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Variance reduction in lattice QCD observables via normalizing flows

Ryan Abbott1,2,3, Denis Boyda2,3, Yang Fu2,3, Daniel C. Hackett4, Gurtej Kanwar5, Fernando Romero-López6, Phiala E. Shanahan2,3, and Julian M. Urban2,3

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Phys. Rev. D 114, 014513 – Published 21 July, 2026

DOI: https://doi.org/10.1103/5xns-7qmj

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 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 volume transfer can be utilized to minimize training costs.

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Phys. Rev. D 114, 014513– Published 21 July, 2026
  • Received 8 April 2026
  • Accepted 1 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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