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arXiv:2509.12431

Neural-Quantum-States Impurity Solver for Quantum Embedding Problems

Zhou, Yinzhanghao, Tsung-Han Lee, Ao Chen, Nicola Lanatà, Hong Yu Guo

Published Sep 15, 2025• 0 citations
arXivHTMLPDF

Abstract

Neural quantum states (NQS) have emerged as a promising approach to solve second-quantized Hamiltonians, because of their scalability and flexibility. In this work, we design and benchmark an NQS impurity solver for the quantum embedding (QE) methods, focusing on the ghost Gutzwiller Approximation (gGA) framework. We introduce a graph transformer-based NQS framework able to represent arbitrarily connected impurity orbitals of the embedding Hamiltonian (EH) and develop an error control mechanism to stabilize iterative updates throughout the QE loops. We validate the accuracy of our approach with benchmark gGA calculations of the Anderson Lattice Model, yielding results in excellent agreement with the exact diagonalisation impurity solver. Finally, our analysis of the computational budget reveals the method's principal bottleneck to be the high-accuracy sampling of physical observables required by the embedding loop, rather than the NQS variational optimization, directly highlighting the critical need for more efficient inference techniques.

DOI
10.48550/arxiv.2509.12431

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Neural quantum states (NQS) have emerged as a promising approach to solve second-quantized Hamiltonians, because of their scalability and flexibility. In this work, we design and benchmark an NQS impurity solver for the quantum embedding (QE) methods, focusing on the ghost Gutzwiller Approximation (gGA) framework. We introduce a graph transformer-based NQS framework able to represent arbitrarily connected impurity orbitals of the embedding Hamiltonian (EH) and develop an error control mechanism to stabilize iterative updates throughout the QE loops. We validate the accuracy of our approach with benchmark gGA calculations of the Anderson Lattice Model, yielding results in excellent agreement with the exact diagonalisation impurity solver. Finally, our analysis of the computational budget reveals the method's principal bottleneck to be the high-accuracy sampling of physical observables required by the embedding loop, rather than the NQS variational optimization, directly highlighting the critical need for more efficient inference techniques.
To understand the intricate exchange between electrons of different bands in strongly correlated materials, it is essential to treat multi-orbital models accurately. For this purpose, dynamical mean-field theory (DMFT) provides an established framework, whose scope crucially hinges on the availability of efficient quantum impurity solvers. Here we present a real-frequency impurity solver based on neural quantum states (NQS) combined with an operator-Lanczos construction. NQS are an asymptotically unbiased variational ground-state ansatz that employs neural networks to capture long-range correlations on complicated graph structures. We leverage this ability to solve multi-orbital impurity problems using a systematically improvable Segmented Commutator Operator-Lanczos (SCOL) construction. Our benchmarks on both the single-orbital Anderson model and the multi-orbital Hubbard-Kanamori impurity Hamiltonian reveal excellent ground-state precision and the capacity to accurately resolve zero temperature spectral functions and self-energies. These results open avenues for extending DMFT to more challenging problems.
Transformer neural networks, known for their ability to recognize complex patterns in high-dimensional data, offer a promising framework for capturing many-body correlations in quantum systems. We employ an adapted vision transformer (ViT) architecture to model quantum impurity models, optimizing it with a subspace expansion scheme that surpasses conventional variational Monte Carlo in both accuracy and efficiency. Benchmarks against matrix product states in single- and three-orbital Anderson impurity models show that these ViT-based neural quantum states achieve comparable or superior accuracy with significantly fewer variational parameters. We further extend our approach to compute dynamical quantities by constructing a restricted excitation space that effectively captures relevant physical processes, yielding accurate core-level x-ray absorption spectra. These findings highlight the potential of ViT-based neural quantum states for accurate and efficient modeling of quantum impurity models.
Abstract To understand the intricate exchange between electrons of different bands in strongly correlated materials, it is essential to treat multi-orbital models accurately. For this purpose, dynamical mean-field theory (DMFT) provides an established framework, whose scope crucially hinges on the availability of efficient quantum impurity solvers. Here we present a real-frequency impurity solver based on neural quantum states (NQS) combined with an operator-Lanczos construction. NQS are an asymptotically unbiased variational ground-state ansatz that employs neural networks to capture long-range correlations on complicated graph structures. We leverage this ability to solve multi-orbital impurity problems using a systematically improvable Segmented Commutator Operator-Lanczos (SCOL) construction. Our benchmarks on both the single-orbital Anderson model and the multi-orbital Hubbard–Kanamori impurity Hamiltonian reveal excellent ground-state precision and the capacity to accurately resolve zero temperature spectral functions and self-energies. These results open avenues for