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arXiv:2605.10265

Expander attention as exchange-correlation

Karim K. Alaa El-Din, Antonius v. Strachwitz, Sam M. Vinko

quant-phSubmitted May 11, 2026• Announced May 12, 2026• 0 citations
arXivHTMLPDF

Abstract

Kohn-Sham density functional theory (DFT) is the workhorse of quantum chemistry, offering an attractive balance between accuracy and computational cost. Although exact in principle, DFT in practice relies on an approximation to the unknown exchange-correlation (XC) functional, which encodes the many-body quantum effects beyond the mean-field treatment. Many such approximations exist, and machine-learned XC functionals have proliferated in recent years. A persistent challenge in this area is the trade-off between accuracy and computational cost: while high-accuracy ML functionals have shown success on strongly correlated systems that are notoriously difficult for conventional approximations, their unfavorable scaling has limited broader adoption. Here, we propose a linearly scaling non-local XC approximation based on an expander graph transformer ansatz, improving the scaling of $O(N^2)$ or worse for previous ML functionals capable of reliably capturing strongly correlated systems. We show that it recovers the correct $\mathrm{H_2}$ dissociation curve in the strongly correlated regime, with promising results on planar $\mathrm{H_4}$, a system where even high-level coupled cluster methods break down. Our approach thus charts a path toward ML functionals that are both accurate on strongly correlated systems and cheap enough to deploy at scale.

