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

TL;DR

This work introduces a physics-informed framework that combines first-principles optical descriptors from LR-TDDFT Casida calculations with both classical and quantum machine learning to distinguish pristine CaF$_2$ from Er-doped CaF$_2$. By constructing Ca$_8$F$_{16}$ and Ca$_7$ErF$_{16}$ clusters and extracting descriptors such as the transition energy $E$, extinction coefficient $κ$, and absorption coefficient $α$, the authors benchmark a classical SVM (achieving $ACC=0.983$, ROC-AUC $=0.999$) against quantum models including QSVMs (statevector and noisy simulators) and a hybrid QNN (3-qubit feature map, depth-4 ansatz) with notable performance: QSVMs show moderate accuracy ($ACC$ up to 0.851 on ideal simulators, down to 0.817 with noise), hardware runs yield $ACC\approx0.733$, while the QNN reaches $ACC=0.93$, $AUC=0.96$. The results demonstrate that physically grounded spectral fingerprints form a robust feature space for evaluating near-term quantum learning against strong classical baselines, and that variational quantum architectures can closely approach classical performance in this materials-classification task. The study highlights the current strengths and limitations of quantum kernels versus trainable quantum circuits for spectral data, informing future algorithm-hardware co-design in quantum materials informatics.

Abstract

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 10~eV were obtained via the Casida formalism and converted into continuous absorption spectra using Gaussian broadening ($σ$=0.1-0.2~eV). 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.

Figures (15)

• Figure 1: UMAP projection of the three selected physical descriptors for CaF$_2$ and CaF$_2$:Er samples from the training set. The two-dimensional embedding is used exclusively for visualization purposes.
• Figure 2: Quantum circuit diagram of the feature map $\mathcal{U}_{\Phi(\bm{x})}$ (ZZFeatureMap) for 3 qubits, used to encode classical data into a quantum feature space for the construction of the QSVM kernel.
• Figure 3: Layout of the qubit connectivity for the 156-qubit IBM Fez device. Accessed from the IBM Quantum Platform on January 19, 2026.
• Figure 4: Schematic of the parameterized TwoLocal ansatz $U(\bm{\theta})$ serving as the variational block of the QNN. The circuit consists of $L=4$ repetitions of rotation layers (containing parameterized $R_y$ and $R_z$ gates) and entanglement layers (using all-to-all CNOT gates). This structure provides the necessary expressibility for the quantum processor to learn the complex decision boundary between pristine and Er-doped spectral signatures.
• Figure 5: High-level architecture of the hybrid quantum-classical neural network. The pipeline proceeds in four stages: (1) Encoding, where classical spectral data is mapped to the quantum processor; (2) Variational Processing, where the ansatz evolves the state based on trainable parameters $\bm{\theta}$; (3) Measurement, where bitstring parity is calculated to derive class probabilities; and (4) Classical Post-Processing, where a linear layer projects probabilities into logits. The entire model is optimized end-to-end using the Adam optimizer to minimize the Cross-Entropy loss.