Abstract
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 nearterm quantum computers for simulations of larger systems, it is desirable to develop hybrid quantumclassical 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 stronglycorrelated 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 nearterm quantum computers.
Introduction
In the last three decades, atomistic simulations based on the solution of the basic equation of quantum mechanics have played an increasingly important role in predicting the properties of functional materials, encompassing catalysts and energy storage systems for energy applications, and materials for quantum information science. Especially in the case of complex, heterogeneous materials, the great majority of firstprinciples simulations are conducted using density functional theory (DFT), which is in principle exact but in practice requires approximations to enable calculations. Within its various approximations, DFT has been extremely successful in predicting numerous properties of solids, liquids, and molecules, and in providing key interpretations to a variety of experimental results; however it is often inadequate to describe socalled stronglycorrelated electronic states^{1,2}. We will use here the intuitive notion of strong correlation as pertaining to electronic states that cannot be described by static meanfield theories. Several theoretical and computational methods have been developed over the years to treat systems exhibiting stronglycorrelated electronic states, including dynamical meanfield theory^{3,4} and quantum MonteCarlo^{5,6}; in addition, ab initio quantum chemistry methods, traditionally developed for molecules, have been recently applied to solid state problems as well^{7}. Unfortunately, these approaches are computationally demanding and it is still challenging to apply them to complex materials containing defects and interfaces, even using highperformance computing architectures.
Quantum computers hold promise to enable efficient quantum mechanical simulations of weakly and stronglycorrelated molecules and materials alike^{8,9,10,11,12,13,14,15,16,17}; in particular when using quantum computers, one is able to simulate systems of interacting electrons exponentially faster than using classical computers. Thanks to decades of successful experimental efforts, we are now entering the noisy intermediatescale quantum (NISQ) era^{18}, with quantum computers expected to have on the order of 100 quantum bits (qubits); unfortunately this limited number of qubits still prevents straightforward quantum simulations of realistic molecules and materials, whose description requires hundreds of atoms and thousands to millions of degrees of freedom to represent the electronic wavefunctions. An important requirement to tackle complex chemistry and material science problems using NISQ computers is the reduction of the number of electrons treated explicitly at the highest level of accuracy^{19,20}. For instance, building on the idea underpinning dynamical mean field theory (DMFT)^{3,4}, one may simplify complex molecular and material science problems by defining active regions (or building blocks) with stronglycorrelated electronic states, embedded in an environment that may be described within meanfield theory^{21,22,23}.
In this work, we present a quantum embedding theory built on DFT, which is scalable to large systems and which includes the effect of exchangecorrelation interactions of the environment on active regions, thus going beyond commonly adopted approximations. In order to demonstrate the effectiveness and accuracy of the theory, we compute ground and excited state properties of several spindefects in solids including the negatively charged nitrogenvacancy (NV) center^{24,25,26,27,28,29,30}, the neutral siliconvacancy (SiV) center^{31,32,33,34,35,36} in diamond, and the Cr impurity (4+) in 4HSiC^{37,38,39}. These spindefects are promising platforms for solidstate quantum information technologies, and they exhibit stronglycorrelated electronic states that are critical for the initialization and readout of their spin states^{40,41,42,43,44,45}. Our quantum embedding theory yields results in good agreement with existing measurements. In addition, we present theoretical predictions for the position and ordering of the singlet states of SiV and of Cr, and we provide an interpretation of experiments that have so far remained unexplained.
Importantly, we report calculations of spindefects using a quantum computer^{46,47}. Based on the effective Hamiltonian derived from the quantum embedding theory, we investigated the stronglycorrelated electronic states of the NV center in diamond using quantum phase estimation algorithm (PEA)^{8,48} and variational quantum eigensolvers (VQE)^{49,50,51}, and we show that quantum simulations yield results in agreement with those obtained with classical full configuration interaction (FCI) calculations. Our findings pave the way to the use of near term quantum computers to investigate the properties of realistic heterogeneous materials with firstprinciples theories.
Results
General strategy
We summarize our strategy in Fig. 1. Starting from an atomistic structural model of materials (e.g., obtained from DFT calculations or molecular dynamics simulations), we identify active regions with stronglycorrelated electrons, which we describe with an effective Hamiltonian that includes the effect of the environment on the active region. This effective Hamiltonian is constructed using the quantum embedding theory described below, and its eigenvalues can be obtained by either classical algorithms such as exact diagonalization (FCI) or quantum algorithms.
