Quantum-Enhanced Neural Exchange-Correlation Functionals

TL;DR

The paper tackles the challenge of learning universal exchange-correlation functionals for KS-DFT by integrating quantum neural networks into differentiable KS-DFT frameworks. It introduces diverse architectures, including fully global QNNs and hybrid quantum-classical models, and demonstrates that quantum-enhanced XC functionals can achieve chemical accuracy for small 1D and 3D systems with far fewer parameters than comparable classical models. FO further improves 3D performance, enabling better extrapolation and stability in strongly correlated regimes. The work provides theoretical error bounds for SCF stability under approximate XC functionals and offers practical resource estimates for quantum hardware, outlining a path toward quantum-enabled, differentiable XC design within KS-DFT and the QEX ecosystem.

Abstract

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 (DQCs) as quantum (hybrid) models for XC in KS-DFT, implemented across various architectures. We assess their performance on 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. 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.

Figures (11)

• Figure 1: (1) Electron density of a molecule is expressed on an integration grid. (2) The neural XC functional takes $L$ neighbouring values of density as input. (3) The output of such functional can be interpreted as local or total XC energy, with respective embeddings in KS-DFT (Eqs. \ref{['eq:ksr-xc']} and \ref{['eq:gl-xc']}). With the global approach, only $\mathcal{O}(1)$ calls to the functional is necessary to obtain XC energy instead of $\mathcal{O}(N_{\mathrm{grid}})$.
• Figure 2: Differentiable chemistry framework for training quantum-enhanced XC functionals. (1) The molecular dataset consists of reference energies $E_{\mathrm{ref}}$ and densities $\mathbf{n}_{\mathrm{ref}}$ generated by an accurate ab initio method. (2) A differentiable (Q)NN $f_{\theta}$ outputs the XC energy. The corresponding potential is obtained via (automatic) differentiation. The Fock equations in KS-DFT are iteratively solved to obtain new orbitals until self-consistency. Each KS iteration contributes to the loss $\mathcal{L}$, resulting in regularization. ksr Trainable parameters $\theta$ are updated by a classical optimizer. (3) The converged parameters $\theta^{*}$ define the trained functional $f_{\theta^{*}}$, which is subsequently used to evaluate the properties of any molecule in stage (4). Brown arrows in (2) indicate the gradient flow through the procedure. CPU and QPU denote classical (i.e., central and graphical) and quantum processing units.
• Figure 3: (a) Method for training quantum-enhanced neural XC functionals within KS-DFT. Data describing a single molecule is passed to KS-DFT. An initial electron density $\mathbf{n}$ estimate is made and potentials $v, v_{\mathrm{H}}$ and $v_{\mathrm{XC}}$ are computed to construct the Fock operator. Neural XC functional $f_{\theta}$ is represented by a quantum and/or classical model, which takes $L$ density values as inputs (i.e., $L=1$ fully local and $L=N_{\mathrm{grid}}$ global). The XC functional can be embedded in DFT locally (outputs the values on every grid point) or globally (outputs the total XC energy). Then, the KS equation and the orbital coefficients $\mathbf{p}$ are updated. The new density is computed and if the change is small enough, the procedure is stopped (self-consistency) and the trained XC functional is obtained. Otherwise, the new orbitals are kept for the next iteration. The loss function is computed and differentiated with respect to the model parameters. The red dashed arrows show the gradient flow. (b) Building blocks of neural XC functionals are defined as compositions of differentiable functions, which can be represented by QNNs. (c) Architectures of classical and (d) quantum (hybrid) XC functionals. Models differ by how many inputs can be taken at a time. The QiCNN architecture allows one to chain classical and quantum models. The orange color indicates that the global energy embedding (see (a)) is used, and the blue color indicates quantum operations. Note that MLPs can be replaced by any classical architecture.
• Figure 4: Results of training neural XC functionals on different geometries of hydrogen molecule in 1D KS-DFT framework ksr. The grey area highlights results within chemical precision ($\leq 1.6$ mHa) and vertical dashed lines mark the training data. Local models improve upon the LDA shape, but still produce a poor qualitative agreement with the reference. Best results are obtained for global models when the whole density is considered as model input. The XC models are defined in Section \ref{['sec:models']}. All neural functionals are trained on eight training data points (circles) and the final model parameters are selected according to the lowest loss on the validation results. The reference (Ref) results are generated using DMRG by Li et al.ksr
• Figure 5: Results of training neural XC functionals on different geometries of H$_2$ in our 3D RKS-DFT framework based on PySCFAD. The same Global QiCNN (Eq. \ref{['eq:qicqnn']}) is trained with and without fractional orbital occupation (FO). Despite the limited training data (3 points), the model accurately extrapolates to distant geometries, demonstrating the strong regularization capacity of our 3D framework. The reference (Ref) results are generated using the FCI/6-31G level of theory.

Theorems & Definitions (4)

• Theorem 1: Iterative SCF Stability
• proof
• Lemma 1: QNN Lipschitz continuity
• proof