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

TL;DR

This paper introduces gradient-based variational methods as a more efficient alternative to stochastic methods for solving quantum few-body problems, demonstrating improved performance through empirical evaluation.

Contribution

It proposes a new family of gradient variational methods that replace stochastic search, and empirically compares their effectiveness against existing stochastic and hybrid approaches.

Findings

Gradient methods outperform stochastic methods in efficiency.

Hybrid methods combine advantages of both approaches.

Performance depends on singularities, oscillations, and optimization strategies.

Abstract

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.