Embedding theory
A number of interesting quantum embedding theories have been proposed over the past decades^{52}. For instance, density functional embedding theory has been developed to improve the accuracy and scalability of DFT calculations^{53,54,55,56,57}. Density matrix embedding theory (DMET)^{58,59,60} and various Green’s function based approaches^{61,62}, e.g., DMFT, have been developed to describe systems with stronglycorrelated electronic states. At present, ab initio calculations of materials using DMET and DMFT have been limited to relatively small unit cells (a few tens of atoms) of pristine crystals, due to their high computational cost^{63,64}. In this work, we present a quantum embedding theory that is applicable to stronglycorrelated electronic states in realistic heterogeneous materials and we apply it to systems with hundreds of atoms. The theory, inspired by the constrained random phase approximation (cRPA) approach^{65,66,67}, does not require the explicit evaluation of virtual electronic states^{68,69}, thus making the method scalable to materials containing thousands of electrons. Furthermore, cRPA approaches contain a specific approximation (RPA) to the screened Coulomb interaction, which neglects exchangecorrelation effects and may lead to inaccuracies in the description of dielectric screening. Our embedding theory goes beyond the RPA by explicitly including exchangecorrelation effects, which are evaluated with a recently developed finitefield algorithm^{70,71}.
The embedding theory developed here aims at constructing an effective Hamiltonian operating on an active space (A), defined as a subspace of the singleparticle Hilbert space:
Here, t^{eff} and V^{eff} are onebody and twobody interaction terms that take into account the effect of all the electrons that are part of the environment (E) in a meanfield fashion, at the DFT level. An active space can be defined, for example, by solving the Kohn–Sham equations of the full system and selecting a subset of eigenstates among which electronic excitations of interest take place (e.g., defect states within the gap of a semiconductor or insulator). To derive an expression for V^{eff} that properly accounts for all effects of the environment including exchange and correlation interactions, we define the environment density response function (reducible polarizability) \({\chi }^{{\rm{E}}}={\chi }_{0}^{E}+{\chi }_{0}^{{\rm{E}}}f{\chi }^{E}\), where \({\chi }_{0}^{{\rm{E}}}={\chi }_{0}{\chi }_{0}^{{\rm{A}}}\) is the difference between the polarizability of the Kohn–Sham system χ_{0} and its projection onto the active space \({\chi }_{0}^{{\rm{A}}}\) (see Supplementary Information (SI)). χ^{E} thus represents the density response outside the active space. The term f = V + f_{xc} is often called the Hartreeexchangecorrelation kernel, where V is the Coulomb interaction and the exchangecorrelation kernel f_{xc} is defined as the derivative of the exchangecorrelation potential with respect to the electron density. We define the effective interactions between electrons in A as
given by the sum of the bare Coulomb potential and a polarization term arising from the density response in the environment E. When the RPA is adopted, the exchangecorrelation kernel f_{xc} is neglected in Eq. (2) and the expression derived here reduces to that used within cRPA. We represent χ^{E} and f on a compact basis obtained from a lowrank decomposition of the dielectric matrix^{68,69} that allows us to avoid the evaluation and summation over virtual electronic states. Once V^{eff} is defined, the onebody term t^{eff} can be computed by subtracting from the Kohn–Sham Hamiltonian a term that accounts for Hartree and exchangecorrelation effects in the active space (see SI).
Embedding theory applied to spindefects
The embedding theory presented above is general and can be applied to a variety of systems for which active regions, or building blocks, with stronglycorrelated electronic states may be identified: for example active sites in inorganic catalysts or organic molecules or defects in solids and liquids (e.g., solvated ions in water). Here we apply the theory to spindefects including NV and SiV in diamond and Cr in 4HSiC. Most of these defects’ excited states are stronglycorrelated (they cannot be represented by a single Slater determinant of singleparticle orbitals), as shown e.g., for the NV center in diamond by Bockstedte et al.^{72} using cRPA calculations. We demonstrate that our embedding theory can successfully describe the manybody electronic structure of different types of defects including transition metal atoms; our results not only confirm existing experimental observations but also provide a detailed description of the electronic structure of defects not presented before, which sheds light into their optical cycles.
