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SlaKoNet-VQD: A universal Slater-Koster tight-binding Hamiltonian for variational quantum band-structure calculations on near-term hardware

This paper introduces SlaKoNet-VQD, a high-throughput workflow that combines a universal deep-learning-based Slater-Koster tight-binding Hamiltonian generator with the variational quantum deflation algorithm to efficiently compute accurate band structures for periodic solids on near-term quantum hardware, demonstrating successful simulations on silicon, various superconductors, and real IBM Quantum devices.

Original authors: Akshaya Ajith, Jaehyung Lee, Charles Rhys Campbell, Kamal Choudhary

Published 2026-10-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Akshaya Ajith, Jaehyung Lee, Charles Rhys Campbell, Kamal Choudhary

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The quest to understand how electrons move through solid materials is one of the most fundamental challenges in modern science. These tiny particles determine whether a substance conducts electricity, glows with light, or becomes a superconductor that carries current without resistance. To predict these behaviors, scientists traditionally rely on complex computer simulations that solve equations describing the energy of every electron in a crystal. While these methods work well for small molecules, they become incredibly difficult and slow when applied to the repeating, infinite patterns found in solid crystals. For decades, researchers have hoped that quantum computers could solve these problems naturally, since they operate using the same quantum rules that govern electrons. However, a major hurdle has remained: before a quantum computer can begin its work, scientists must first translate the physical material into a specific mathematical format the machine can understand. This translation step has been slow, expensive, and required a human expert to manually tune the settings for every single new material, effectively blocking the path to testing thousands of different substances.

A team of researchers at Johns Hopkins University has now built a bridge across this bottleneck. They developed a new workflow that combines a powerful artificial intelligence model with a quantum algorithm to calculate the energy bands of solid materials. Instead of asking a human to manually construct the mathematical description for each new crystal, they trained a neural network to learn the rules of electron movement across the periodic table. Once trained, this AI model can instantly generate the necessary mathematical description for any material made from 65 different elements, from simple metals to complex compounds. The researchers then fed these AI-generated descriptions directly into a quantum algorithm running on a real quantum computer. The result is a system that can predict the electronic properties of a wide variety of materials without the need for slow, manual setup, opening the door to screening vast libraries of materials for future technologies.

The core of this achievement lies in replacing the old, labor-intensive method of building these mathematical models. Previously, to study a new solid, a scientist would run a heavy simulation to get a rough picture, then manually adjust parameters to fit the data, a process that could take days or weeks for a single material. The new approach uses a tool called SlaKoNet, which acts as a universal translator. It was trained on a massive database of known materials to learn how electrons hop between atoms based on their distance and type. Once this learning phase is complete, the model can instantly produce a faithful mathematical map for any new crystal structure it encounters. This map is then handed to a quantum algorithm known as variational quantum deflation. This algorithm is designed to find the energy levels of a system, much like finding the specific notes a musical instrument can play, but it does so by iteratively searching for the lowest energy states and then "deflating" them out of the way to find the next ones.

To test if this new pipeline worked, the researchers first applied it to silicon, a material that is the foundation of modern electronics. They asked the system to calculate the full range of energy levels for silicon across a standard path through its internal structure. Using a simulator that mimics a quantum computer with three qubits, the system successfully reproduced all eight energy bands with an average error of just 1.78 millielectronvolts. This level of precision is remarkably close to the exact theoretical values, proving that the AI-generated map was accurate enough for the quantum computer to work with. The researchers then took the next critical step: they ran the same calculation on actual quantum hardware provided by IBM. They focused on aluminum, a common metal, and asked the machine to find its ground state energy. While the real hardware introduced more noise and error than the perfect simulator, the system still produced a result that was only about 0.37 electronvolts away from the exact value. This demonstrated that the entire process, from the AI generating the map to the quantum computer solving it, could function on real, imperfect machines available today.

