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Few-Body Systems

The Steepest Slope toward a Quantum Few-Body Solution: Gradient Variational Methods for the Quantum Few-Body Problem

Paolo Recchia, Debabrota Basu, M. Gattobigio, Christian Miniatura, Stéphane Bressan

Published Nov 18, 2024• 1 citation
Publisher

Abstract

Abstract Quantum few-body systems are deceptively simple. Indeed, with the notable exception of a few special cases, their associated Schrödinger equation cannot be solved analytically for more than two particles. One has to resort to approximation methods to tackle quantum few-body problems. In particular, variational methods have been proposed to ease numerical calculations and obtain precise solutions. One such method is the Stochastic Variational Method, which employs a stochastic search to determine the number and parameters of correlated Gaussian basis functions used to construct an ansatz of the wave function. Stochastic methods, however, face numerical and optimization challenges as the number of particles increases. We introduce a family of gradient variational methods that replace stochastic search with gradient optimization. We comparatively and empirically evaluate the performance of the baseline Stochastic Variational Method, several instances of the gradient variational method family, and some hybrid methods for selected few-body problems. We show that gradient and hybrid methods can be more efficient and effective than the Stochastic Variational Method. We discuss the role of singularities, oscillations, and gradient optimization strategies in the performance of the respective methods.

DOI
10.1007/s00601-024-01965-7

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Abstract Quantum few-body systems are deceptively simple. Indeed, with the notable exception of a few special cases, their associated Schrödinger equation cannot be solved analytically for more than two particles. One has to resort to approximation methods to tackle quantum few-body problems. In particular, variational methods have been proposed to ease numerical calculations and obtain precise solutions. One such method is the Stochastic Variational Method, which employs a stochastic search to determine the number and parameters of correlated Gaussian basis functions used to construct an ansatz of the wave function. Stochastic methods, however, face numerical and optimization challenges as the number of particles increases. We introduce a family of gradient variational methods that replace stochastic search with gradient optimization. We comparatively and empirically evaluate the performance of the baseline Stochastic Variational Method, several instances of the gradient variational method family, and some hybrid methods for selected few-body problems. We show that gradient and hybrid methods can be more efficient and effective than the Stochastic Variational Method. We discuss the role of singularities, oscillations, and gradient optimization strategies in the performance of the respective methods.
Quantum few-body systems are deceptively simple. Indeed, with the notable exception of a few special cases, their associated Schrodinger equation cannot be solved analytically for more than two particles. One has to resort to approximation methods to tackle quantum few-body problems. In particular, variational methods have been proposed to ease numerical calculations and obtain precise solutions. One such method is the Stochastic Variational Method, which employs a stochastic search to determine the number and parameters of correlated Gaussian basis functions used to construct an ansatz of the wave function. Stochastic methods, however, face numerical and optimization challenges as the number of particles increases. We introduce a family of gradient variational methods that replace stochastic search with gradient optimization. We comparatively and empirically evaluate the performance of the baseline Stochastic Variational Method, several instances of the gradient variational method family, and some hybrid methods for selected few-body problems. We show that gradient and hybrid methods can be more efficient and effective than the Stochastic Variational Method. We discuss the role of singularities, oscillations, and gradient optimization strategies in the performance of the respective methods.
Variational solutions are given for problems involving diverse fermionic and bosonic N=2--7-body systems. The trial wave functions are chosen to be combinations of correlated Gaussians, which facilitate a fully analytical calculation of the matrix elements. The nonlinear parameters of the trial function are selected by a stochastic technique. The method has proved efficient, accurate, and seems feasible for any few-body bound-state problems emerging in atomic or nuclear physics. \textcopyright{} 1996 The American Physical Society.
Precise variational solutions are given for problems involving diverse fermionic and bosonic (N=2--7)-body systems. The trial wave functions are chosen to be combinations of correlated Gaussians, which are constructed from products of the single-particle Gaussian wave packets through an integral transformation, thereby facilitating fully analytical calculations of the matrix elements. The nonlinear parameters of the trial function are chosen by a stochastic technique. The method has proved very efficient, virtually exact, and it seems feasible for any few-body bound-state problems emerging in nuclear or atomic physics.
Solving the quantum many-body Schrödinger equation is a fundamental and challenging problem in the fields of quantum physics, quantum chemistry, and material sciences. One of the common computational approaches to this problem is Quantum Variational Monte Carlo (QVMC), in which ground-state solutions are obtained by minimizing the energy of the system within a restricted family of parameterized wave functions. Deep learning methods partially address the limitations of traditional QVMC by representing a rich family of wave functions in terms of neural networks. However, the optimization objective in QVMC remains notoriously hard to minimize and requires second-order optimization methods such as natural gradient. In this paper, we first reformulate energy functional minimization in the space of Born distributions corresponding to particle-permutation (anti-)symmetric wave functions, rather than the space of wave functions. We then interpret QVMC as the Fisher-Rao gradient flow in this distributional space, followed by a projection step onto the variational manifold. This perspective provides us with a principled framework to derive new QMC algorithms, by endowing the distributional space with better metrics, and following the projected gradient flow induced by those metrics. More specifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which uses the gradient flow induced by the Wasserstein metric, rather than Fisher-Rao metric, and corresponds to transporting the probability mass, rather than teleporting it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.
Many variational descriptions of quantum many-body systems rest on an expansion over basis functions, and their practical limit is often set by the number of basis functions required. We propose the Bayesian variational method (BVM), in which the basis functions are selected by Bayesian optimization: a Gaussian-process surrogate model, conditioned on the candidates evaluated so far, predicts which candidates are most likely to lower the energy, and the candidate evaluations, being mutually independent, are distributed over many nodes. Two further ingredients make the method practical. An incremental diagonalization evaluates each candidate by reusing the previous diagonalization of the accepted basis instead of solving the full eigenvalue problem anew. A trimming procedure continually removes basis functions that have become nearly linearly dependent, keeping the accepted basis small while guiding it toward the optimal solution. The BVM applies broadly to energy variational problems based on basis-function expansions in quantum mechanics; here we apply it to the Gaussian expansion method (GEM), a standard approach in few-body physics. Because the GEM basis is nonorthogonal, its linear dependence is strong, so the basis reduction achieved by the BVM is large. The reduction both accelerates the computation and, more importantly, greatly reduces the memory requirement, one of the central bottlenecks of the variational method: the reference energy of the full 32,000-dimensional GEM diagonalization is reproduced to within 0.01 K with only 705 basis functions and to within 0.001 K with 2,127, corresponding to memory reductions of 99.95% and 99.56%, since the matrix storage grows as the square of the basis dimension. Within the GEM, this opens a path to the precision study of six- and seven-body systems, and beyond, that has so far been difficult to reach.