Comments
9 pages (16 with references and appendices), 5 figures

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Kohn-Sham density functional theory (DFT) is the workhorse of quantum chemistry, offering an attractive balance between accuracy and computational cost. Although exact in principle, DFT in practice relies on an approximation to the unknown exchange-correlation (XC) functional, which encodes the many-body quantum effects beyond the mean-field treatment. Many such approximations exist, and machine-learned XC functionals have proliferated in recent years. A persistent challenge in this area is the trade-off between accuracy and computational cost: while high-accuracy ML functionals have shown success on strongly correlated systems that are notoriously difficult for conventional approximations, their unfavorable scaling has limited broader adoption. Here, we propose a linearly scaling non-local XC approximation based on an expander graph transformer ansatz, improving the scaling of O(N2)O(N^2)O(N2) or worse for previous ML functionals capable of reliably capturing strongly correlated systems. We show that it recovers the correct H2\mathrm{H_2}H2​ dissociation curve in the strongly correlated regime, with promising results on planar H4\mathrm{H_4}H4​, a system where even high-level coupled cluster methods break down. Our approach thus charts a path toward ML functionals that are both accurate on strongly correlated systems and cheap enough to deploy at scale.
Kohn-Sham density functional theory (KS-DFT) provides the exact ground-state energy and electron density of a molecule, contingent on the as-yet-unknown universal exchange-correlation (XC) functional. Recent research has demonstrated that neural networks can efficiently learn to represent approximations to that functional, offering accurate generalizations to molecules not present during the training process. With the latest advancements in quantum-enhanced machine learning (ML), evidence is growing that quantum neural network (QNN) models may offer advantages in ML applications. In this work, we explore the use of QNNs for representing XC functionals, enhancing and comparing them to classical ML techniques. We present QNNs based on differentiable quantum circuits as quantum (hybrid) models for XC in KS-DFT, implemented across various architectures. We assess their performance on one-dimensional (1D) and 3D systems. To that end, we expand existing differentiable KS-DFT frameworks and propose strategies for efficient training of such functionals, highlighting the importance of fractional orbital occupation for accurate results. We prove that, under specific assumptions, noise-induced errors do not accumulate across SCF iterations, and we confirm this result through numerical experiments. Our best QNN-based XC functional yields energy profiles of the H 2 and planar H 4 molecules that deviate by no more than 1 mHa from the reference DMRG and FCI/6-31G results, respectively. Moreover, they reach chemical precision on a system, H 2 H 2 , not present in the training dataset, using only a few variational parameters. This work lays the foundation for the integration of quantum models in KS-DFT, thereby opening new avenues for expressing XC functionals in a differentiable way and facilitating computations of various properties.
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Accurate exchange-correlation (XC) potentials are essential for density functional theory, yet reliable approximations remain challenging for strongly correlated systems. In this work, we present a quantum algorithmic framework to determine spin-resolved XC potentials using a variational quantum eigensolver. Using the Hubbard model as a prototypical strongly correlated lattice system, we prepare ground states in fixed spin sectors through a Hamiltonian variational ansatz initialized from the non-interacting (U=0)(U=0)(U=0) Slater-determinant ground state. From the resulting many-body ground states, we extract the XC energy and compute the corresponding spin-resolved XC potentials via finite differences. The accuracy of the approach is benchmarked against exact diagonalization for one- and two-dimensional Hubbard systems of various lattice sizes. We demonstrate that the variational ansatz reproduces the ground-state energies and densities with high fidelity, enabling accurate construction of both magnetic and non-magnetic XC potentials. We analyzed the dependence of the XC potentials on the interaction strength, charge, spin densities, and magnetization. We also present an empirical complexity scaling relation for the computational cost of the method at a fixed fidelity. These results illustrate how quantum computing can be used to construct spin-resolved XC functionals for correlated lattice models, providing a potential pathway for improving density functional approximations in strongly correlated materials.
Standard approximations for the exchange-correlation functional in Kohn-Sham density functional theory (KS-DFT) typically lead to unacceptably large errors when applied to strongly correlated electronic systems. Partition-DFT (PDFT) is a formally exact reformulation of KS-DFT in which the ground-state density and energy of a system are obtained through self-consistent calculations on isolated fragments, with a partition energy representing inter-fragment interactions. Here, we show how typical errors of the local density approximation (LDA) in KS-DFT can be largely suppressed through a simple approximation, the multi-fragment overlap approximation (MFOA), for the partition energy in PDFT. Our method is illustrated on simple models of one-dimensional strongly correlated linear hydrogen chains. The MFOA, when used in combination with the LDA for the fragments, improves LDA dissociation curves of hydrogen chains and produces results that are comparable to those of spin-unrestricted LDA, but without breaking the spin symmetry. MFOA also induces a correction to the LDA electron density that partially captures the correct density dimerization in strongly correlated hydrogen chains. Moreover, with an additional correction to the partition energy that is specific to the one-dimensional LDA, the approximation is shown to produce dissociation energies in quantitative agreement with calculations based on the density matrix renormalization group method.
We introduce ΣΣΣ-Attention, a Transformer-based operator-learning framework to address a key computational challenge in correlated materials. Our approach utilizes an Encoder-Only Transformer as an ansatz to approximate the self-energy operator of strongly correlated electronic systems. By creating a batched dataset that combines results from three complementary approaches: many-body perturbation theory, strong-coupling expansion, and exact diagonalization, each effective in specific parameter regimes, ΣΣΣ-Attention is applied to learn a universal approximation for the self-energy operator that is valid across all regimes. This hybrid strategy leverages the strengths of existing methods while relying on the transformer's ability to generalize beyond individual limitations. More importantly, the scalability of the Transformer architecture allows the learned self-energy to be extended to systems with larger sizes, leading to much improved computational scaling. Using the 1D Hubbard model, we demonstrate that ΣΣΣ-Attention can accurately predict the Matsubara Green's function and hence effectively captures the Mott transition at finite temperature. Our framework offers a promising and scalable pathway for studying strongly correlated systems with many possible generalizations.
Density functional theory (DFT) is routinely employed in material science and quantum chemistry to simulate weakly correlated electronic systems. Recently, deep learning (DL) techniques have been adopted to develop promising functionals for the strongly correlated regime. DFT can be applied to quantum spin models too, but functionals based on DL have not been developed yet. Here, we investigate DL-based DFTs for random quantum Ising chains, both with nearest-neighbor and up to next-nearest-neighbor couplings. Our neural functionals are trained and tested on data produced via the Jordan-Wigner transformation, exact diagonalization, and tensor-network methods. An economical gradient-descent optimization is used to find the ground-state properties of previously unseen Hamiltonian instances. Notably, our nonlocal functionals drastically improve upon the common local density approximations, and they are designed to be scalable, allowing us to simulate chain sizes much larger than those used for training. The prediction accuracy is analyzed, paying attention to the spatial correlations of the learned functionals and to the role of quantum criticality. Our findings indicate a suitable strategy to extend the reach of other computational methods with a controllable accuracy.