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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 $ 330 MeV (domain-wall quark) and 2.6-20 times for the hadronic vacuum polarization at M $ 315 MeV (Asqtad quark). These cost reductions should improve with decreasing quark mass and increasing lattice sizes.As nonperturbative computations using lattice gauge theory are applied to a wider range of physically interesting observables, it is increasingly important to find numerical strategies that provide precise results. In Monte Carlo simulations our reach to important physics is still often limited by statistical uncertainties. Examples include hadronic contributions to the muon's anomalous magnetic moment [1], nucleon form factors and structure functions [2], including nucleon electric dipole moments [3-6], hadron matrix elements relevant to flavor physics (e.g., K ! amplitudes) [7], and multihadron state physics [8], to name only a few. In addition, there are many examples of Monte Carlo simulation applied to condensed matter physics [9,10], many-body systems [11,12] and cold gases [13]. As a generalization of low-mode averaging (LMA) [14,15], we present a class of unbiased statistical error reduction techniques, utilizing approximations that are covariant under lattice symmetry transformations. LMA has worked well in cases where low eigenmodes of the Dirac operator dominate [16]: low energy constants in the " regime [14,17-20], pseudoscalar meson masses, decay constants [21-23], and so on. With a modest increase in computational cost, the generalized method can reduce statistical errors by an order of magnitude, or more, even in cases where LMA fails. Unlike LMA, we account for all modes of the Dirac operator, averaging over (most of) the lattice volume, with modest additional computational cost. The all-to-all methods [24,25] implement this stochastically for the higher modes, while treating the low modes exactly. For expectation values invariant under translations, statistics effectively increase by averaging over the whole lattice.The all-to-all method is advantageous when the stochastic noise introduced in the target observable is comparable to, or smaller than, the gauge field fluctuations of the ensemble [26], which typically holds only for many random source vectors per measurement. The error reduction techniques presented here, which do not rely on stochastic noise, are potentially more effective, provided an inexpensive approximation can be found for the desired observable.In lattice ...

We present an improved result of lattice computation of the proton decay matrix elements in N f = 2 + 1 QCD. In this study, the significant improvement of statistical accuracy by adopting the error reduction technique of All-mode-averaging, is achieved for relevant form factor to proton (and also neutron) decay on the gauge ensemble of N f = 2 + 1 domain-wall fermions in m π = 0.34-0.69 GeV on 2.7 fm 3 lattice as used in our previous work [1]. We improve total accuracy of matrix elements to 10-15% from 30-40% for p → πe + or from 20-40% for p → Kν. The accuracy of the low energy constants α and β in the leading-order baryon chiral perturbation theory (BChPT) of proton decay are also improved. The relevant form factors of p → π estimated through the "direct" lattice calculation from three-point function appear to be 1.4 times smaller than those from the "indirect" method using BChPT with α and β. It turns out that the utilization of our result will provide a factor 2-3 larger proton partial lifetime than that obtained using BChPT. We also discuss the use of these parameters in a dark matter model.

We present a new class of statistical error reduction techniques for Monte-Carlo simulations. Using covariant symmetries, we show that correlation functions can be constructed from inexpensive approximations without introducing any systematic bias in the final result. We introduce a new class of covariant approximation averaging techniques, known as all-mode averaging (AMA), in which the approximation takes account of contributions of all eigenmodes through the inverse of the Dirac operator computed from the conjugate gradient method with a relaxed stopping condition. In this paper we compare the performance and computational cost of our new method with traditional methods using correlation functions and masses of the pion, nucleon, and vector meson in N f = 2 + 1 lattice QCD using domain-wall fermions. This comparison indicates that AMA significantly reduces statistical errors in Monte-Carlo calculations over conventional methods for the same cost.