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arXiv:2604.25792

Exact emulation of few-body systems at low cost

Sven Heihoff, Arseniy A. Filin, E. Epelbaum

Published Apr 28, 2026• 0 citations
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Abstract

Effective field theories have established themselves as key pillars of modern nuclear physics. They enable a quantitative understanding of the strong nuclear force, provided low-energy constants that parametrize short-distance physics can be determined from experimental data. This, however, often becomes prohibitively expensive due to a significant computational cost of solving the A-body problem. The computational challenge is particularly severe for three-body forces, which are at the frontier of nuclear and atomic physics and play an important role in the equation of state of neutron stars. Here we prove that for a parametric low-rank update of a Hamiltonian, the A-body problem at a fixed energy exactly reduces to a low-dimensional matrix equation regardless of the size of the Hilbert space. As a proof-of-principle, we present exact and computationally cheap snapshot-based emulators for few-body scattering and bound states. Unlike alternatives, our emulators can be used far away from the snapshot region without loss of precision and yield accurate results for parameter values not accessible using conventional solution techniques. Our approach is not restricted by the interaction type, number of particles, and methods for generating snapshots and can be applied to mitigate the computational burden of the A-body problem to a broad class of problems in nuclear, atomic, and molecular physics.

DOI
10.48550/arxiv.2604.25792

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Effective field theories have established themselves as key pillars of modern nuclear physics. They enable a quantitative understanding of the strong nuclear force, provided low-energy constants that parametrize short-distance physics can be determined from experimental data. This, however, often becomes prohibitively expensive due to a significant computational cost of solving the A-body problem. The computational challenge is particularly severe for three-body forces, which are at the frontier of nuclear and atomic physics and play an important role in the equation of state of neutron stars. Here we prove that for a parametric low-rank update of a Hamiltonian, the A-body problem at a fixed energy exactly reduces to a low-dimensional matrix equation regardless of the size of the Hilbert space. As a proof-of-principle, we present exact and computationally cheap snapshot-based emulators for few-body scattering and bound states. Unlike alternatives, our emulators can be used far away from the snapshot region without loss of precision and yield accurate results for parameter values not accessible using conventional solution techniques. Our approach is not restricted by the interaction type, number of particles, and methods for generating snapshots and can be applied to mitigate the computational burden of the A-body problem to a broad class of problems in nuclear, atomic, and molecular physics.
Recent advances in both theoretical and computational methods have enabled large-scale, precision calculations of the properties of atomic nuclei. With the growing complexity of modern nuclear theory, however, also comes the need for novel methods to perform systematic studies and quantify the uncertainties of models when confronted with experimental data. This study presents an application of such an approach, the reduced basis method, to substantially lower computational costs by constructing a significantly smaller Hamiltonian subspace informed by previous solutions. Our method shows comparable efficiency and accuracy to other dimensionality reduction techniques on an artificial three-body bound system while providing a richer representation of physical information in its projection and training subspace. This methodological advancement can be applied in other contexts and has the potential to greatly improve our ability to systematically explore theoretical models and thus enhance our understanding of the fundamental properties of nuclear systems.
We develop a class of emulators for solving quantum three-body scattering problems. They are based on combining the variational method for scattering observables and the recently proposed eigenvector continuation concept. The emulators are first trained by the exact scattering solutions of the governing Hamiltonian at a small number of points in its parameter space, and then employed to make interpolations and extrapolations in that space. Through a schematic nuclear-physics model with finite-range two and three-body interactions, we demonstrate the emulators to be extremely accurate and efficient. The computing time for emulation is on the scale of milliseconds (on a laptop), with relative errors ranging from ${10}^{\ensuremath{-}13}$ to ${10}^{\ensuremath{-}4}$ depending on the case. The emulators also require little memory. We argue that these emulators can be generalized to even more challenging scattering problems. Furthermore, this general strategy may be applicable for building the same type of emulators in other fields, wherever variational methods can be developed for evaluating physical models.
In recent years, reduced-basis methods (RBMs) have been adapted to the many-body eigenvalue problem and they have been used, largely in nuclear physics, as fast emulators able to bypass expensive direct computations while still providing highly accurate results. This work is meant to show that the RBM is an efficient and accurate emulator for the strong correlations induced by the pairing interaction in a variety of finite systems like ultrasmall superconducting grains, interacting topological superfluids, and mesoscopic hybrid superconductor-semiconductor devices, all of which require an expensive, beyond-mean-field, particle-number conserving description. These systems are modeled by the number-conserving Richardson pairing Hamiltonian and its appropriate generalizations. Their ground state is solved for exactly using the density matrix renormalization group. The reduced basis is assembled iteratively from a small number of exact ground-state vectors, well-chosen from across the relevant parameter space using a fast estimate of the emulation error and a greedy local optimization algorithm. The RBM emulation is found to accurately describe the weak-to-strong pairing crossover in small grains, the third-order topological phase transition of the interacting Richardson-Kitaev chain, and the complex charge stability diagram of a hybrid quantum dot--superconductor device. For all considered systems the emulation error decreases exponentially with the reduced-basis dimensionality. The number of basis vectors necessary to reach a fixed emulation accuracy only shows a modest linear growth with increasing system size. RBMs are confirmed to be cheap and accurate emulators for the widely encountered superconducting phenomena. Capable of providing orders of magnitude computational speedup with respect to approaches based only on traditional many-body solvers, they open new possibilities in building and solving models of interacting many-body systems and in better interfacing them with experimental design and data analysis.
