Applications of flow models to the generation of correlated lattice QCD ensembles

TL;DR

This work addresses high-variance derivative estimates in lattice QCD by using normalizing flows to map configurations between distributions $p_\alpha$ and $p_{\alpha'}$, producing correlated ensembles that cancel fluctuations in observables across action-parameter changes. The proposed residual-flow architecture implements gauge-equivariant transformations $U'_{\mu}(x) = e^{g_x(U)} U_{\mu}(x)$ with masking patterns to enable tractable Jacobians, while maintaining exactness through reweighting when needed. The authors demonstrate three proof-of-principle applications—the continuum limit via gradient-flow scales, a Feynman-Hellmann calculation of the gluon momentum fraction $\langle x \rangle_g^{\rm latt}$, and the mass dependence of observables in $N_f=2$ QCD—showing substantial variance reductions relative to uncorrelated ensembles or direct reweighting (ESS improvements up to $>20$). They report that, after accounting for flow costs, a substantial computational advantage is achieved (e.g., fewer configurations by factors of several to ten) and discuss extensions to full QCD with pseudofermions and to QED+QCD, as well as potential applications to sigma terms and sign-problem contexts.

Abstract

Machine-learned normalizing flows can be used in the context of lattice quantum field theory to generate statistically correlated ensembles of lattice gauge fields at different action parameters. This work demonstrates how these correlations can be exploited for variance reduction in the computation of observables. Three different proof-of-concept applications are demonstrated using a novel residual flow architecture: continuum limits of gauge theories, the mass dependence of QCD observables, and hadronic matrix elements based on the Feynman-Hellmann approach. In all three cases, it is shown that statistical uncertainties are significantly reduced when machine-learned flows are incorporated as compared with the same calculations performed with uncorrelated ensembles or direct reweighting.

Figures (5)

• Figure 1: Sketch of the recursive transformation, \ref{['eq:PTconv']}, to build generic Wilson loops in the residual layers.
• Figure 2: Continuum extrapolation of the ratio of two gradient flow scales $t_{0.3}/t_{0.35}$, using the quantity in the numerator to set the scale. Two methods are shown: $\epsilon$ reweighting (dotted grey line), and using a flowed ensemble (solid orange line). Statistical uncertainties are displayed as bands.
• Figure 3: (a) Pion mass in lattice units as a function of the coupling to the gluonic energy-momentum tensor $\lambda$. Marker shapes denote how the ensembles were obtained: orange circles for heatbath ensembles at fixed values of $\lambda$, blue squares for ensembles flowed from $\lambda=0$, and red triangles when using configurations generated at $\lambda=0$ and reweighted to ${\lambda=\epsilon=10^{-4}}$. The pion mass is evaluated in quenched lattice QCD at $\beta=6.0$, $\kappa=0.132$, $L=8$ and $T=16$. (b) Bare gluon momentum fraction of the pion from \ref{['eq:mastereqFH']} using a finite-difference approximation computed using the three different methods: independent heatbath ensembles, $\epsilon$ reweighting, and correlated flowed ensembles.
• Figure 4: Illustration of the error reduction in derivatives of observables with respect to the action parameter $\kappa$. $W_{n \times n}$ is the average square Wilson loop of size $n$, $Q^2$ is the squared topological charge defined via the gradient flow, and $t_c$ labels gradient flow scales, as in \ref{['eq:tcdef']}. The y-axis shows the values of the observables and their statistical errors normalized to the value obtained with flows. Results that incorporate flows are shown as blue squares, while the errors with $\epsilon$ reweighting are denoted by red triangles.
• Figure 5: Summary of the variance reduction in observables computed from derivatives with respect to the action parameters when using flows compared with $\epsilon$ reweighting. The improvement factor is defined as the ratio of variances of the observables computed with $\epsilon$ reweighting over flows. The label "$N_f=2$ QCD" denotes derivatives of observables with respect to $\kappa$ in two-flavor QCD, the label "Pure Gauge" corresponds to the result for the continuum limit extrapolation of gradient flow scales in the pure gauge theory, and the label "Feynman-Hellmann" indicates observables computed using the Feynman-Hellmann approach in quenched QCD.