Filtered Rayleigh-Ritz is all you need

TL;DR

The paper establishes that Prony, GPOF, GEVP, and Lanczos analyses of lattice QCD correlators all belong to a Prony-Ritz equivalence class: they yield identical Ritz values and vectors when data exactly represent a finite-dimensional exponential spectrum. It then shows that filtering methods, notably Hermitian subspace filtering, allow these RR-based approaches to extract physically meaningful spectra from noisy data while retaining optimal subspace accuracy (KPS-type convergence) within the Hermitian subspace. The work provides practical implementations by recasting oblique RR in terms of standard GEVPs and Hankel-based formulations, enabling numerically stable, Lanczos-free calculations that preserve spectral outputs. Finally, it discusses convergence, residual bounds, and the limits of equivalence under noise, arguing that FRR offers a principled, robust framework for extracting excited states and matrix elements from large correlator matrices with clear physical interpretation and error control.

Abstract

Recent work has shown that the (block) Lanczos algorithm can be used to extract approximate energy spectra and matrix elements from (matrices of) correlation functions in quantum field theory, and identified exact coincidences between Lanczos analysis methods and others. In this work, we note another coincidence: the Lanczos algorithm is equivalent to the well-known Rayleigh-Ritz method applied to Krylov subspaces. Rayleigh-Ritz provides optimal eigenvalue approximations within subspaces; we find that spurious-state filtering allows these optimality guarantees to be retained in the presence of statistical noise. We explore the relation between Lanczos and Prony's method, their block generalizations, generalized pencil of functions (GPOF), and methods based on the generalized eigenvalue problem (GEVP), and find they all fall into a larger "Prony-Ritz equivalence class", identified as all methods which solve a finite-dimensional spectrum exactly given sufficient correlation function (matrix) data. This equivalence allows simpler and more numerically stable implementations of (block) Lanczos analyses.