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

TL;DR

This paper presents a novel application of the Lanczos algorithm combined with a bootstrap method to efficiently and accurately determine the energy spectrum in lattice QCD, improving convergence and error estimation.

Contribution

It introduces a new approach using Lanczos and bootstrap techniques for spectral analysis in lattice QCD, enhancing accuracy and convergence over traditional methods.

Findings

Faster ground-state convergence than effective mass methods.

More accurate energy estimates than multi-state fits.

Provides two-sided error bounds for results.

Abstract

This work introduces a method for determining the energy spectrum of lattice quantum chromodynamics (LQCD) by applying the Lanczos algorithm to the transfer matrix and using a bootstrap generalization of the Cullum-Willoughby method to filter out spurious eigenvalues. Proof-of-principle analyses of the simple harmonic oscillator and the LQCD proton mass demonstrate that this method provides faster ground-state convergence than the "effective mass," which is 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.