We first performed spinrestricted DFT calculations using hybrid functionals^{73} to obtain a meanfield description of the defects and of the whole host solid. The spin restriction ensures that both spin channels are treated on an equal footing and that there is no spincontamination when building effective Hamiltonians. Based on our DFT results, we then selected active spaces that include singleparticle defect wavefunctions and relevant resonant and band edge states. We verified that the size of the chosen active spaces yields converged excitation energies (see SI). We then constructed effective Hamiltonians (Eqs. (1)–(2)) by taking into account exchangecorrelation effects, and we obtained manybody ground and excited states using classical (FCI) and, for selected cases, quantum algorithms (PEA, VQE). All calculations were performed at the spin triplet ground state geometries obtained by spinunrestricted DFT calculations, thus obtaining vertical excitation energies (equal to the sums of zero phonon line (ZPL) and Stokes energies). It is straightforward to extend the current approach to compute potential energy surfaces at additional geometries^{Footnote 1}, so as to include relaxations and Jahn–Teller effects^{36,72}. In Fig. 2 we present atomistic structures, singleparticle defect levels, and the manybody electronic structure of three spindefects. Several relevant vertical excitation energies are reported in Table 1, and additional ones are given in the SI. In the following discussion, lowercase symbols represent singleparticle states obtained from DFT and uppercase symbols represent manybody states.
For the NV in diamond, we constructed effective Hamiltonians (Eq. (1)) by using an active space that includes a_{1} and e singleparticle defect levels in the band gap and states near the valence band maximum (VBM). Our FCI calculations correctly yield the symmetry and ordering of the lowlying ^{3}A_{2}, ^{3}E, ^{1}E and ^{1}A_{1} states. The vertical excitation energies reported in Table 1 show that including exchangecorrelation effects yields results in better agreement with experiments than those obtained within the RPA. The results obtained within RPA (0.476/1.376/1.921 eV for ^{1}E/^{1}A_{1}/^{3}E states) are in good agreement with cRPA results reported in^{72} (0.47/1.41/2.02 eV).
In the case of the SiV in diamond, we built effective Hamiltonians using an active space with the e_{u} and e_{g} defect levels and states near the VBM, including resonant \({e}_{u}^{\prime}\) and \({e}_{g}^{\prime}\) states. Effective Hamiltonians including or neglecting exchangecorrelation effects yield similar results, with the excitation energies obtained beyond RPA being slightly higher. We predicted the first opticallyallowed excited state to be a ^{3}E_{u} state with vertical excitation energy of 1.59 eV, in good agreement with the sum of 1.31 eV ZPL measured experimentally^{31} and 0.258 eV Stokes shift estimated using an electronphonon model^{36}. Our calculations predicted a ^{3}A_{2u} state 11 meV below the ^{3}E_{u} state, in qualitative agreement with a recent experimental observation by Green et al.^{35}, which proposed a ^{3}A_{2u}^{3}E_{u} manifold with 7 meV separation in energy. The small difference in energy splitting between our results and experiment is likely due to geometry relaxation effects are not yet taken into account in our study. In addition to states of u symmetry generated by e_{u} → e_{g} excitations, we observed a number of optically dark states of g symmetry (gray levels in Fig. 2b) originating from the excitation from the \({e}_{g}^{\prime}\) level and the VBM states to the e_{g} level.
Despite significant efforts^{33,34,35,36}, several important questions on the singlet states of SiV remain open. These states are crucial for a complete understanding of the optical cycle of the SiV center. Our predicted ordering of singlet states of SiV is shown in Fig. 2b. We find the vertical excitation energies of the ^{1}A_{1u} state to be slightly higher than that of the ^{3}A_{2u}^{3}E_{u} triplet manifold, suggesting that the intersystem crossing (ISC) from ^{3}A_{2u} or ^{3}E_{u} to singlet states may be energetically unfavorable (firstorder ISC to lower ^{1}E_{g} and ^{1}A_{1g} states are forbidden). We note that the ^{1}E_{u} and ^{1}A_{2u} states are much higher in energy than ^{1}A_{1u} and are not expected to play a significant role in the optical cycle. In addition the firstorder ISC process from the lowest energy singlet state ^{1}E_{g} to the ^{3}A_{2g} ground state is forbidden by symmetry. Overall our results indicate that the ^{3}A_{2g} state is populated through higherorder processes and therefore the spinselectivity of the full optical cycle is expected to be low. While more detailed studies including spinorbit coupling are required for definitive conclusions, our predictions shed light on the stronglycorrelated singlet states of SiV and provide a possible explanation for the experimental difficulties in measuring opticallydetected magnetic resonance of SiV.
