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arXiv:2604.10230

A Scalable Configuration-Interaction Impurity Solver via Active Learning

Jeongmoo Lee, Ara Go

Published Apr 11, 2026• 0 citations
arXivHTMLPDF

Abstract

Finite-Hamiltonian impurity solvers provide direct real-frequency spectra and a natural route to enlarged impurity Hamiltonians, but their applicability is limited by the rapid Hilbert-space growth with the number of bath or other added one-particle orbitals. We introduce an active-learning extension of adaptive-truncation configuration interaction (AL-ATCI) that identifies the determinant manifold relevant to the correlated state. The approximation is systematically controlled by the query size N_query, which also provides an internal convergence parameter when no external benchmark is available. Over the benchmark range studied here, the computational cost grows only weakly with bath size, because enlarging the bath mainly expands the combinatorial determinant space rather than the physically relevant manifold. In dynamical mean-field-theory benchmarks for the one-dimensional Hubbard model, AL-ATCI reproduces exact-diagonalization accuracy and extends cellular calculations to clusters as large as N_c = 10. For a three-orbital rotationally invariant Sr2RuO4 impurity problem, we demonstrate systematic convergence of dynamical quantities and a highly compressed configuration space as N_b is increased from 9 to 18. These results substantially alleviate the bath-discretization bottleneck of exact-diagonalization- and configuration-interaction-based impurity solvers and make large-bath and enlarged-orbital calculations more practical.

DOI
10.48550/arxiv.2604.10230

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Finite-Hamiltonian impurity solvers provide direct real-frequency spectra and a natural route to enlarged impurity Hamiltonians, but their applicability is limited by the rapid Hilbert-space growth with the number of bath or other added one-particle orbitals. We introduce an active-learning extension of adaptive-truncation configuration interaction (AL-ATCI) that identifies the determinant manifold relevant to the correlated state. The approximation is systematically controlled by the query size N_query, which also provides an internal convergence parameter when no external benchmark is available. Over the benchmark range studied here, the computational cost grows only weakly with bath size, because enlarging the bath mainly expands the combinatorial determinant space rather than the physically relevant manifold. In dynamical mean-field-theory benchmarks for the one-dimensional Hubbard model, AL-ATCI reproduces exact-diagonalization accuracy and extends cellular calculations to clusters as large as N_c = 10. For a three-orbital rotationally invariant Sr2RuO4 impurity problem, we demonstrate systematic convergence of dynamical quantities and a highly compressed configuration space as N_b is increased from 9 to 18. These results substantially alleviate the bath-discretization bottleneck of exact-diagonalization- and configuration-interaction-based impurity solvers and make large-bath and enlarged-orbital calculations more practical.
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We extend a previously proposed rotation and truncation scheme to optimize quantum Anderson impurity calculations with exact diagonalization [Y. Lu, M. H\"oppner, O. Gunnarsson, and M. W. Haverkort, Phys. Rev. B 90, 085102 (2014)] to density-matrix renormalization group (DMRG) calculations. The method reduces the solution of a full impurity problem with virtually unlimited bath sites to that of a small subsystem based on a natural impurity orbital basis set. The later is solved by DMRG in combination with a restricted-active-space truncation scheme. The method allows one to compute Green's functions directly on the real frequency or time axis. We critically test the convergence of the truncation scheme using a one-band Hubbard model solved in the dynamical mean-field theory. The projection is exact in the limit of both infinitely large and small Coulomb interactions. For all parameter ranges, the accuracy of the projected solution converges exponentially to the exact solution with increasing subsystem size.
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We here present how a self-consistent solution of the dynamical mean field theory equations can be obtained using exact diagonalization of an Anderson impurity model with accuracies comparable to those found using renormalization group or quantum Monte Carlo methods. We show how one can solve a correlated quantum impurity coupled to several hundred uncorrelated bath sites, using a restricted active basis set. The number of bath sites determines the resolution of the obtained spectral function, which consists of peaks roughly spaced by the band width divided by the number of bath sites. The self-consistency cycles are fully performed on the real frequency axis and expressed as numerical stable matrix operations. The same impurity solver has been used on Ligand Field and finite size cluster calculations and is capable of treating involved Hamiltonians including the full rotational invariant Coulomb interaction, spin-orbit coupling and low symmetry crystal-fields. The proposed method allows for the calculation of a variety of correlation functions at little extra cost.
Quantum impurity models, which describe the coupling between interacting orbitals and a non-interacting bath, play a central role in the physics of strongly correlated electron systems. Solving a quantum impurity model in general requires the use of non-perturbative numerical methods. Hamiltonian-based approaches, which rely on an explicit bath discretization, are typically limited to a small number of bath sites or small entanglement, and diagrammatic methods suffer from sign problems, slow convergence, or diagram truncation approximations. Here we show that these two classes of methods can be combined: augmenting diagrammatic methods with a small auxiliary bath can reduce the residual problem to a regime where low-order perturbation theory is highly accurate and rapidly converging. In a simple benchmark, the precision of the hybrid approach surpasses bold-line calculations by several orders of magnitude; for a strongly interacting two-orbital model with a severe sign problem, convergence is achieved at three orders of magnitude lower computational cost than competing methods; and convergence to the unknown exact result is rapidly accelerated in a difficult realistic problem. Our results establish a practical route to high-precision quantum impurity solutions in correlated quantum systems.
High Resolution Image Download MS PowerPoint Slide Strongly correlated molecules remain out of reach for most electronic-structure solvers because the exact wave function spans a determinant space that grows exponentially with the number of orbitals. State-of-the-art selected-CI (e.g., CIPSI and HCI) and ML-CI methods mitigate search by exploiting Hamiltonian sparsity and learned prescreening. Nonetheless, in strongly multireference regimes, the excitation-based candidate lists can grow rapidly, which may increase memory and screening costs. We present the Hamiltonian-guided autoregressive selected-configuration interaction (HAAR-SCI), a learn–sample–compress workflow that runs on a single GPU. A gated Transformer samples determinants autoregressively; Gumbel Top-K noise encourages exploration, and GPU min-heap kernels keep only configurations with the largest Hamiltonian couplings. The network is retrained after each expansion, and iterations stop when successive energies differ by ≤1 mHa. Across an 18-molecule benchmark set reaching 116 spin–orbitals, HAAR-SCI attains a mean absolute error of 0.51 mHa while using much fewer determinants than heat-bath CI. It traces the entire N 2 dissociation curve within 0.67 mHa and achieves HCI accuracy for the 40-spin-orbital [Fe 2 S 2 (SCH 3 ) 4 ] 2– cluster with a determinant reduction of 72%, demonstrating its power on systems considered intractable for conventional selected-CI solvers. A final probability-mass pruning compresses typical wave functions by a further 10–50×, retaining <0.01% of the Hilbert space yet still capturing >99.9% of the correlation energy. HAAR-SCI thus offers a compact and truly scalable route to chemical-accuracy quantum chemistry on commodity hardware.