  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.
