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arXiv:2110.04269 · Physical Review C

Fast emulation of quantum three-body scattering

Xilin Zhang, R. J. Furnstahl

Published Jun 28, 2022• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevc.105.064004

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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{10}^{\ensuremath{-}13}10\ensuremath−13 to 10\ensuremath−4{10}^{\ensuremath{-}4}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.
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.
Quantum Monte Carlo simulations are powerful and versatile tools for the quantum many-body problem. In addition to the usual calculations of energies and eigenstate observables, quantum Monte Carlo simulations can in principle be used to build fast and accurate many-body emulators using eigenvector continuation or design time-dependent Hamiltonians for adiabatic quantum computing. These new applications require something that is missing from the published literature, an efficient quantum Monte Carlo scheme for computing the inner product of ground state eigenvectors corresponding to different Hamiltonians. In this work, we introduce an algorithm called the floating block method, which solves the problem by performing Euclidean time evolution with two different Hamiltonians and interleaving the corresponding time blocks. We use the floating block method and nuclear lattice simulations to build eigenvector continuation emulators for energies of ^{4}He, ^{8}Be, ^{12}C, and ^{16}O nuclei over a range of local and nonlocal interaction couplings. From the emulator data, we identify the quantum phase transition line from a Bose gas of alpha particles to a nuclear liquid.
The development of emulators for the evaluation of many-body observables has gained increasing attention over the last years. In particular the framework of eigenvector continuation (EC) has been identified as a powerful tool when the Hamiltonian admits for a parametric dependence. By training the emulator on a set of training data the many-body solution for arbitrary parameter values can be robustly predicted in many cases. Furthermore, it can be used to resum perturbative expansions that otherwise diverge. In this work, we apply EC to the pairing Hamiltonian and show that EC-resummed perturbation theory is in qualitative agreement with the exact solution and that EC-based emulators robustly predict the ground-state energy once the training data are chosen appropriately. In particular the phase transition from the normal to the superfluid regime is quantitatively predicted using a very low number of training points. Published by the American Physical Society 2024
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.
An atomic nucleus is a quantum system of interacting nucleons and constitutes a problem difficult to solve exactly. For this reason, a diversity of approximate resolution methods has been designed, and Projected Generator Coordinate Method (PGCM) is one of them. The strong point of PGCM is to construct a physically inspired small dimensional space, in which an approximate solution of the nuclear many-body problem is easily found. However the numerical cost of PGCM space computation make this method inadapted for sensibility analysis of nuclear observables with restect to parametrisation of the interaction model, this analysis requiring an huge number of PGCM computations. In order to make this type of study possible, this thesis explore the concept of PGCM emulator. In this work, a combination of PGCM with Eigenvector Continuation (EC) is constructed and studied. This combination (the PGCM-EC emulator) takes advantage of mathematical similarities between PGCM and EC, and above all of the decomposition of the hamiltonian as a linear combination of parameter-independent terms. The latter property is used to concentrate the heavier numerical cost in the computation of parameter-independent quantities (the elementary kernels), and open the feasability of massive PGCM emulations, the price being having first-handedly computed the costly elementary kernels. Limits of the emulator are also explored, by introducing the concept of over-training, which is exactly a consequence of the aproximativeness of a PGCM computation. Eventually this thesis demonstrates the possibility to emulate millions of PGCM computations with an error on collective spectroscopy less than 3%, and with a low numerical cost fraction of 1% of the million PGCM calculations cost.
Emulating the Green's function using reduced-order equations is challenging because of the amount of information it contains. This information includes the one-particle spectroscopy over the entire energy spectrum, the one-body density matrix, and the ground-state energy. Nevertheless, the computational speedup offered by emulators is essential for applications requiring repeated model evaluations, which are ubiquitous in many-body physics. These include parameter calibration, uncertainty quantification, phase-diagram mapping, and geometry optimization. In this Letter, we introduce a systematically improvable emulator based on eigenvector continuation and moment-matching. The resulting approximation, which we refer to as an ``eigenvector-continued Green's function,'' can be obtained by solving linear equations, enabling an efficient separation of the computational workload into offline and online stages.