Lanczos, the transfer matrix, and the signal-to-noise problem

This work introduces a method for determining the energy spectrum of lattice quantum chromodynamics (LQCD) by applying the Lanczos algorithm to the transfer matrix. Proof-of-principle analyses of the simple harmonic oscillator and the LQCD proton mass demonstrate that this method provides faster ground-state convergence than standard energy estimators, which are related to the power-iteration algorithm. Lanczos provides more accurate energy estimates than multi-state fits to correlation functions with small imaginary times while achieving comparable statistical precision. Two-sided error bounds are computed for Lanczos results and guarantee that excited-state effects cannot shift Lanczos results far outside their statistical uncertainties. Further, Lanczos results avoid the exponential signal-to-noise degradation present in standard energy estimators.