A Scalable Configuration-Interaction Impurity Solver via Active Learning

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.

• The paper introduces AL-ATCI, which leverages active learning to pinpoint key determinants in finite-Hamiltonian impurity solvers.
• It employs a random-forest classifier to pre-screen and rank determinants, significantly reducing the Hamiltonian diagonalization workload.
• The approach achieves reliable ED-level accuracy with fewer determinants, enabling efficient large-bath and multi-orbital DMFT simulations.

Active-Learning Configuration-Interaction Impurity Solver: Principles and Performance

Introduction

The exponential scaling of Hilbert space remains the primary bottleneck for finite-Hamiltonian impurity solvers—those that diagonalize explicit, finite Anderson models within DMFT frameworks—especially as the number of bath or orbital degrees of freedom increases. This paper introduces an active learning extension of adaptive-truncation configuration interaction (AL-ATCI), addressing this challenge by employing machine learning techniques to identify the physically relevant subspace of high-importance Slater determinants. The primary findings are: (1) the computational complexity of AL-ATCI grows only weakly with bath or orbital dimension, and (2) $N_{\text{query}}$ provides a direct, systematically improvable handle on solution accuracy in the absence of external benchmarks.

AL-ATCI Algorithm Overview

AL-ATCI builds upon the adaptive-truncation configuration interaction (ATCI) solver, itself an iterative selected CI method that employs reference configurations and particle-hole substitutions in conjunction with natural orbital transformations. In AL-ATCI, a random-forest classifier is trained during the iterative solver cycle using previously accumulated data to predict the importance of candidate determinants, which are then ranked and the top $N_{\text{query}}$ retained for Hamiltonian construction and diagonalization. This pre-screening step sharply reduces the dimension of the matrix diagonalization task, the dominant computational cost in the ED/CI context.

Figure 2: AL-ATCI flowchart, indicating the classifier training, determinant ranking, and selective querying loop.

The classifier operates on binary occupation representations, and importance labels are determined by wavefunction coefficients exceeding a prescribed threshold. Careful dataset management, symmetry partner construction, and class-imbalance corrections ensure stability and generalization within DMFT cycles.

Systematic Control and Convergence

The query size $N_{\text{query}}$ constitutes a direct and fine-grained convergence knob for AL-ATCI, enabling systematic approach to ED-level accuracy across both static and dynamical observables. For the one-dimensional half-filled Hubbard model, clear monotonic convergence of total energy, occupancies, and (matrix-valued) self-energy is observed as $N_{\text{query}}$ increases, with best agreement achieved for orders-of-magnitude fewer determinants than the full Hilbert space.

Figure 3: Convergence of various observables (energy, occupancy, self-energy) with increasing $N_{\text{query}}$ in a cluster DMFT setup.

This behavior is robust to physical regime—convergence in insulators (large $U/t$) is achieved at lower $N_{\text{query}}$ than in metals, consistent with the smaller entanglement and more compact determinant manifolds in insulating ground states.

Scaling and Computational Cost

A pronounced advantage of AL-ATCI is its weak scaling with bath size $N_b$. Unlike ATCI and conventional ED—which see exponential increases in determinant counts and wall-clock time as bath orbitals are added—AL-ATCI exhibits markedly slower growth, essentially decoupling the cost of Hamiltonian enlargement from the nominal orbital count. The physically relevant manifold, as discovered by active learning, expands far more slowly than the total determinant space.

Figure 4: Scaling of selected determinant count and computational cost with cluster ($N_c$) and bath size ($N_b$) for ATCI and AL-ATCI.

This enables regime access (e.g., large $N_{\text{query}}$0 cellular DMFT, large $N_{\text{query}}$1 for hybridization-fit convergence, or inclusion of spectroscopically relevant noninteracting orbitals) that is otherwise prohibitive for standard CI methods.

Timing analyses indicate the speedup emerges from the reduction in ground-state diagonalization size. Overheads incurred from classifier training and prediction are negligible by comparison.

Figure 6: Breakdown of cumulative wall-clock time as a function of bath size, illustrating the dominance and suppression of diagonalization time with AL-ATCI.

Physical Benchmarks: Cluster and Multi-Orbital Calculations

One-Dimensional Hubbard Model

Applying AL-ATCI to the 1D Hubbard model within cluster DMFT, the solver achieves ED-equivalent spectral functions for clusters as large as $N_{\text{query}}$2, with significantly reduced computational effort and memory.

Figure 1: Spectral function evolution with increasing cluster size, showing improved short-range correlation capture and approach to Bethe-ansatz spinon/holon bands as $N_{\text{query}}$3 increases.

Momentum-resolved spectra illustrate systematic convergence and accurate reproduction of Mott gaps, particle-hole symmetry, and fine structure absent at low cluster sizes.

Figure 8: Momentum-resolved spectral function $N_{\text{query}}$4 for small clusters, documenting approach to correct spectral symmetry and gap.

Figure 10: Convergence of $N_{\text{query}}$5 at larger clusters $N_{\text{query}}$6, showing near-indistinguishable spectra with increasing $N_{\text{query}}$7.

Sr$N_{\text{query}}$8RuO$N_{\text{query}}$9 Three-Orbital Problem

AL-ATCI is also applied to Sr$N_{\text{query}}$0RuO$N_{\text{query}}$1 with a fully rotationally invariant three-orbital impurity model—a regime intractable for ED at moderate bath size. The rapid decay of CI coefficient weights across all $N_{\text{query}}$2 demonstrates the compressibility of the many-body wavefunction and justifies truncated Hilbert space techniques. Convergence of self-energy and spectral functions with bath size is achieved up to $N_{\text{query}}$3.

Figure 11: Self-energy convergence with query size and CI coefficient weight decay for increasing $N_{\text{query}}$4 in the three-orbital impurity problem.

Figure 5: Sr$N_{\text{query}}$5RuO$N_{\text{query}}$6 spectral function for $N_{\text{query}}$7 to $N_{\text{query}}$8, showing systematic convergence and robustness to bath discretization.

Implications and Future Directions

The paper establishes that the physically relevant determinant manifold for correlated impurity solvers does not scale exponentially with one-particle orbital dimension in physically meaningful regimes, instead being dictated by underlying correlation structure and entanglement. For cluster extensions/truncation-based solvers, this is a decisive shift: large-bath, enlarged orbital, or multi-orbital calculations become accessible while retaining the real-frequency, sign-problem-free advantages of finite-Hamiltonian approaches.

The $N_{\text{query}}$9 parameter provides not merely computational control, but a practical error bar and internal convergence metric in the absence of external benchmarks—essential for realistic material calculations where exact solutions are lacking.

Beyond quantum impurity models, this paradigm of actively discovering and compressing the physically relevant many-body space with machine learning can be extended to more general correlated lattice problems, quantum chemistry, and spectroscopic simulations (e.g., large-orbital RIXS/XAS).

Conclusion

AL-ATCI marks a significant advance in the tractability of configuration-interaction-based impurity solvers for DMFT and related quantum embedding frameworks. By introducing an active learning loop that systematically targets the subspace of relevant Slater determinants, the method decouples computational complexity from nominal orbital count and renders large-bath and multi-orbital DMFT calculations feasible without loss of real-frequency precision. This scalability, supported by extensive numerical validation across single- and multi-band regimes, positions AL-ATCI as an essential tool for advancing the frontier of numerically exact many-body quantum solvers.