Quantum Defect Embedding Theory (QDET)

• Quantum Defect Embedding Theory (QDET) is a rigorously formulated framework that separates an active region from its environment to accurately model strongly correlated electronic states.
• It employs advanced double-counting corrections, frequency-dependent screening, and hybridization techniques to derive effective Hamiltonians for precise predictions of excitation energies and optical responses.
• QDET is applied to point defects in semiconductors and molecular qubits, offering scalable solutions for quantum sensing and quantum computing simulations with chemical accuracy.

Quantum Defect Embedding Theory (QDET) is a rigorously formulated many-body embedding framework designed to provide accurate, scalable, and systematically improvable descriptions of strongly correlated electronic states in condensed matter systems, with primary applications to point defects in wide-gap semiconductors, insulators, and molecular qubits. QDET leverages the separation of a system into an “active” region, wherein strong electron correlation prevails (such as defect-localized orbitals), and an “environment,” treated at a lower level of theory, to derive effective Hamiltonians for quantitative prediction of excitation energies, optical properties, and correlated ground and excited states. The methodology advances earlier quantum embedding formalisms by incorporating exact treatments for double counting, frequency-dependent screening, and active space construction, yielding robust agreement with experimental results and enabling applications both in classical and quantum computing contexts.

1. Theoretical Foundations and Formulation

QDET is built upon Green’s function techniques and many-body perturbation theory, resulting in an explicit derivation of the effective Hamiltonian for the active region (“defect” or “fragment”) embedded within an extended environment. The effective Hamiltonian for the active space $A$ is:

$$H^\text{eff} = \sum_{i,j\in A} t_{ij}^\text{eff} a_i^\dagger a_j + \frac{1}{2} \sum_{i,j,k,l\in A} v_{ijkl}^\text{eff} a_i^\dagger a_j^\dagger a_l a_k$$

• $t_{ij}^\text{eff}$ are renormalized one-electron integrals that include subtraction of environmental Hartree and exchange–correlation contributions.
• $v_{ijkl}^\text{eff}$ are two-electron integrals incorporating environmental screening (frequency-dependent within the $G_0W_0$ approximation or beyond).

Crucially, QDET introduces a rigorous double-counting correction (e.g., the DC2025 protocol (Chen et al., 8 Apr 2025)) that aligns the treatment of electron–electron interactions between the active space and the environment. This ensures that neither correlations nor screenings are spuriously counted more than once. The effective screened Coulomb interaction, $W_A^{(R)}(\omega)$, is evaluated at zero or multiple frequencies to capture dynamic screening. The double-counting term is constructed as:

$$t^\text{eff} \approx h_A + \Sigma_A^\text{LOW} - \Sigma_A^\text{DC}$$

where $h_A$ is the bare Hamiltonian, $\Sigma_A^\text{LOW}$ is the low-level (e.g., $G_0W_0$) self-energy projected onto the active space, and $\Sigma_A^\text{DC}$ ensures exact cancellation of environmental contributions included both in the full system and in the active-region many-body solver.

Hybridization between the active space and environment is accounted for via a frequency-dependent hybridization function, $\Delta_{ij}(\omega)$, mapped to a static (auxiliary) Hamiltonian in an enlarged orbital space (active + bath), thereby facilitating numerically tractable solution by high-level solvers (Chen et al., 8 Apr 2025).

2. Active Space Construction and Systematic Improvability

A haLLMark of QDET is its active space selection protocol, which is critical to both physical accuracy and computational tractability. Active space orbitals are derived from first-principles calculations, typically:

• Kohn–Sham (KS) orbitals from DFT, selected by a localization criterion:

$L_n = \int_V |\phi_n(\mathbf{r})|^2 d\mathbf{r}$

where $V$ is a volume surrounding the defect or site of interest (Casares et al., 18 Aug 2025).

• Maximally localized Wannier functions, Wannierization of KS manifolds, or projected atomic orbitals, enabling both localized and symmetry-adapted choices for the correlated subspace (Muechler et al., 2021).
• Expansion to unoccupied states: The inclusion of low-lying unoccupied orbitals in the active space, particularly for systems like NV$^-$ in diamond, has been shown to shift excitation energies by up to 0.1 eV, signifying non-negligible polarization and correlation effects contributed by virtual bands (Chen et al., 8 Apr 2025). In other systems, e.g., neutral group-IV vacancies, these effects are weaker ($\sim$0.02 eV).

The convergence of computed observables with respect to active space size is assessed by incrementally including orbitals (e.g., ordered by energy relative to the band edge), with final results demonstrably insensitive to further expansion beyond a physically motivated threshold—typically a few tenths of an eV for excitation energies and other defect properties (Otis et al., 27 Jan 2025).

3. Many-body Solvers and Computational Protocol

QDET supports a variety of many-body solvers for its embedded effective Hamiltonians:

• Full configuration interaction (FCI) or “exact diagonalization” for small active + bath spaces.
• Selected CI, CISD/CIS(D) (truncated singles, doubles, perturbative triples), suitable for larger orbital sets with truncated residual interactions outside the active space.
• Auxiliary-field quantum Monte Carlo (AFQMC), which scales favorably and delivers accuracy within 0.04 eV for defect vertical excitation energies, enabling systematically improvable treatment of correlated states in large-scale systems.
• Equation-of-motion coupled-cluster (EOM-CCSD) for excitation spectra, harnessing active-space orbital selection schemes (canonical, local, and natural transition orbitals) to efficiently approach experimental accuracy with $O(10^2)$ orbitals out of $O(10^3-10^4)$ in large supercells (Lau et al., 2023).

