Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yash Palan

TL;DR

This work analyzes the relationship between imaginary-time evolution (ITE) and gradient descent (GD) within Gaussian variational states for ground-state search. It shows exact equivalence in the single-particle case, with updates of the form $\delta \psi = -\alpha' [\hat H - E] \psi$, but reveals that for bosonic many-body systems GD converges faster than ITE on the Gaussian manifold. The authors derive GD updates for both fermionic and bosonic sectors, including projections onto the tangent spaces of $O(2N_f)$ and $Sp(n,\mathbb{R})$, and compare convergence paths on a quadratic bosonic Hamiltonian $H=\sum_{ij} b_i^{\dagger} \omega_{ij} b_j$, showing ITE can be slower due to misalignment with the steepest-descent direction when $\Gamma_b \neq I$. The findings suggest leveraging gradient-based optimization from machine learning to accelerate variational simulations and motivate further study of classical versus quantum algorithmic speedups in this context.

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.

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

TL;DR

This work analyzes the relationship between imaginary-time evolution (ITE) and gradient descent (GD) within Gaussian variational states for ground-state search. It shows exact equivalence in the single-particle case, with updates of the form , but reveals that for bosonic many-body systems GD converges faster than ITE on the Gaussian manifold. The authors derive GD updates for both fermionic and bosonic sectors, including projections onto the tangent spaces of and , and compare convergence paths on a quadratic bosonic Hamiltonian , showing ITE can be slower due to misalignment with the steepest-descent direction when . The findings suggest leveraging gradient-based optimization from machine learning to accelerate variational simulations and motivate further study of classical versus quantum algorithmic speedups in this context.

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.

Paper Structure

This paper contains 11 sections, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Single particle Hamiltonians
  3. Many body Hamiltonians: Gaussian states
  4. Gaussian state
  5. Imaginary time evolution equations
  6. Gradient Descent
  7. Fermions
  8. Bosons
  9. Comparing to Imaginary-Time Evolution
  10. Conclusion
  11. Acknowledgements

Figures (2)

  • Figure 1: The variation of the length of the complete path of convergence for $\Delta_{R}$ and $\Gamma_b$, with different initial boson covariance matrices $\Gamma_b(a,b)$ for (a-b) GD and (c-d)ITE, respectively.
  • Figure 2: Difference of the length of convergence path for (a)$\Delta_R$ and (b)$\Gamma_b$ between ITE and GD for various initial boson covariance matrices $\Gamma_b(a,b)$ .