High-degree Polynomial Noise Subtraction

Abstract: In lattice QCD, the calculation of physical quantities from disconnected quark loop calculations have large variance due to the use of Monte Carlo methods for the estimation of the trace of the inverse lattice Dirac operator. In this work, we build upon our POLY and HFPOLY variance reduction methods by using high-degree polynomials. Previously, the GMRES polynomials used were only stable for low-degree polynomials, but through application of a new, stable form of the GMRES polynomial, we have achieved higher polynomial degrees than previously used. While the variance is not dependent on the trace correction term within the methods, the evaluation of this term will be necessary for forming the vacuum expectation value estimates. This requires computing the trace of high-degree polynomials, which can be evaluated stochastically through our new Multipolynomial Monte Carlo method. With these new high-degree noise subtraction polynomials, we obtained a variance reduction for the scalar operator of nearly an order of magnitude over that of no subtraction on a $24^3 \times 32$ quenched lattice at $\beta = 6.0$ and $\kappa = 0.1570 \approx \kappa_{crit}$. Additionally, we observe that for sufficiently high polynomial degrees, POLY and HFPOLY approach the same level of effectiveness. We also explore the viability of using double polynomials for variance reduction as a means of reducing the required orthogonalization and memory costs associated with forming high-degree GMRES polynomials.