Using orthogonal projectors in multigrid multilevel Monte Carlo for trace estimation in lattice QCD

Abstract: We introduce a multigrid multilevel Monte Carlo method for stochastic trace estimation in lattice QCD based on orthogonal projections. This formulation extends the previously proposed oblique decomposition and it is assessed on three representative problems: the connected pseudoscalar correlator, the trace of the full Dirac operator's inverse $\mathrm{tr}(D^{-1})$, and disconnected fermion loops. For the connected correlator, variance reductions grow systematically with the time separation and lead to cost savings of up to a factor of 30 at large separations, outperforming both the plain Hutchinson's estimator and the oblique formulation. For $\mathrm{tr}(D^{-1})$, reductions are more modest but remain systematic, with stronger effects on more ill-conditioned systems. Disconnected loops show no improvement, since their variance is dominated by local same-slice contributions not targeted by the decomposition.