We now turn to Cr in 4HSiC, where we considered the hexagonal configuration. We constructed effective Hamiltonians with the halffilling e level in the band gap and states near the conduction band minimum including resonance states. Upon solving the effective Hamiltonian, we predict the lowest excited state to be a ^{1}E state arising from e → e spinflip transition, with excitation energy of 1.09 (0.86) eV based on embedding calculations beyond (within) the RPA. Results including exchangecorrelation effects are in better agreement with the measured ZPL of 1.19 eV^{37}, where the Stokes energy is expected to be small given the large DebyeWaller factor^{39}. There is currently no experimental report for the triplet excitation energies of Cr in 4HSiC, but our results are in good agreement with existing experimental measurements for Cr in GaN, a host material with a crystal field strength similar to that of 4HSiC^{38}. We predict the existence of a ^{3}E + ^{3}A_{1} manifold at ≃ 1.4 eV and a \({}^{3}E^{\prime} {+}^{3}{A}_{2}^{\prime}\) manifold at ≃ 1.7 eV above the ground state (Fig. 2c), resembling the ^{3}T_{2} manifold (1.2 eV) and ^{3}T_{1} manifold (1.6 eV) for Cr in GaN observed experimentally^{74}. We note that in many cases it is challenging to study materials containing transition metal elements with DFT^{75}. The agreement between FCI results and experimental measurements clearly demonstrates that the embedding theory developed here can effectively describe the stronglycorrelated part of the system, while yielding at the same time a quantitatively correct description of the environment.
Quantum simulations of spindefects
The results presented in the previous section were obtained using classical algorithms. We now turn to the use of quantum algorithms. To perform quantum simulations with PEA and VQE, we constructed a minimum model of an NV center including only a_{1} and e orbitals in the band gap. This model (four electrons in six spin orbitals) yields excitation energies within 0.2 eV of the converged results using a larger active space. In Fig. 3 we show the results of quantum simulations.
We first performed PEA simulations with a quantum simulator (without noise)^{46} to compute the energy of ^{3}A_{2}, ^{3}E, ^{1}E, and ^{1}A_{1} states. We used molecular orbital approximations of these states derived from group theory^{26} as initial states for PEA, which are single Slater determinant for ^{3}A_{2} (M_{S} = 1) and ^{3}E (M_{S} = 1) states, and superpositions of two Slater determinants for ^{1}E and ^{1}A_{1} states. As shown in Fig. 3a, PEA results converge to classical FCI results with an increasing number of ancilla qubits.
We then performed VQE simulations with a quantum simulator and with the IBM Q 5 Yorktown quantum computer^{47}. We estimated the energy of the ^{3}A_{2} ground state manifold by performing VQE calculations for both the singleSlaterdeterminant M_{S} = 1 component and the stronglycorrelated M_{S} = 0 component. Within a molecular orbital notation, M_{S} = 1 and M_{S} = 0 ground states can be represented as \(\lefta\bar{a}{e}_{x}{e}_{y}\right\rangle\) and \(\frac{1}{\sqrt{2}}\left(\lefta\bar{a}{e}_{x}{\bar{e}}_{y}\right\rangle +\lefta\bar{a}{\bar{e}}_{x}{e}_{y}\right\rangle \right)\), respectively, where a, e_{x}, e_{y} (spinup) and \(\bar{a}\), \({\bar{e}}_{x}\), \({\bar{e}}_{y}\) (spindown) denote a_{1} and e orbitals. To obtain the M_{S} = 0 ground state, we used a closedshell Hartree–Fock state \(\lefta\bar{a}{e}_{x}{\bar{e}}_{x}\right\rangle\) as reference; the M_{S} = 1 ground state is itself an openshell Hartree–Fock state, so we started with a higher energy reference state \(\lefta{e}_{x}{\bar{e}}_{x}{e}_{y}\right\rangle\) in the ^{3}E manifold. We used unitary coupledcluster single and double (UCCSD) ansatzes^{49} to represent the trial wavefunctions. Fig. 3b and c shows the estimated ground state energy as a function of the number of VQE iterations, where VQE calculations performed with the quantum simulator correctly converges to the ground state energy in both the M_{S} = 1 and M_{S} = 0 case. Despite the presence of noise, whose characterization and study will be critical to improve the use of quantum algorithms^{76}, the results obtained with the quantum computer converge to the ground state energy within a 0.2 eV error. Calculations of excited states with quantum algorithms will be the focus of future works.
