Quantum simulations of materials on near-term quantum computers

The study addresses how to simulate realistic, heterogeneous materials on near-term (NISQ) quantum computers with limited qubits. While density functional theory (DFT) has been successful for many properties, it struggles with strongly-correlated electronic states. Existing high-accuracy many-body methods (e.g., DMFT, QMC, ab initio quantum chemistry) are computationally demanding for complex materials. The authors propose a scalable, hybrid quantum-classical embedding strategy that treats a small active region with strong correlations at high accuracy while describing the environment at the DFT level, enabling efficient simulations relevant to quantum information materials (e.g., defect qubits in semiconductors).

Prior work highlights DFT’s limitations for strongly correlated states and the development of advanced methods such as dynamical mean-field theory (DMFT), quantum Monte Carlo, and ab initio quantum chemistry approaches adapted to solids. Embedding frameworks including density functional embedding theory, density matrix embedding theory (DMET), and Green’s-function-based methods (e.g., DMFT) have been proposed but remain computationally intensive for large, heterogeneous systems. Constrained RPA (cRPA) constructs effective interactions but neglects exchange-correlation (XC) effects in screening. With the advent of NISQ devices, hybrid quantum-classical approaches and problem decomposition ideas have emerged, aiming to restrict high-level treatment to small active regions while treating the environment at mean-field level.

The authors develop a DFT-based quantum embedding theory to construct an effective many-body Hamiltonian for an active space A embedded in an environment E. The effective Hamiltonian on A is H_eff = ∑_i ε_i a_i† a_i + (1/2) ∑_{ijkl} V_{ijkl} a_i† a_j† a_k a_l, where t_eff and V_eff incorporate mean-field effects of E at the DFT level. The active space is defined from Kohn–Sham (KS) solutions (e.g., selecting defect and relevant band-edge/resonant states). To include dielectric screening and XC effects from E beyond RPA, the environment’s reducible polarizability χ is defined as χ = χ̄_0 + χ̄_0 f χ, where χ̄_0 is the difference between the KS polarizability and its projection onto A, and f = V + f_xc is the Hartree–exchange–correlation kernel. Effective interactions within A are V_eff = V + f χ f, explicitly including XC via a finite-field algorithm to evaluate f_xc. χ and f are represented on a compact basis built from a low-rank decomposition of the dielectric matrix, avoiding explicit virtual-state summations. The one-body term t_eff is obtained by subtracting Hartree and XC effects already accounted for in A from the KS Hamiltonian. Active spaces are chosen to include defect gap states and nearby band/resonant states, verifying convergence of excitation energies. Many-body spectra are computed by exact diagonalization (FCI) of H_eff; for selected cases, quantum algorithms (PEA, VQE) are used. Calculations target vertical excitations at ground-state geometries (spin-unrestricted DFT), with extension to relaxed excited-state surfaces and Jahn–Teller effects discussed as straightforward. Computational details: Ground-state DFT uses Quantum ESPRESSO with norm-conserving SG15 pseudopotentials, 50 Ry cutoff, Γ-point sampling; geometries relaxed with spin-unrestricted PBE to force < 0.013 eV/Å. Defects modeled in supercells (diamond: 216 atoms; 4H-SiC: 128 atoms). Spin-restricted hybrid DFT (dielectric-dependent hybrid, DDH) provides accurate mean-field defect levels; exact exchange fractions are 17.8% (diamond) and 15.2% (4H-SiC). Effective Hamiltonian construction uses WEST; density response and f_xc are expanded on the first 512 eigenpotentials of χ̄_0; finite-field evaluations couple WEST and Qbox. FCI is performed with PySCF. Quantum simulations: A minimum NV model with a1 and e orbitals (four electrons in six spin orbitals) is used. PEA employs Jordan–Wigner mapping; Pauli terms with coefficients < 10 a.u. are dropped (≤ 0.003 eV effect). The Hamiltonian is scaled so 0–2.5 eV maps to ancilla phases 0–1; first-order Trotter with 4 time slices is used. VQE adopts the parity mapping; for M_s=1, a 4-qubit Hamiltonian with UCCSD ansatz (two parameters); for M_s=0, occupation of the a orbital is fixed leading to a 2-qubit Hamiltonian and a replicated excitation operator (six parameters total). Parameters optimized with COBYLA. Circuits are executed on QASM simulator and IBM Q 5 Yorktown (8192 shots per circuit).

