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arXiv:2104.03409 · RSC Advances

A systematic variational approach to band theory in a quantum computer

Kyle Sherbert, Frank Cerasoli, Marco Buongiorno Nardelli

Published Jan 1, 2021• 20 citations
arXivHTMLPDF

Abstract

of tight-binding orbitals per unit-cell, similarly to its classical counterparts. Our simulations offer a new insight into the "quantum" mindset and demonstrate how the algorithms under active development today can be optimized in special cases, such as band structure calculations.
DOI
10.1039/d1ra07451b

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arXiv:2607.09761•Oct 13, 2026•0 citations
Abstract While the main thrust of quantum computing research in materials science is to accurately measure the classically intractable electron correlation effects due to Coulomb repulsion, designing optimal quantum algorithms for simpler problems with well-understood solutions is a useful tactic to advance our quantum ``toolbox''. With this in mind, we consider the quantum calculation of a periodic system's single-electron band structure over a path through reciprocal space. Previous efforts have used the Variational Quantum Eigensolver algorithm to solve the energy of each band, which involves numerically optimizing the parameters of a variational quantum circuit to minimize a cost function, constructed as the expectation value of a Hamiltonian operator. Traditionally, a unique Hamiltonian operator is constructed for each k-point, so that many cost functions, each with their own parameter space, must be optimized to generate a single band. Similarly, calculating higher bands than the first has traditionally involved modifying the cost function with additional overlap terms to ensure higher-energy eigenstates are orthogonal to those of lower bands. In this paper, we adopt a direct space approach, using a novel hybrid first/second-quantized qubit mapping which allows us to construct a single Hamiltonian, and a single cost-function, suitable for solving the entire band-structure. In contrast to previous approaches, the k-point and the band index are selected by additional parameters in our quantum circuit, rather than through modifications to the cost function. The result is a technically and conceptually simpler approach to band structure calculations on a quantum computer. Moreover, we expect that the tools developed herein will motivate new strategies for tackling highly-correlated materials beyond the grasp of classical computing.
While the main thrust of quantum computing research in materials science is to accurately measure the classically intractable electron correlation effects due to Coulomb repulsion, designing optimal quantum algorithms for simpler problems with well-understood solutions is a useful tactic to advance our quantum "toolbox". With this in mind, we consider the quantum calculation of a periodic system's single-electron band structure over a path through reciprocal space. Previous efforts have used the Variational Quantum Eigensolver algorithm to solve the energy of each band, which involves numerically optimizing the parameters of a variational quantum circuit to minimize a cost function, constructed as the expectation value of a Hamiltonian operator. Traditionally, a unique Hamiltonian operator is constructed for each k-point, so that many cost functions, each with their own parameter space, must be optimized to generate a single band. Similarly, calculating higher bands than the first has traditionally involved modifying the cost function with additional overlap terms to ensure higher-energy eigenstates are orthogonal to those of lower bands. In this paper, we adopt a direct space approach, using a novel hybrid first/second-quantized qubit mapping which allows us to construct a single Hamiltonian, and a single cost-function, suitable for solving the entire electronic band structure. In contrast to previous approaches, the k-point and the band index are selected by additional parameters in our quantum circuit, rather than through modifications to the cost function. The result is a technically and conceptually simpler approach to band structure calculations on a quantum computer. Moreover, we expect that the tools developed herein will motivate new strategies for tackling highly-correlated materials beyond the grasp of classical computing.
Abstract Variational quantum algorithms such as the variational quantum eigensolver (VQE) and variational quantum deflation (VQD) have emerged as leading candidates for computing electronic structure on near-term quantum hardware. For periodic solids, however, their reach has been limited not by the quantum algorithm itself but by the cost of constructing a faithful second-quantized Hamiltonian for each material, typically via density-functional theory (DFT) followed by Wannierization or by hand-fitted empirical tight-binding parameters. SlaKoNet, a recently developed method, significantly reduces the computational cost of electronic Hamiltonian construction with a small accuracy tradeoff. It integrates modern deep-learning methods and the Slater-Koster tight-binding formalism to fit thousands of onsite and distance-dependent hopping and overlap tight-binding parameters across 65 elements of the periodic table. After fitting, these parameters can be used to deterministically construct an electronic Hamiltonian for any crystal made up of any of the 65 elements in the training set, and the crystal's band structure can subsequently be computed. In this work, we combine a SlaKoNet model trained on the JARVIS-TBmBJ dataset with a Qiskit-based variational quantum deflation algorithm, replacing the traditional and costly Hamiltonian construction step with a universal neural Hamiltonian generator. Within the 65-element set, the resulting SlaKoNet-VQD workflow is structure-agnostic, fully differentiable, fast, and suited to high-throughput screening of band structures. We benchmark the workflow on silicon, recovering the full eight-band band structure along the standard high-symmetry k-path with a mean absolute deviation of 1.78 meV from exact diagonalization on a 3-qubit statevector simulator. We further compute the full band structure for five additional conventional superconductors, Al, Ta, Nb, V, and ZrN, achieving a mean absolute deviation within approximately 1.8 meV, and demonstrate full execution of SlaKoNet-VQD on IBM Quantum hardware for a representative k-point ground-state calculation on aluminum with a mean absolute error of roughly 0.37 eV. Beyond the single-particle level, we promote the same SlaKoNet Hamiltonian to a correlated Hubbard model and solve it with dynamical mean-field theory, recovering the weakening of correlations across the group-5 metals and strong quasiparticle renormalization in the cuprate La₂CuO₄, and we identify the resulting impurity problem as the natural target for a quantum solver. The pipeline opens a path to high-throughput benchmarking of variational quantum algorithms across the periodic table and to gradient-based co-optimization of ansatz and Hamiltonian for materials discovery. An interactive web app is available at https://atomgpt.org/quantum.