Quantum-inspired Chemical Rule for Discovering Topological Materials

Technical Summary: Quantum-Inspired Chemical Rule for Discovering Topological Materials

Problem Statement
The discovery of topological materials (TMs), including topological insulators (TIs) and topological semimetals (TSMs), is currently hindered by the high computational cost of first-principles calculations and the limitations of existing screening methods. While symmetry-based diagnostics (e.g., topological quantum chemistry and symmetry indicators) have enabled large-scale screening, they are inherently incomplete. They fail to identify topological phases arising from accidental band inversions, low-symmetry distortions, or strong spin-orbit interactions that lack clear symmetry indicators (e.g., Weyl semimetals like TaAs). Furthermore, existing machine learning approaches often rely on complex, high-dimensional models that lack interpretability, making it difficult to extract clear physical intuition or simple chemical rules for material design.

Methodology
The authors propose a Hybrid Quantum-Classical Neural Network (HQCNN) to derive a "quantum-inspired chemical rule," denoted as $g_Q(M)$. This approach aims to bridge atomic chemistry and topological physics by embedding quantum mechanical concepts—specifically probability amplitudes and interference—into a data-driven framework.

1. Model Architecture: The HQCNN utilizes a parameterized variational quantum circuit (VQC) to encode classical compositional data (elemental fractions) into quantum states. These states are measured, and the resulting probability amplitudes feed into a classical fully connected neural network. Both quantum and classical parameters are jointly optimized via backpropagation.
2. The Quantum-Inspired Rule ($g_Q(M)$): Unlike classical heuristics (e.g., the "topogivity" rule by Ma et al.) that assign independent scalar propensities to elements, the HQCNN naturally generates pairwise correlation terms. The resulting rule is expressed as:
   $$g_Q(M) = \sum_{E} f_E(M)\tau_E + \sum_{E \neq E'} \sqrt{f_E(M)f_{E'}(M)}\tau_{EE'}$$
   Where $f_E(M)$ is the fractional abundance of element $E$, $\tau_E$ represents the element-specific topogivity, and $\tau_{EE'}$ captures pairwise inter-element interactions.
3. Validation of Formulation: To ensure the physical validity and interpretability of the quantum-derived correlations, the authors constructed an equivalent Complex-Valued Neural Network (CVNN). The CVNN independently reproduces the same analytical form, confirming that the pairwise terms are consistent and interpretable rather than artifacts of the quantum circuit simulation.
4. Workflow: The rule classifies a material as topologically nontrivial if $g_Q(M) > 0$ and trivial if $g_Q(M) < 0$. The magnitude $|g_Q(M)|$ serves as a confidence metric, with $|g_Q(M)| \ge 20$ identified as a high-confidence threshold.

Key Contributions

• Novel Formulation: The paper introduces a chemical rule that extends classical topogivity by incorporating pairwise quantum interaction terms ($\tau_{EE'}$), which are inaccessible to classical heuristics treating elements independently.
• Interpretability: By deriving an analytical expression from a hybrid model and validating it with a CVNN, the authors provide a transparent, physically motivated rule that retains the simplicity of chemical reasoning while embedding quantum-native features.
• High-Throughput Screening: The rule is applied to a discovery set of 1,433 materials, prioritizing candidates for first-principles validation based on high-confidence scores.

Results

• Model Performance: On an independent test set of 1,026 materials, the HQCNN-derived rule achieved 83.5% accuracy, 79.2% recall, and 86.2% precision. Performance was robust for binary and ternary compounds.
• Confidence Thresholding: Analysis revealed that predictions with $|g_Q(M)| \ge 20$ yielded a classification accuracy of 92.8% for topological materials, significantly outperforming the overall average.
• Discovery of New Materials: Applying the rule to the discovery set and filtering for high-confidence candidates ($g_Q(M) \ge 20$) led to the identification of five previously unreported topological compounds:
  1. Ag$_3$Sb (SG 59)
  2. Ca (SG 64)
  3. Pd$_3$(PbS)$_2$ (SG 166)
  4. P$_3$Sc$_7$ (SG 186)
  5. As$_2$Ta (SG 12)
     Note: One candidate, PdS$_4$U$_2$, was identified as topologically trivial.
• Validation: These five new candidates were validated using Density Functional Theory (DFT) calculations (VASP) including spin-orbit coupling. The results confirmed their topological characteristics, supported by symmetry indicators from the Topological Materials Database where available. The overall prediction accuracy for the refined screening workflow reached 84.7%, reducing false positives compared to the classical counterpart.

Significance and Claims
The authors claim that this work demonstrates how a classical compositional rule transforms when embedded within a quantum framework, revealing specific interaction terms that enhance the representation of inter-element correlations. The significance lies in providing a physically motivated extension of the classical topogivity framework. By incorporating inter-element correlations derived from quantum interference effects, the method offers enhanced predictive power and physical insight without requiring complex, uninterpretable models.

The paper positions this approach as an efficient high-throughput screening tool that requires only compositional information, significantly reducing the computational and experimental effort needed to explore systems beyond the scope of symmetry-indicator methods. The authors view this as a promising pathway for integrating quantum computing concepts with materials informatics, laying a foundation for future implementation on actual quantum hardware to explore complex correlations in materials systems. They modestly note that while the mathematical form is clear, the specific quantitative contributions and underlying physicochemical interpretations of the interference terms remain subjects for further investigation.