Probing and graph coloring techniques for trace estimation in Lattice QCD

2026-05-29

AI Summary

Estimating the trace of the inverse Wilson–Dirac operator in lattice QCD is highly challenging due to the large size and dense structure of the matrix. This work proposes a novel distance-$d$ graph coloring algorithm based on multipliers, which efficiently generates structured probing vectors for arbitrary distances while significantly reducing the required number of colors. When combined with the Hutchinson stochastic estimator, the method effectively suppresses variance induced by short-range off-diagonal elements and achieves even lower variance at intermediate distances. Numerical experiments demonstrate that the trace estimation variance decreases smoothly and monotonically with the number of colors, thereby avoiding the irregular behavior typical of conventional hierarchical probing and substantially improving relative accuracy.

Abstract

The computation of $\mathrm{Tr}[D^{-1}]$, where $D$ is the Wilson-Dirac matrix of Lattice QCD, is a fundamental and computationally demanding task with applications to disconnected hadronic correlation functions. Since $D^{-1}$ is a dense matrix of prohibitive size, its trace cannot be computed exactly, and one must resort to stochastic estimation via the Hutchinson estimator. The variance of the resulting estimation, however, can be large, as it is dominated by the off-diagonal entries of $D^{-1}$. We review the stochastic probing technique, which reduces the variance by constructing structured sampling vectors from distance-$d$ colorings of the graph associated with $D$, exploiting the exponential off-diagonal decay of $D^{-1}$ to eliminate dominant short-range contributions to the variance. We then present a novel multiplier-based coloring scheme, which achieves valid distance-$d$ colorings at arbitrary distances with significantly fewer colors than the established hierarchical probing construction. We prove that at any intermediate coloring falling between two consecutive hierarchical levels, the multiplier-based estimator achieves strictly lower variance than the partial hierarchical estimator, for large enough $d$. This is confirmed by numerical experiments showing that the multiplier-based variance decreases smoothly and monotonically with the number of colors, avoiding the irregular behavior affecting hierarchical probing at intermediate colorings, and achieving a substantial improvement in relative accuracy.

Problem

• trace estimation
• Lattice QCD
• variance reduction
• stochastic probing
• graph coloring

Innovation

• multiplier-based coloring
• stochastic trace estimation
• distance-d coloring
• Lattice QCD
• variance reduction