New class of variance-reduction techniques using lattice symmetries

We present a general class of unbiased improved estimators for physical observables in lattice gauge theory computations which significantly reduces statistical errors at modest computational cost. The idea can be easily adapted to other branches of physics and computational science that employ Monte Carlo methods. The error reduction techniques, referred to as covariant approximation averaging, utilize approximations which are covariant under lattice symmetry transformations. We observe cost reductions from the new method compared to the traditional one, for fixed statistical error, of 16 times for the nucleon mass at ${M}_{\ensuremath{\pi}}\ensuremath{\sim}330\text{ }\text{ }\mathrm{MeV}$ (domain-wall quark) and 2.6--20 times for the hadronic vacuum polarization at ${M}_{\ensuremath{\pi}}\ensuremath{\sim}315\text{ }\text{ }\mathrm{MeV}$ (Asqtad quark). These cost reductions should improve with decreasing quark mass and increasing lattice sizes.