the performance of p-DMET on the two-dimensional Hubbard model.

Quantum embedding theory in the screened Coulomb interaction: Combining configuration interaction withGW/BSE

We present a quantum embedding theory called dynamical configuration interaction (DCI) that combines wave function and Green's function theories. DCI captures static correlation in a correlated subspace with configuration interaction and couples to high-energy, dynamic correlation outside the subspace with many-body perturbation theory based on Green's functions. DCI takes the strengths of both theories to balance static and dynamic correlation in a single, fully ab initio embedding concept. The theory adds dynamic correlation around a fixed active space of orbitals with efficient $\mathcal{O}({N}^{5})$ scaling, while maintaining a multireference treatment of the active space. We show that treating high-energy correlation up to the $GW$ and Bethe-Salpeter equation level is sufficient even for challenging multireference problems. Our theory treats ground and excited states on equal footing, and we compute the dissociation curve of ${\mathrm{N}}_{2}$, the vertical excitation energies of small molecules, and the ionization spectrum of benzene in excellent agreement with high-level quantum chemistry methods and experiment.