Quantum-enhanced neural exchange-correlation functionals

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi mathvariant="normal">H</a:mi> <a:mn>2</a:mn> </a:msub> </a:math> and planar <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi mathvariant="normal">H</c:mi> <c:mn>4</c:mn> </c:msub> </c:math> 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, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:msub> <e:mi mathvariant="normal">H</e:mi> <e:mn>2</e:mn> </e:msub> <e:msub> <e:mi mathvariant="normal">H</e:mi> <e:mn>2</e:mn> </e:msub> </e:mrow> </e:math> , 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.