extending DMFT to more challenging problems. Repository contents This repository contains the figures, data, and plotting script used to recreate Fig. 2 and Fig. 3 of “Operator Lanczos Approach enabling Neural Quantum States as Real Frequency Impurity Solvers”. The notebook reproduces spectral functions and self-energies for the Single Anderson Impurity Model (SIAM) and a Hubbard–Kanamori system with two and three orbitals (MO2, MO3). The files in this repsitory are: `plot.ipynb` – matplotlib script that reads the HDF5 files and generate the publication figures. `data/SIAM_plot.h5` – SIAM spectral/self-energy data grouped into three panels (`panel_left`, `panel_mid`, `panel_right`) with metadata for interaction strength `U`, segment depth `l_s`, and method `kind` (`NQS` or `NRG` -> for reference). `data/MO_plot.h5` – MO2 and MO3 results. `MO2` contains frequency `omega`, spectral function `A` and self-energy arrays (`omega_Sigma`, `Sigma_im_nqs` and NRG references). `MO3` holds NRG spectra plus frequncies `omega` and spectral function `A_nqs` for `l_s = 2,3,4`. `siam_square.pdf` – rendered SIAM figure (Fig. 2 in the paper). `mo.pdf` – rendered MO figure (Fig. 3 in the paper). Requirements Python 3 with `numpy`, `matplotlib`, and `h5py`. Jupyter (if you want to run the notebook interactively). A LaTeX installation (e.g., TeX Live) so matplotlib can render TeX labels (`text.usetex=True` is enabled). Install the Python packages locally (examples): pip install numpy matplotlib h5py jupyter Notes - The notebook uses Computer Modern fonts via LaTeX; if TeX is missing you can disable `text.usetex` inside the first code cell. - Data are already curated for plotting, no additional preprocessing is needed beyond loading the HDF5 datasets described above. - Tested versions (newer should also work): `numpy==1.26.4`, `matplotlib==3.10.0`, `h5py==3.13.0`, `jupyter==1.1.1`.
The ghost Gutzwiller ansatz (gGut) embedding technique was shown to achieve comparable accuracy to the gold standard dynamical mean-field theory method in simulating real material properties, yet at a much lower computational cost. Despite that, gGut is limited by the algorithmic bottleneck of computing the density matrix of the underlying effective embedding model, a quantity which must be converged within a self-consistent embedding loop. We develop a hybrid quantum-classical gGut technique which computes the ground state properties of embedding Hamiltonians with the help of a quantum computer, using the sample-based quantum-selected configuration interaction (QSCI) algorithm. We study the applicability of SCI-based methods to the evaluation of the density of states for single-band Anderson impurity models within gGut and find that such ground states of interest become sufficiently sparse in the CI basis as the number of ghost orbitals is increased. Further, we investigate the performance of QSCI using local unitary cluster Jastrow (LUCJ) variational quantum states in combination with a circuit cutting technique, prepared on IQM's quantum hardware for system sizes of up to 11 ghost orbitals, equivalent to 24 qubits. We report converged gGut calculations which correctly capture the metal-to-insulator phase transition in the Fermi-Hubbard model on the Bethe lattice by using quantum samples to build an SCI basis with as little as 1%1\%1% of the total CI basis states.
Quantum embedding methods, such as dynamical mean-field theory (DMFT), provide a powerful framework for investigating strongly correlated materials. A central computational bottleneck in DMFT is in solving the Anderson impurity model (AIM), whose exact solution is classically intractable for large bath sizes. In this work, we develop and benchmark a quantum-classical hybrid solver tailored for DMFT applications, using the variational quantum eigensolver (VQE) to prepare the ground state of the AIM with shallow quantum circuits. The solver uses a unified ansatz framework to prepare the particle and hole excitations of the ground-state from parameter-shifted circuits, enabling the reconstruction of the impurity Green's function through a continued-fraction expansion. We evaluate the performance of this approach across a few bath sizes and interaction strengths under noisy, shot-limited conditions. We compare three optimization routines (COBYLA, Adam, and L-BFGS-B) in terms of convergence and fidelity, assess the benefits of estimating a quantum-computed moment (QCM) correction to the variational energies, and benchmark the approach by comparing the reconstructed density of states (DOS) against that obtained using a classical pipeline. Our results demonstrate the feasibility of Green's function reconstruction on near-term devices and establish practical benchmarks for quantum impurity solvers embedded within self-consistent DMFT loops.
Quantum impurity solvers are the computational bottleneck of quantum embedding approaches to correlated materials, such as dynamical mean-field theory (DMFT). We show that neural networks trained on synthetic, material-agnostic data learn the impurity mapping from hybridization functions and local interactions to Green's functions with quantitative accuracy for both model systems and real materials, providing fast solvers for single- and multi-orbital models. Benchmarks against numerically controlled quantum Monte Carlo show that the method reproduces the Mott transition, multi-orbital phase diagrams of Hubbard-Kanamori models, and the electronic properties of SrVO3_33​ and SrMnO3_33​. The learned solvers achieve orders-of-magnitude speedup and can initialize controlled calculations, dramatically accelerating DMFT while preserving accuracy.