Discussion
With the goal of providing a strategy to solve complex materials problems on NISQ computers, we proposed a firstprinciples quantum embedding theory where appropriate active regions of a material and their environment are described with different levels of accuracy, and the whole system is treated quantum mechanically. In particular, we used hybrid DFT for the environment, and we built a manybody Hamiltonian for the active space with effective electronelectron interactions that include dielectric screening and exchangecorrelation effects from the environment. Our method overcomes the commonly used random phase approximation, which neglects exchangecorrelation effects; importantly it is applicable to heterogeneous materials and scalable to large systems, due to the algorithms used here to compute response functions^{70,71}. We emphasize that the embedding theory presented here provides a flexible framework where multiple effects of the environment may be easily incorporated. For instance, dynamical screening effects can be included by considering a frequencydependent screened Coulomb interaction, evaluated using the same procedure as the one outlined here for static screening; electronphonon coupling effects can be incorporated by including phonon contributions in the screened Coulomb interactions. Furthermore, for systems where the electronic structure of the active region is expected to influence that of the host material, a selfconsistent cycle in the calculation of the screened Coulomb interaction of the environment can be easily added to the approach.
We presented results for spindefects in semiconductors obtained with both classical and quantum algorithms, and we showed excellent agreement between the two sets of techniques. Importantly, for selected cases we showed results obtained using a quantum simulator and a quantum computer, which agree within a relatively small error, in spite of the presence of noise in the quantum hardware. We made several predictions for excited states of SiV in diamond and Cr in SiC, which provide important insights into their full optical cycle. We also demonstrated that a treatment of the dielectric screening beyond the random phase approximation leads to an accurate prediction of excitation energies.
The method proposed in our work enables calculations of realistic, heterogeneous materials using the resources of NISQ computers. We demonstrated quantum simulations of stronglycorrelated electronic states in considerably larger systems (with hundreds of atoms) than previous studies combining quantum simulation and quantum embedding^{19,20,21,22,23}. We have studied solids with defects, not just pristine materials, which are of great interest for quantum technologies. The strategy adopted here is general and may be applied to a variety of problems, including the simulation of active regions in molecules and materials for the understanding and discovery of catalysts and new drugs, and of aqueous solutions containing complex dissolved species. We finally note that our approach is not restricted to stronglycorrelated active regions and will be useful also in the case of weakly correlated systems, where different regions of a material may be treated with varying levels of accuracy. Hence we expect the strategy presented here to be widely applicable to carry out quantum simulations of materials on nearterm quantum computers.
Methods
Density functional theory
All ground state DFT calculations are performed with the Quantum Espresso code^{77} using the planewave pseudopotential formalism. Electronion interactions are modeled with normconserving pseudopotentials from the SG15 library^{78}. A kinetic energy cutoff of 50 Ry is used. All geometries are relaxed with spinunrestricted DFT calculations using the PerdewBurkeErnzerhof (PBE) functional^{79} until forces acting on atoms are smaller than 0.013 eV / Å. NV and SiV in diamond are modeled with 216atom supercells; Cr in 4HSiC is modeled with a 128atom supercell. The Brillouin zone is sampled with the Γ point.
