First-Principles Optical Descriptors and Hybrid Classical-Quantum Classification of Er-Doped CaF$_2$

We present a physics-informed classical-quantum machine learning framework for discriminating pristine CaF$2$ from Er-doped CaF$2$ using first-principles optical descriptors. Finite Ca$8$F${16}$ and Ca$7$ErF${16}$ clusters were constructed from the fluorite structure (a=5.46~$Å$) and treated using density functional theory (DFT) and linear-response time-dependent DFT (LR-TDDFT) within the GPAW code. Geometry optimization was performed in LCAO mode with a DZP basis and PBE exchange-correlation functional, followed by real-space finite-difference ground-state calculations with grid spacing h=0.30~$Å$ and N${bands}$=N${occ}$+20. Optical excitations up to 10eV were obtained via the Casida formalism and converted into continuous absorption spectra using Gaussian broadening ($σ$=0.1-0.2eV). From 1,589 energy-resolved points per system, physically interpretable descriptors including transition energy $E$, extinction coefficient $κ$, and absorption coefficient $α$ were extracted. A classical RBF-kernel support vector machine (SVM) achieves a test accuracy (ACC) of 0.983 and ROC-AUC of 0.999. Quantum support vector machines (QSVMs) evaluated on statevector and noisy simulators reach accuracies of 0.851 and 0.817, respectively, while execution on IBM quantum hardware yields a test-slice accuracy of 0.733 under finite-shot and decoherence constraints. A hybrid quantum neural network (QNN) with a 3-qubit feature map and depth-4 ansatz achieves a test accuracy of 0.93 and AUC of 0.96. Results here demonstrate that dopant-induced optical fingerprints form a robust, physically grounded feature space for benchmarking near-term quantum learning models against strong classical baselines.

💡 Research Summary

This paper introduces a physics‑informed machine‑learning framework that distinguishes pristine calcium fluoride (CaF₂) from erbium‑doped CaF₂ (CaF₂:Er) using optical descriptors derived from first‑principles calculations. The authors construct finite Ca₈F₁₆ and Ca₇ErF₁₆ clusters based on the fluorite crystal structure (a = 5.46 Å) and perform density‑functional theory (DFT) geometry optimizations with a double‑ζ polarized (DZP) basis and the PBE exchange‑correlation functional. After relaxation, ground‑state electronic structures are recomputed on a real‑space finite‑difference grid (spacing h = 0.30 Å) with 20 virtual bands added to ensure convergence of excited‑state properties.

Linear‑response time‑dependent DFT (LR‑TDDFT) in the Casida formalism is employed to obtain excitation energies up to 10 eV and corresponding oscillator strengths. The discrete excitation data are broadened with Gaussian functions (σ = 0.1–0.2 eV) to generate continuous absorption spectra. From each spectrum the authors extract 1,589 energy‑resolved points and compute physically meaningful quantities: transition energy (E), extinction coefficient (κ), absorption coefficient (α), as well as real/imaginary dielectric functions, refractive index, etc. These quantities constitute a high‑dimensional, physics‑grounded feature set.

To prepare the data for machine learning, a Box‑Cox power transformation mitigates skewness, and a linear support‑vector machine (SVM) is used to rank feature importance. The three most discriminative descriptors—α, κ, and E—are selected for downstream modeling. A two‑dimensional UMAP visualization of these descriptors already shows clear separation between the two material classes.

A classical SVM with a radial‑basis‑function (RBF) kernel (default C = 1, γ = 1/(n_features·Var)) serves as the baseline. Using only the three selected descriptors, the model achieves a test accuracy of 0.983 and a ROC‑AUC of 0.999, demonstrating that the optical fingerprints are highly informative for this binary classification task.

For the quantum side, the authors implement a quantum kernel SVM (QSVM). They encode the three features into a three‑qubit ZZFeatureMap (single repetition) and compute the kernel matrix via the FidelityQuantumKernel in Qiskit. Two simulation environments are explored: an ideal state‑vector simulator (accuracy = 0.851) and a shot‑based noisy simulator with depolarizing error probability p = 0.05 (accuracy = 0.817). When executed on IBM Quantum hardware (a 3‑qubit device) under realistic shot limits and decoherence, the quantum kernel SVM reaches a test‑slice accuracy of 0.733, reflecting current NISQ hardware constraints.

In addition, a hybrid quantum neural network (QNN) is built with a three‑qubit feature map followed by a depth‑4 variational ansatz. Training with the Adam optimizer and cross‑entropy loss yields a test accuracy of 0.93 and an AUC of 0.96, indicating that quantum circuits can capture non‑linear relationships in the data with fewer parameters than a classical deep network.

The authors discuss why the classical SVM outperforms the quantum models: the low dimensionality (three features) limits the advantage of mapping into a high‑dimensional Hilbert space; current gate errors and limited shot numbers degrade kernel fidelity; and the feature map parameters have not been fully optimized. Nevertheless, the QNN’s competitive performance suggests that quantum‑enhanced non‑linear embeddings may become valuable as hardware improves.

The paper concludes by highlighting three main contributions: (1) a systematic workflow for extracting interpretable optical descriptors from LR‑TDDFT calculations; (2) a demonstration that these descriptors enable near‑perfect classification with a classical kernel SVM; (3) an exploratory benchmark of quantum kernel methods and variational quantum classifiers, showing that while present‑day quantum devices lag behind classical baselines, they hold promise for future materials‑science applications once qubit counts, error rates, and circuit design are advanced.

Future directions include expanding the descriptor set to incorporate multiple transitions and band‑structure information, employing error‑mitigation and more expressive quantum feature maps, and testing the approach on other rare‑earth dopants (e.g., Yb) and experimental spectra to assess generalizability.