Equivalence and Divergence of Imaginary-Time Evolution and Gradient Descent for Gaussian Variational States

TL;DR

This paper compares imaginary-time evolution and gradient descent for Gaussian states, revealing their equivalence in fermionic systems but faster convergence of GD in bosonic systems, challenging previous assumptions.

Contribution

It demonstrates the formal equivalence of ITE and GD for fermionic systems and shows GD's superior convergence in bosonic systems, providing new insights into their relationship.

Findings

ITE and GD are equivalent for fermionic systems

GD converges faster than ITE for bosonic systems

Challenges the assumption of complete equivalence between ITE and GD

Abstract

Imaginary-time evolution (ITE) is one of the most widely used numerical techniques for obtaining ground states of many-body Hamiltonians. In this work, we compare ITE with gradient descent (GD) within the framework of Gaussian wavefunction ansatze. We show that while ITE and GD are formally equivalent for fermionic systems, GD exhibits consistently faster convergence for bosonic systems, challenging the common assumption of their complete equivalence.