- The embedding theory constructs V_eff = V + f χ f including exchange–correlation effects of the environment via finite-field evaluation of f_xc, going beyond RPA/cRPA while avoiding explicit virtual states through a low-rank dielectric basis. - NV center in diamond: FCI on H_eff reproduces correct symmetries and ordering of low-lying states (3A2, 3E, 1E, 1A1). Vertical excitation energies (beyond RPA) show improved agreement with experiment compared to RPA. Examples: 3A2→3E: 2.001 eV (RPA: 1.921 eV; expt ZPL 1.945 eV, vertical 2.180 eV); 3A2→1A1: 1.759 eV (RPA: 1.376 eV); 3A2→1E: 0.561 eV (RPA: 0.476 eV). The RPA results align with prior CRPA values. - SiV center in diamond (neutral): Predicted first optically allowed excited state is Eg with vertical energy 1.59 eV, consistent with 1.31 eV ZPL plus ~0.258 eV Stokes shift. An A2u state is predicted 11 meV below Eg, in qualitative agreement with experiment (7 meV splitting). Several dark g-symmetry states are identified. Predicted singlet ordering: 1A1g slightly above the A2u/3Eu manifold; 1Eg and 1A2u much higher, suggesting limited roles in the optical cycle. - Cr4+ in 4H-SiC (hexagonal site): Lowest excited state predicted as a 3E state from eg→eu spin-flip. Beyond-RPA embedding yields 1.09 eV (RPA: 0.86 eV), closer to measured ZPL 1.19 eV; Stokes expected small due to large Debye–Waller factor. Additional manifolds predicted: 3E+3A1 at ~1.4 eV and ~1.7 eV above ground, consistent with analogous Cr centers in GaN (experimental 3T2 at 1.2 eV and 3T1 at 1.6 eV). - Quantum simulations: A minimal NV model (4e in 6 spin orbitals) reproduces excitation energies within ~0.2 eV of larger active-space FCI. PEA (noise-free simulator) converges to FCI energies for 3A2, 3E, 1E, 1A1 as ancilla qubits increase. VQE on simulator converges for both M_s=1 and strongly correlated M_s=0 components of the 3A2 ground-state manifold. VQE on IBM Q 5 Yorktown converges within ~0.2 eV of the ground-state energy despite hardware noise. Dropping small Pauli terms (<10 a.u.) changes eigenvalues by <0.003 eV. Each circuit used 8192 shots.

The results demonstrate that a quantum embedding approach that includes exchange–correlation screening from the environment enables accurate predictions of strongly-correlated defect states in realistic solids using tractable active spaces. By going beyond RPA in the screened interactions, excitation energies for key defect qubits (NV, SiV in diamond; Cr in 4H-SiC) are brought into close agreement with experimental benchmarks, addressing DFT’s limitations for strongly correlated electronic states. The agreement between classical FCI and quantum algorithm results (PEA and VQE), including execution on a noisy 5-qubit device within ~0.2 eV error, validates the feasibility of leveraging NISQ hardware for materials simulations when combined with effective Hamiltonians derived from first-principles embedding. The approach offers mechanistic insight into optical cycles (e.g., singlet ordering and intersystem crossing pathways in SiV) and accurately captures transition-metal defect energetics (Cr in SiC), highlighting its relevance for quantum information materials.

The study introduces a scalable, first-principles quantum embedding theory for materials that constructs effective Hamiltonians for strongly correlated active regions, while treating the environment at the hybrid-DFT level with dielectric screening and exchange–correlation effects included beyond RPA. Applied to defect qubits in diamond and SiC, the method yields excitation energies and state orderings consistent with experiments and provides new predictions (e.g., SiV singlet ordering, Cr excitation manifolds). Quantum simulations (PEA, VQE) on a minimal NV model agree with classical FCI and demonstrate viability on NISQ hardware. The framework is flexible and extensible: frequency-dependent (dynamical) screening, electron–phonon coupling, and self-consistent environment screening can be incorporated. Future work includes excited-state quantum simulations on hardware, inclusion of geometry relaxation and Jahn–Teller effects, and spin–orbit coupling, broadening applicability to catalysts, molecular active sites, and complex solutions.

- Reported excitation energies are vertical (computed at ground-state geometries); geometry relaxation and Jahn–Teller effects are not included in the main results, potentially affecting fine splittings (e.g., SiV Eg–A2u separation). - Spin–orbit coupling is not included; more detailed studies are needed for definitive conclusions on intersystem crossing pathways and spin selectivity. - Quantum hardware results are affected by noise; while VQE converged within ~0.2 eV, error mitigation and noise characterization are needed for higher accuracy. - Quantum algorithm demonstrations used a minimal active-space model (NV: 4 electrons in 6 spin orbitals); larger active spaces and excited-state quantum simulations are deferred to future work. - Current embedding uses static screening; dynamical screening and explicit electron–phonon effects are discussed as extensions but not implemented in the presented results.