The protocol for a QDET calculation is as follows:

1. DFT (or mean-field) electronic structure calculation of supercell hosting the point defect/qubit/molecule.
2. Localization and selection of relevant orbitals for the active space (by $L_n$ or other strategies).
3. Screened interaction computation (using $G_0W_0$, cRPA, or advanced DFT-based screening) for the projection onto the active space and extraction of $t^\text{eff}$, $v^\text{eff}$.
4. Double-counting correction with frequency-dependent self-energy and screening.
5. Definition of hybridization bath if strong coupling to the environment exists.
6. Solution of impurity (embedded) Hamiltonian with chosen many-body solver; validation via convergence with respect to active space dimension and supercell size.
7. Direct computation of experimental observables: vertical excitation energies, multiplet gaps, optical properties, with chemical accuracy (typically $<$0.1–0.2 eV) (Otis et al., 27 Jan 2025, Chen et al., 8 Apr 2025).

4. Applications: Defects in Solids and Molecular Qubits

QDET is extensively applied to a variety of key physical systems:

• Point Defects in Semiconductors and Insulators: NV$^-$ and group-IV centers in diamond, iron impurities in AlN (Ma et al., 2020, Otis et al., 27 Jan 2025). QDET achieves close agreement with measured optical gaps, excitation energies, and photoluminescence.
• Molecular Spin Qubits: e.g., Cr(o-tolyl)$_4$, where QDET, with unoccupied $d$-orbital expansion, computes VEEs within chemical accuracy of measured zero-phonon lines (Chen et al., 8 Apr 2025).
• Quantum Sensing and ODMR: For two-dimensional materials, QDET enables construction of minimal yet correlated active spaces (e.g., 18 spatial orbitals in hBN), dramatically reducing the computational cost for quantum simulation algorithms, as in the detection of ISC-rate imbalances needed for ODMR-active defects (Casares et al., 18 Aug 2025).
• Materials Design and Quantum Technologies: Systematic convergence protocols and benchmarking against ab initio quantum Monte Carlo enable application to new defect systems, critical for the rational design of quantum sensors, memory elements, and transducer materials.

The table below summarizes exemplary systems and QDET outcomes:

System	Active Space Examples	Key Outcome
NV$^-$ in diamond	5+ orbitals (localized + unocc.)	VEE and state ordering converged (<0.1 eV variation)
Fe in AlN	Minimal: 5 $d$-orbitals	Agreement with experiment; negligible further corrections with active space expansion
Molecular qubit Cr(o-tolyl)$_4$	Min. + occupied + $d$ unocc.	VEEs in close agreement ($\sim$0.03 eV) with spectroscopy
hBN $V_B^-$ defect	6–18 orbitals	Quantum resource-efficient ODMR response algorithms leveraging QDET embedding

5. Recent Methodological Advances and Validation

Significant recent advances solidify QDET’s systematic improvability and practical impact:

• Double-counting corrections (DC2025) fully account for frequency dependence in self-energies, eliminating ambiguities and ensuring Hermitian effective Hamiltonians (Chen et al., 8 Apr 2025).
• Hybridization functions incorporated via auxiliary Hamiltonians with bath orbitals, enabling treatment of defects where environmental coupling modulates excited state structure.
• Active space flexibility: Expansion to unoccupied orbitals is empirically system dependent; essential for capturing high-fidelity spectra in select cases (e.g., NV$^-$), less so in group-IV centers or molecular systems.
• Impurity solver comparison demonstrates agreement among FCI, selected CI, CIS(D), and AFQMC within chemical accuracy, even in challenging systems, reinforcing the reliability of physical conclusions (Chen et al., 8 Apr 2025).
• Benchmarking against quantum Monte Carlo exposes sensitivity to DFT reference and active space construction, emphasizing that accurate downfolding is critical and that existing mean-field inaccuracies are difficult to rectify post hoc by DC or perturbative corrections (Kleiner et al., 1 May 2025).

6. Practical and Quantum Computing Implications

QDET has been established as an efficient interface to quantum algorithms in quantum simulation, by virtue of yielding compact, correlated Hamiltonians amenable to representation with $O(100)$ logical qubits. For ODMR-active defect identification, the use of QDET embeddings enables quantum algorithms to efficiently simulate intersystem crossing contrasts and optical responses without direct rate calculations, attaining significant resource savings ($10^5$ logical qubits, $2.2 \times 10^8$ Toffoli gates for the $V_B^-$ in hBN) (Casares et al., 18 Aug 2025). This scalable approach positions QDET as a key tool in the in silico screening pipeline for future quantum sensors and quantum information processors.

7. Limitations, Validation, and Future Directions

While QDET can systematically achieve chemical accuracy for many defect properties, its results are contingent on the fidelity of the underlying mean-field electronic structure and the careful choice of active space. In cases where the mean-field DFT orbitals inadequately represent the localized defect character—as evidenced in Fe/AlN (Kleiner et al., 1 May 2025)—embedded model predictions may deviate significantly from ab initio quantum Monte Carlo results, particularly in orbital occupations and symmetry assignments of excited states. Thus, ongoing development is directed at improving mean-field reference selection, active space optimization (possibly self-consistent or post-DFT refined), and impurity solver scalability.

Continued integration of QDET with quantum Monte Carlo benchmarking, expanding hybridization treatments, and automation of active space protocols are anticipated to further extend QDET applicability to increasingly complex, strongly correlated systems in solid-state, molecular, and hybrid quantum technologies.