# Prony methods for extracting excited states

## TL;DR

This paper introduces a novel algebraic approach based on Prony's method for extracting multiple excited states from lattice gauge theory correlation functions, offering an alternative to traditional exponential fitting.

## Contribution

It develops a Prony-based algebraic technique to efficiently extract energies and amplitudes, along with a clustering method to identify states amidst overlapping errors.

## Key findings

- Successfully extracts multiple excited states from correlation functions.
- Addresses ambiguity in state identification due to overlapping error ellipses.
- Provides a clustering approach to improve state assignment.

## Abstract

We propose an algebraic method for extracting excited states from lattice gauge theory correlation functions. Instead of fitting to a sum of decaying exponentials, we adopt a variant of Prony's method to obtain $M$ energies (exponential decay rates) by finding the roots of an $M^{\rm th}$ order polynomial, and then solving for the amplitudes linearly. The resulting states tend to have overlapping error ellipses, making identification of states ambiguous. This is especially problematic at large Euclidean times where the signal to noise may be low, as well as when many states are extracted. We propose a variation of K-means clustering to identify each extracted state.