Construction of effective Hamiltonians
Construction of effective Hamiltonians is performed with the WEST code^{69}, starting from wavefunctions of spinrestricted DFT calculations. For this step, we remark that the use of hybrid functional is important for an accurate meanfield description of defect levels, even though the geometry of defects are well represented at the PBE level. We used a dielectric dependent hybrid (DDH) functional^{73} which selfconsistently determines the fraction of exact exchange based on the dielectric constant of the host material. In particular, 17.8% and 15.2% of exact exchange were used for the calculations of defects in diamond and 4HSiC, respectively. The DDH functional was shown to yield accurate band gaps of diamond and silicon carbide, as well as optical properties of defects^{41,42,80,81,82}. After hybrid functional solutions of the Kohn–Sham equations are obtained, iterative diagonalizations of χ_{0} are performed, and density response functions and f_{xc} of the system are represented on a basis consisting of the first 512 eigenpotentials of χ_{0}. Finite field calculations of f_{xc} are performed by coupling the WEST code with the Qbox^{83} code. FCI calculations^{84} on the effective Hamiltonian are carried out using the PySCF^{7} code.
Quantum simulations
In order to carry out quantum simulations, a minimum model of the NV center is constructed by applying the embedding theory with a_{1} and e orbitals beyond the RPA.
In PEA simulations, the Jordan–Wigner transformation^{85} is used to map the fermionic effective Hamiltonian to a qubit Hamiltonian, and Pauli operators with prefactors smaller than 10^{−6} a.u. are neglected to reduce the circuit depth, which results in less than 10^{−4} a.u. (0.003 eV) change in eigenvalues. In order to achieve optimal precision, the Hamiltonian is scaled such that 0 and 2.5 eV are mapped to phases ϕ = 0 and ϕ = 1 of the ancilla qubits, respectively. We used the firstorder Trotter formula to split time evolution operators into 4 time slices.
In VQE simulations, the parity transformation^{9} is adopted. For the simulation of the M_{S} = 1 state, the resulting qubit Hamiltonian acts on four qubits and there are two variational parameters in the UCCSD ansatz. For the simulation of the M_{S} = 0 state, we fixed the occupation of the a orbital and the resulting qubit Hamiltonian acts on 2 qubits. We replicated the exponential excitation operator twice, with parameters in both replicas variationally optimized. Such a choice results in six variational parameters, providing a sufficient number of degrees of freedom for an accurate representation of the stronglycorrelated M_{S} = 0 state. Parameters in the ansatz are optimized with the COBYLA algorithm^{86}.
Quantum simulations are performed with the QASM simulator and the IBM Q 5 Yorktown quantum computer using the IBM Qiskit package^{46}. Each quantum circuit is executed 8192 times to obtain statistically reliable sampling of the measurement results.
Data availability
Data that support the findings of this study are available through the Qresp^{87} curator at https://doi.org/10.6084/m9.figshare.12501407.
Code availability
The quantum embedding theory is implemented in the open source code WEST (westcode.org/).
Notes
For example, one may follow the strategy of ref. ^{72} and compute excited states of defects along given normal modes, which are usually obtained from deltaSCF calculations. This type of treatment, albeit approximate, provides valuable insights into the vibrational properties of defects in excited states.
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Acknowledgements
We thank C. P. Anderson, D. D. Awschalom, T. C. Berkelbach, B. Diler, S. Dong, D. Freedman, L. Gagliardi, F. Gygi, F. J. Heremans, L. Jiang, A. M. Lewis, A. Mezzacapo, P. J. Mintun, H. Seo, S. Sullivan, S. J. Whiteley, and G. Wolfowicz for fruitful discussions and comments on the manuscript. We also thank the Qiskit Slack channel for generous help. This work was supported by MICCoM, as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences, and Engineering Division through Argonne National Laboratory, under contract number DEAC0206CH11357 and by AFOSR FA95501910358. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under Contract No. DEAC0205CH11231, resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DEAC0206CH11357, and resources of the University of Chicago Research Computing Center.
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H.M., M.G., and G.G. designed the research. H.M. implemented the quantum embedding theory and performed simulations with classical and quantum algorithms, with supervision by M.G. and G.G. All authors wrote the manuscript.
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Ma, H., Govoni, M. & Galli, G. Quantum simulations of materials on nearterm quantum computers. npj Comput Mater 6, 85 (2020). https://doi.org/10.1038/s4152402000353z
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DOI: https://doi.org/10.1038/s4152402000353z
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