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.

翻译：计算 $\mathrm{Tr}[D^{-1}]$（其中 $D$ 为格点QCD中的Wilson-Dirac矩阵）是基本但计算量巨大的任务，广泛应用于断连强子关联函数。由于 $D^{-1}$ 是密集且规模庞大的矩阵，其迹无法精确计算，必须借助Hutchinson估计器进行随机估计。然而，此类估计的方差可能很大，主要源于 $D^{-1}$ 的非对角元。本研究回顾了随机探针技术，该技术通过从与 $D$ 关联的图的距离-$d$ 染色构建结构化采样向量，利用 $D^{-1}$ 的指数级非对角衰减消除方差中占主导的短程贡献。我们进一步提出一种新颖的乘子基染色方案，能在任意距离下实现有效的距离-$d$ 染色，且所需颜色数显著少于已有分层探针构造方法。我们证明，对于任意介于两个连续分层层级之间的中间染色，当 $d$ 足够大时，乘子基估计器的方差严格低于部分分层估计器。数值实验证实了这一结论：乘子基方差随颜色数增加呈现平滑单调递减趋势，避免了分层探针在中间染色处的不规则波动，并在相对精度上实现了显著提升。