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Variance reduction in lattice QCD observables via normalizing flows

Ryan Abbott, Denis Boyda, Yang Fu, Daniel C. Hackett, Gurtej Kanwar, Fernando Romero-López, Phiala E. Shanahan, Julian M. Urban

Abstract

Normalizing flows can be used to construct unbiased, reduced-variance estimators for lattice field theory observables that are defined by a derivative with respect to action parameters. This work implements the approach for observables involving gluonic operator insertions in the SU(3) Yang-Mills theory and two-flavor Quantum Chromodynamics (QCD) in four space-time dimensions. Variance reduction by factors of $10$-$60$ is achieved in glueball correlation functions and in gluonic matrix elements related to hadron structure, with demonstrated computational advantages. The observed variance reduction is found to be approximately independent of the lattice volume, so that volume transfer can be utilized to minimize training costs.

Variance reduction in lattice QCD observables via normalizing flows

Abstract

Normalizing flows can be used to construct unbiased, reduced-variance estimators for lattice field theory observables that are defined by a derivative with respect to action parameters. This work implements the approach for observables involving gluonic operator insertions in the SU(3) Yang-Mills theory and two-flavor Quantum Chromodynamics (QCD) in four space-time dimensions. Variance reduction by factors of - is achieved in glueball correlation functions and in gluonic matrix elements related to hadron structure, with demonstrated computational advantages. The observed variance reduction is found to be approximately independent of the lattice volume, so that volume transfer can be utilized to minimize training costs.
Paper Structure (22 sections, 58 equations, 4 figures, 2 tables)

This paper contains 22 sections, 58 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Flows for variance reduction
  3. Derivative observables
  4. Vacuum-subtracted correlation functions via the derivative trick.
  5. Hadronic matrix elements via the Feynman--Hellmann theorem.
  6. Variance of derivative observables
  7. Flows
  8. Derivative observables via flows
  9. Determinant ratio estimation
  10. Linearization of the models
  11. Summary of the workflow
  12. Workflow for flowed derivative observables
  13. Using a finite-difference approximation.
  14. Using linearization.
  15. Cost comparison between linearization and finite differences
  16. ...and 7 more sections

Figures (4)

  • Figure 1: Illustration of variance reduction in the glueball correlator in the SU(3) Yang-Mills theory with $\beta=6$ for a $16^3 \times 32$ lattice geometry. The upper panel displays the correlator itself, evaluated on an ensemble of 3200 configurations. The baseline calculation using the standard two-point function evaluation is compared against the derivative-trick estimate evaluated using both the finite flow with $\lambda = 10^{-3}$ and the unbiased linearized estimate. In the lower panel, the ratio between the variance of the baseline and flow-improved estimates is shown.
  • Figure 2: Variance reduction in the glueball correlator achieved by the flow approach at timeslice $t=2a$ as a function of the $L^3 \times 32$ lattice geometry. Results for both the finite-$\lambda$ and linearized approach are shown. Data for the $L=16$ case is obtained from an ensemble of 3200 configurations while the remaining cases are evaluated using ensembles of approximately 1600 configurations each.
  • Figure 3: Illustration of the variance reduction in the glueball correlator in QCD with $N_f=2$ Wilson fermions using 1280 configurations. The upper panel displays the correlator itself; the flow-improved estimates using both naive pseudofermions and a pseudofermion flow are compared against a baseline calculation using the standard computation of the two-point function. A single pseudofermion hit for both flow estimates. In the lower panel, the ratio of variances is shown with and without employing a pseudofermion flow.
  • Figure 4: Illustration of the variance reduction in the gluon momentum fraction in QCD with $N_f=2$ Wilson fermions. The upper panel compares the gluon momentum fraction of the pion computed using flows (orange squares) versus the baseline $\epsilon$-reweighting method (blue circles), both evaluated on the same ensemble of 8576 configurations. In the lower panel, the ratio between the variance of the baseline and the flow-improved estimate is shown. The results are shown as a function of the lower end of the fit range used for the extraction of the mass, $t_{\rm min}$.