We present the reduced basis method as a tool for developing emulators for equations with tunable parameters within the context of the nuclear many-body problem. The method uses a basis expansion informed by a set of solutions for a few values of the model parameters and then projects the equations over a well-chosen low-dimensional subspace. We connect some of the results in the eigenvector continuation literature to the formalism of reduced basis methods and show how these methods can be applied to a broad set of problems. As we illustrate, the possible success of the formalism on such problems can be diagnosed beforehand by a principal component analysis. We apply the reduced basis method to the one-dimensional Gross-Pitaevskii equation with a harmonic trapping potential and to nuclear density functional theory for 48Ca^{48}\mathrm{Ca}48Ca, achieving speed-ups of more than $\ifmmode\times\else\texttimes\fi{}150$ and $\ifmmode\times\else\texttimes\fi{}250$, respectively, when compared to traditional solvers. The outstanding performance of the approach, together with its straightforward implementation, show promise for its application to the emulation of computationally demanding calculations, including uncertainty quantification.
First principles calculations of atomic nuclei based on microscopic nuclear forces derived from chiral effective field theory (EFT) have blossomed in the past years. A key element of such ab initio studies is the understanding and quantification of systematic and statistical errors arising from the omission of higher-order terms in the chiral expansion as well as the model calibration. While there has been significant progress in analyzing theoretical uncertainties for nucleon-nucleon scattering observables, the generalization to multi-nucleon systems has not been feasible yet due to the high computational cost of evaluating observables for a large set of low-energy couplings. In this Letter we show that a new method called eigenvector continuation (EC) can be used for constructing an efficient and accurate emulator for nuclear many-body observables, thereby enabling uncertainty quantification in multi-nucleon systems. We demonstrate the power of EC emulation with a proof-of-principle calculation that lays out all correlations between bulk ground-state observables in the few-nucleon sector. On the basis of ab initio calculations for the ground-state energy and radius in 4He, we demonstrate that EC is more accurate and efficient compared to established methods like Gaussian processes.
Background: The computational resources needed to generate the ab initio solution of the nuclear many-body problem for increasing mass number and/or accuracy necessitates innovative developments to improve upon (i) the storage of many-body operators and (ii) the scaling of many-body methods used to evaluate nuclear observables. The storing and efficient handling of many-body operators with high particle ranks is currently one of the major bottlenecks limiting the applicability range of ab initio studies with respect to mass number and accuracy. Recently, the application of tensor decomposition techniques to many-body tensors has proven highly beneficial to reduce the computational cost of ab initio calculations in quantum chemistry and solid-state physics.Purpose: The impact of applying state-of-the-art tensor factorization techniques to modern nuclear Hamiltonians derived from chiral effective field theory is investigated. Subsequently, the error induced by the tensor decomposition of the input Hamiltonian on ground-state energies of closed-shell nuclei calculated via second-order many-body perturbation theory is benchmarked.Methods: The first proof-of-principles application of tensor-decomposition techniques to the nuclear Hamiltonian is performed. Two different tensor formats are investigated by systematically benchmarking the approximation error on matrix elements stored in various bases of interest. The analysis is achieved while including normal-ordered three-nucleon interactions that are nowadays used as input to the most advanced ab initio calculations in medium-mass nuclei. With the aid of the factorized Hamiltonian, the second-order perturbative correction to ground-state energies is decomposed and the scaling properties of the underlying tensor network are discussed.Results: The employed tensor formats are found to lead to an efficient data compression of two-body matrix elements of the nuclear Hamiltonian. In particular, the sophisticated tensor hypercontraction scheme yields low tensor ranks with respect to both harmonic-oscillator and Hartree-Fock single-particle bases. It is found that the tensor rank depends on the two-body total angular momentum JJJ for which one performs the decomposition, which is itself directly related to the sparsity of the corresponding tensor. Furthermore, including normal-ordered two-body contributions originating from three-body interactions does not compromise the efficient data compression. Ultimately, the use of factorized matrix elements authorizes controlled approximations of the exact second-order ground-state energy corrections. In particular, a small enough error is obtained from low-rank factorizations in 4He,16O^{4}\mathrm{He},^{16}\mathrm{O}4He,16O, and 40Ca^{40}\mathrm{Ca}40Ca.Conclusions: It is presently demonstrated that tensor-decomposition techniques can be efficiently applied to systematically approximate the nuclear many-body Hamiltonian in terms of lower-rank tensors. Beyond the input Hamiltonian, tensor-decomposition techniques can be envisioned to scale down the cost of state-of-the-art nonperturbative many-body methods in order to extend ab initio studies to (i) higher precisions, (ii) larger masses, and (iii) nuclei of doubly open-shell character.