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Waveflow: boundary-conditioned normalizing flows applied to fermionic wavefunctions

Luca Thiede, Chong Sun, Alán Aspuru-Guzik

TL;DR

Waveflow is introduced, an innovative framework for learning many-body fermionic wave functions using boundary-conditioned normalizing flows and using O-spline priors and I-spline bijections to handle the topological mismatch between the prior and target distributions.

Abstract

An efficient and expressive wavefunction ansatz is key to scalable solutions for complex many-body electronic structures. While Slater determinants are predominantly used for constructing antisymmetric electronic wavefunction ansätze, this construction can result in limited expressiveness when the targeted wavefunction is highly complex. In this work, we introduce Waveflow, an innovative framework for learning many-body fermionic wavefunctions using boundary-conditioned normalizing flows. Instead of relying on Slater determinants, Waveflow imposes antisymmetry by defining the fundamental domain of the wavefunction and applying necessary boundary conditions. A key challenge in using normalizing flows for this purpose is addressing the topological mismatch between the prior and target distributions. We propose using O-spline priors and I-spline bijections to handle this mismatch, which allows for flexibility in the node number of the distribution while automatically maintaining its square-normalization property. We apply Waveflow to a one-dimensional many-electron system, where we variationally minimize the system's energy using variational quantum Monte Carlo (VQMC). Our experiments demonstrate that Waveflow can effectively resolve topological mismatches and faithfully learn the ground-state wavefunction.

Waveflow: boundary-conditioned normalizing flows applied to fermionic wavefunctions

TL;DR

Waveflow is introduced, an innovative framework for learning many-body fermionic wave functions using boundary-conditioned normalizing flows and using O-spline priors and I-spline bijections to handle the topological mismatch between the prior and target distributions.

Abstract

An efficient and expressive wavefunction ansatz is key to scalable solutions for complex many-body electronic structures. While Slater determinants are predominantly used for constructing antisymmetric electronic wavefunction ansätze, this construction can result in limited expressiveness when the targeted wavefunction is highly complex. In this work, we introduce Waveflow, an innovative framework for learning many-body fermionic wavefunctions using boundary-conditioned normalizing flows. Instead of relying on Slater determinants, Waveflow imposes antisymmetry by defining the fundamental domain of the wavefunction and applying necessary boundary conditions. A key challenge in using normalizing flows for this purpose is addressing the topological mismatch between the prior and target distributions. We propose using O-spline priors and I-spline bijections to handle this mismatch, which allows for flexibility in the node number of the distribution while automatically maintaining its square-normalization property. We apply Waveflow to a one-dimensional many-electron system, where we variationally minimize the system's energy using variational quantum Monte Carlo (VQMC). Our experiments demonstrate that Waveflow can effectively resolve topological mismatches and faithfully learn the ground-state wavefunction.
Paper Structure (19 sections, 4 theorems, 30 equations, 7 figures, 4 algorithms)

This paper contains 19 sections, 4 theorems, 30 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. Electronic wavefunction
  4. Energy and gradient evaluation
  5. Normalizing flows
  6. Bijections and priors
  7. Topological mismatch
  8. Boundary conditions
  9. Antisymmetry
  10. Waveflow algorithm
  11. Experiments
  12. Conclusions
  13. Normalizing flow algorithmss
  14. Complexity Analysis
  15. Splines
  16. ...and 4 more sections

Key Result

Theorem 1

Let $g^{-1}_{l}(\mathbf{x})$ be a composition of autoregressive bijections with parameters $\bm{\theta}_{\mathrm{B}}^l$. Assume that $g^{-1}_l(x_i = x^*|\bm{\theta}_{\mathrm{B},i}^l) = x^*$, for all $l$ and $\bm{\theta}_{\mathrm{B}}$, and that $p_\mathbf{z}(\mathbf{z})$ is an autoregressive prior di

Figures (7)

  • Figure 1: Learning the ground-state fermionic wavefunction using Waveflow and variational quantum Monte Carlo (VQMC).
  • Figure 2: Topological mismatch with reproducing the double-circle distribution. (a) The target double-circle distribution. (b) Distribution learned by a normalizing flow using affine coupling layers. (c) Distribution learned by Waveflow.
  • Figure 3: An example illustrating the node number mismatch between prior and target distributions.
  • Figure 4: The wavefunction of two particles in a box, including the relative coordinate systems and the fundamental domain. The blue and red regions are equivalent, with either capable of serving as the fundamental domain.
  • Figure 5: Training progress of Waveflow for the ground state of a one-dimensional helium-like system. The blue region corresponds to the wavefunction with positive values and the red region with negative values. The $x$-axis and $y$-axis denote $x_0$ and $x_1$, respectively. The dots indicate samples drawn from the fundamental domain, chosen as the region where $d_1 = x_1 - x_0 \geq 0$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Definition 2.1: Orbit
  • Definition 2.2: Fundamental domain
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof