Atomistic simulations, crucial for predicting material properties, often rely on density functional theory (DFT). While successful, DFT struggles with strongly-correlated electronic states. Methods like dynamical mean-field theory (DMFT) and quantum Monte-Carlo offer improvements but are computationally expensive. Quantum computers promise efficient simulations of both weakly and strongly-correlated systems, but current limitations in qubit number restrict simulations to small systems. This paper addresses this by developing a hybrid quantum-classical method. This method focuses on a smaller, 'active region' within the larger system, treating this region with high accuracy while describing the environment using mean-field theory. The focus is on the development and application of a quantum embedding theory, tested on spin-defects in semiconductors—systems crucial for quantum information science, exhibiting strongly-correlated electronic states essential for spin state initialization and readout. The ultimate goal is to demonstrate the potential of near-term quantum computers for materials science simulations by comparing quantum and classical computations of these defects.

The introduction provides a comprehensive overview of existing computational methods for simulating materials, highlighting the limitations of Density Functional Theory (DFT) in accurately describing strongly-correlated electronic states. It mentions alternative methods like Dynamical Mean-Field Theory (DMFT) and Quantum Monte-Carlo, acknowledging their high computational cost. The review then transitions to the promise of quantum computing for tackling these challenges, contrasting the potential for exponential speedup with the limitations of current Noisy Intermediate-Scale Quantum (NISQ) devices. The discussion emphasizes the necessity for hybrid quantum-classical approaches that selectively treat strongly-correlated regions with higher accuracy, citing prior work on similar embedding techniques and their limitations in scalability and applicability to complex materials.

The core methodology centers on a novel quantum embedding theory. This theory partitions a material into an 'active region' with strongly-correlated electrons and an environment described by DFT. An effective Hamiltonian is constructed for the active region, incorporating environmental effects via dielectric screening and exchange-correlation interactions. This goes beyond the Random Phase Approximation (RPA) by explicitly including exchange-correlation effects, evaluated using a recently developed finite-field algorithm. The effective Hamiltonian, described by equations (1) and (2), includes one-body and two-body interaction terms. The authors explain how these terms are calculated, highlighting the use of a low-rank decomposition of the dielectric matrix to ensure scalability. The methodology then details the application of this theory to spin defects (NV and SiV in diamond, Cr in 4H-SiC), outlining the DFT calculations used to obtain initial wavefunctions and select active spaces. The calculation of many-body ground and excited states involves both classical methods (Full Configuration Interaction, FCI) and quantum algorithms (Phase Estimation Algorithm, PEA, and Variational Quantum Eigensolver, VQE). The authors carefully describe the specifics of the quantum simulations, including the mapping of fermionic Hamiltonians to qubit Hamiltonians, the choice of ansatzes for VQE, and the optimization algorithms used. The use of both classical and quantum simulations enables a direct comparison to validate the embedding theory and the feasibility of quantum computing for these types of problems. The experimental data are compared with the computed results to show the accuracy and potential of the developed embedding theory. The DFT calculations utilize the Quantum Espresso code with norm-conserving pseudopotentials. Geometries are relaxed using the Perdew-Burke-Ernzerhof (PBE) functional. Effective Hamiltonians are constructed using the WEST code, employing a dielectric-dependent hybrid (DDH) functional to accurately capture defect levels. The FCI calculations are performed using the PySCF code. Quantum simulations leverage the Qiskit package, employing both a quantum simulator and the IBM Q 5 Yorktown quantum computer. Specific details are given on the Jordan-Wigner transformation used in PEA, the parity transformation in VQE, and the ansatzes employed. The authors emphasize the importance of mitigating noise in the quantum hardware results.

The study demonstrates the successful application of the quantum embedding theory to several spin-defects, achieving excellent agreement between classical and quantum computations. Specific key findings include: * **Accurate prediction of excitation energies:** The embedding theory, going beyond RPA by incorporating exchange-correlation effects, accurately predicts excitation energies for NV⁻, SiV⁰, and Cr⁴⁺ defects, showing better agreement with experiments compared to RPA-based results. Table 1 provides a detailed comparison of calculated and experimental vertical excitation energies for these defects. * **Electronic structure of SiV⁰:** The theory predicts the ordering of singlet states for SiV⁰, providing insights into the optical cycle and explaining experimental difficulties in measuring optically-detected magnetic resonance (ODMR). The authors highlight the energy ordering of the excited states, showing that the intersystem crossing (ISC) from triplet to singlet states is energetically unfavorable. * **Electronic structure of Cr⁴⁺ in 4H-SiC:** The work predicts the existence of ³E and ³A₁ manifolds for Cr⁴⁺, consistent with experimental measurements in similar materials (Cr in GaN). * **Successful quantum simulations:** The authors demonstrate quantum simulations using both a simulator and the IBM Q 5 Yorktown quantum computer for the NV⁻ center, obtaining results that agree with classical FCI calculations within a relatively small error margin, despite the noise present in the quantum hardware. Figure 3 shows the comparison of results obtained with the Phase Estimation Algorithm (PEA) and Variational Quantum Eigensolver (VQE) methods, and with classical calculations. * **Scalability:** The methodology emphasizes scalability to larger systems. By avoiding explicit evaluation of virtual electronic states, the method is applicable to materials containing thousands of electrons, and the choice of algorithm and the method itself is suitable for NISQ algorithms, with its hybrid nature allowing for larger systems to be analyzed. These findings underscore the potential of the quantum embedding theory for simulating complex materials using near-term quantum computers.

The agreement between classical and quantum computations validates the quantum embedding theory and showcases its potential for simulating realistic heterogeneous materials on NISQ computers. The ability to accurately predict excitation energies, especially those of SiV⁰ and Cr⁴⁺ where some experimental data was lacking, highlights the method’s power to provide insights into systems relevant to quantum technologies. The successful quantum simulations, despite the noise inherent in current quantum hardware, demonstrate a significant advancement in combining quantum simulation with quantum embedding theories. The inclusion of exchange-correlation effects in the calculation of the effective Hamiltonian is shown to provide more accurate results than calculations that only consider the RPA approximation, emphasizing the importance of these corrections. The work opens avenues for future research, including extending the approach to simulate dynamical screening effects, electron-phonon coupling, and the development of self-consistent calculations to capture feedback between the active region and the environment.

This work presents a scalable quantum embedding theory capable of simulating strongly-correlated electronic states in complex materials on near-term quantum computers. The theory's success in accurately reproducing experimental results for spin-defects and its validation via quantum simulations represents a significant step towards harnessing the power of quantum computing for materials science. Future research could focus on incorporating dynamical effects, studying larger and more complex systems, and developing error mitigation techniques to improve the accuracy of quantum simulations on noisy hardware.

The current study focuses on static calculations and does not explicitly include dynamic effects like phonon coupling or dynamical screening. While the authors mention the potential for incorporating these effects, it remains a direction for future work. Another limitation is the use of relatively small active spaces in some simulations; larger active spaces can be considered for future works. Although the results on noisy quantum hardware are promising, further research is needed to fully mitigate the impact of noise on the accuracy of quantum simulations. The authors only explore a limited number of defects, and expanding the scope to include a wider range of materials would provide a more comprehensive evaluation of the method's effectiveness.