A Joint Quantum Computing, Neural Network and Embedding Theory Approach for the Derivation of the Universal Functional
Martin J. Uttendorfer, Daniel Barragan-Yani, Matthias Sperl, Marc Landmann
TL;DR
The paper introduces a quantum-classical framework that derives a universal functional for interacting quantum systems by fusing Reduced Density Matrix Functional Theory (RDM-FT) with Density Matrix Embedding Theory (DMET) and a neural-network surrogate. It leverages variational quantum eigensolver (VQE) data to train a deep neural network surrogate $\mathcal{F}_{\text{DNN}}$ that, via a Legendre transform $\mathcal{F}_{\text{RDM}}[\gamma]=E_0(h)-\sum_{ij} h_{ij}\gamma_{ij}$ and Hellmann–Feynman relations, reproduces the full functional information including off-diagonal terms; this enables a transferable embedding framework $E_0^{\text{emb}}=\min_{\gamma}\{\mathcal{F}_{\text{DNN}}[\gamma]+\sum_{i,j\in\text{emb}} h^{\text{emb}}_{ij}\gamma^{\text{emb}}_{ij}\}$ within DMET. The FT-DMET approach is demonstrated on Bose-Hubbard and Fermi-Hubbard models, including a two-band fermionic case, showing energies and observables in good agreement with exact or RHF baselines and illustrating robustness to noisy quantum data. This framework promises a cumulative quantum advantage by reusing a learned universal functional across systems with identical interactions, while keeping quantum resources manageable and scalable for larger fragments and molecules, aided by potential error mitigation and advanced neural architectures in future work.
Abstract
We introduce a novel approach that exploits the intersection of quantum computing, machine learning and reduced density matrix functional theory to leverage the potential of quantum computing to improve simulations of interacting quantum particles. Our method focuses on obtaining the universal functional using a deep neural network trained with quantum algorithms. We also use fragment-bath systems defined by density matrix embedding theory to strengthen our approach by substantially expanding the space of Hamiltonians for which the obtained functional can be applied without the need for additional quantum resources. Given the fact that once obtained, the same universal functional can be reused for any system where the interactions within the embedded fragment are identical, our work demonstrates a way to potentially achieve a cumulative quantum advantage within quantum computing applications for quantum chemistry and condensed matter physics.