The true power of this method, however, is its ability to generalize. The researchers did not stop at silicon and aluminum. They applied the exact same workflow to five other materials, including several metals that become superconductors at low temperatures and a hard ceramic compound. Without changing the settings or retraining the model, the system successfully generated the energy band structures for all of them. The accuracy remained high, with errors staying below 1.8 millielectronvolts for the more complex materials. This shows that the AI model has learned a universal set of rules that apply across different types of chemistry, from simple free-electron metals to transition metals and nitrides. It means that a single, pre-trained model can now serve as the starting point for studying thousands of different materials, removing the need for a human expert to build a new model for each one.

Looking beyond simple energy levels, the researchers also explored how this method could tackle more difficult problems where electrons interact strongly with one another. In many materials, the behavior of one electron depends heavily on the others, creating complex effects that are hard to simulate. The team showed that their AI-generated model could be upgraded to include these interactions and then solved using a technique called dynamical mean-field theory. They applied this to a famous copper-based material known for its strong electron correlations. The system successfully captured the weakening of these correlations as they moved down a group of metals and identified the specific mathematical problem that a quantum computer would need to solve to understand the material's behavior. This suggests that the pipeline is not just a tool for simple calculations but a complete framework that can guide quantum computers toward the most challenging problems in materials science.

The implications of this work extend far beyond the specific materials tested. By removing the manual setup step, the researchers have created a path for high-throughput screening, where quantum algorithms could be tested against thousands of materials in a single sweep. This allows scientists to study how different quantum algorithms perform across a wide range of chemical environments, something that was previously impossible due to the time required to prepare each input. Furthermore, because the entire process is differentiable, meaning the mathematical steps can be traced backward to see how changes in the material affect the final result, it opens the door to optimizing both the material design and the quantum algorithm simultaneously. The researchers have made their tools and data available to the public, inviting others to build upon this foundation. While current quantum computers are still limited by noise and the number of qubits they can control, this work demonstrates that a universal, AI-driven approach can make the best use of today's hardware. It turns the quantum computer from a machine that requires a custom-built key for every door into a tool that can unlock a vast library of materials, accelerating the discovery of new superconductors, semiconductors, and other advanced substances.

Technical Summary: SlaKoNet-VQD

Problem Statement
Variational quantum algorithms, specifically the Variational Quantum Eigensolver (VQE) and Variational Quantum Deflation (VQD), are leading candidates for computing electronic structures on near-term quantum hardware. While these methods have matured for molecular systems, their application to periodic solids remains limited. The primary bottleneck is not the quantum algorithm itself, but the high cost of constructing a faithful second-quantized Hamiltonian for each specific material. Current workflows rely on either Density Functional Theory (DFT) followed by Wannierization or hand-fitted empirical Slater-Koster (SK) parameters. Both approaches require significant human expertise per material, are difficult to compose into high-throughput or differentiable pipelines, and lack transferability across different chemistries without re-fitting.

Methodology
The authors propose SlaKoNet-VQD, a workflow that integrates a universal neural network Hamiltonian generator with a Qiskit-based VQD algorithm.

SlaKoNet Hamiltonian Generation: The pipeline utilizes SlaKoNet, a deep-learning framework trained on the JARVIS-TBmBJ dataset (approx. 20,000 materials with atomic numbers $Z \le 65$ ). Unlike graph neural networks that predict matrix elements directly, SlaKoNet learns the distance-dependent Slater-Koster tight-binding parameters (onsite energies, hopping integrals, and overlap integrals) via automatic differentiation. Once trained, it deterministically constructs the real-space Hamiltonian $H(R)$ and overlap matrix $S(R)$ for any crystal composition within its training domain. These are transformed into $k$ -resolved Hamiltonians $H(k)$ via Bloch sums.
Hermitization and Decomposition: To ensure compatibility with quantum hardware, the generated $H(k)$ matrices (which may suffer from finite-precision symmetry breaking) are symmetrized via $(H + H^\dagger)/2$ . The resulting Hermitian matrix is then decomposed into a sum of weighted Pauli strings ( $H = \sum c_P P$ ) using a tensor-product Pauli projector, mapping the problem to $n = \lceil \log_2 N \rceil$ qubits, where $N$ is the number of orbitals.
Variational Quantum Deflation (VQD): The quantum algorithm solves for eigenvalues. The ground state is found via standard VQE minimization. Excited states are obtained by augmenting the cost function with overlap penalties against previously found eigenstates to enforce orthogonality.
Ansatz and Optimization: The study employs the EfficientSU2 ansatz (circuit6) with five repetitions, consisting of alternating $R_Y/R_Z$ rotation layers and linear entanglement chains. The classical outer loop uses the COBYLA optimizer.

Key Contributions
The paper makes five specific contributions:

Pipeline Construction: A clean interface between the SlaKoNet Hamiltonian generator and the Qiskit Estimator primitive, including a necessary Hermitization step to handle numerical noise.
Validation: A full-Brillouin-zone benchmark on silicon, comparing 3-qubit VQD eigenvalues against exact diagonalization of the same neural Hamiltonian.
Hardware Execution: Demonstration of the workflow on IBM Quantum hardware (ibm_boston), including error mitigation and quantification of discrepancies between simulated and real hardware.
Generalization: Application of the same workflow, without retraining, to a panel of five conventional superconductors (Al, Ta, V, Nb, and ZrN), demonstrating transferability across elemental free-electron metals, transition metals, and transition-metal nitrides.
Correlated Extension: A conceptual extension promoting the SlaKoNet Hamiltonian to a correlated Hubbard model solved via Dynamical Mean-Field Theory (DMFT). This recovers known correlation trends in Group-5 metals and strong quasiparticle renormalization in the cuprate La $_2$ CuO $_4$ , identifying the resulting impurity problem as a natural target for quantum solvers.

Results

Silicon Benchmark: On a 3-qubit statevector simulator, the SlaKoNet-VQD workflow recovered the full eight-band structure of silicon along the standard high-symmetry path with a Mean Absolute Deviation (MAD) of 1.78 meV from exact diagonalization. The largest deviations (29.32 meV) occurred in the highest excited states due to the accumulation of deflation penalties.
Hardware Execution: A ground-state calculation for aluminum on IBM Quantum hardware yielded a mean absolute error of approximately 0.37 eV relative to exact diagonalization (after error mitigation). The authors note this error is comparable to typical differences between DFT functionals and larger than chemical accuracy, attributing it to current hardware noise rather than pipeline flaws.
Generalization: For the five additional superconductors, the workflow achieved MADs within approximately 1.8 meV per band. Elemental metals (Al, Ta, V, Nb) showed errors $\le 0.05$ meV, while the three-qubit ZrN system showed 1.74 meV.
Correlated Systems: In the DMFT extension, the pipeline successfully reproduced the weakening of correlations down the Group-5 column (V $\to$ Nb $\to$ Ta) and identified strong quasiparticle renormalization in La $_2$ CuO $_4$ when restricting the bath to the low-energy manifold.

Significance and Claims
The paper claims that SlaKoNet-VQD removes the "per-material setup cost" that has historically constrained quantum band-structure studies. By replacing DFT/Wannierization with a universal, differentiable neural Hamiltonian, the workflow enables:

High-throughput screening: The ability to run VQE/VQD on thousands of materials in a single sweep without manual intervention.
Differentiable pipelines: The entire chain from crystal structure to quantum eigenvalue is differentiable, allowing for gradient-based co-optimization of ansatz and Hamiltonian.
Algorithmic benchmarking: The pipeline provides a reproducible, fixed-structure substrate for testing new variational algorithms, analogous to the role of FCIDUMP in molecular chemistry.

The authors explicitly state that the accuracy of the results is bounded by the underlying SlaKoNet model (trained against TBmBJ data) rather than the quantum solver itself. They acknowledge limitations, including the lack of spin-orbit coupling and repulsive pair potentials in the current model, and the degradation of excited-state fidelity as the spectrum is climbed. However, they posit that this pipeline establishes a foundation for high-throughput benchmarking of variational algorithms across the periodic table and paves the way for ML-accelerated quantum dynamical mean-field theory.

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