Accelerating Resonant Spectroscopy Simulations Using Multi-Shifted Bi-Conjugate Gradient

TL;DR

This paper introduces a multi-shifted biconjugate gradient algorithm that significantly accelerates resonant spectroscopy simulations in quantum materials by reducing computational complexity and maintaining accuracy.

Contribution

The paper presents a novel multi-shifted biconjugate gradient method that exploits shared Krylov subspaces to efficiently simulate resonant spectroscopies for large systems.

Findings

Algorithm achieves constant complexity regardless of incident energies

Mathematical proofs confirm stability and accuracy

Numerical benchmarks demonstrate substantial acceleration

Abstract

Resonant spectroscopies, which involve intermediate states with finite lifetimes, provide essential insights into collective excitations in quantum materials that are otherwise inaccessible. However, theoretical understanding in this area is often limited by the numerical challenges of solving Kramers-Heisenberg-type response functions for large-scale systems. To address this, we introduce a multi-shifted biconjugate gradient algorithm that exploits the shared structure of Krylov subspaces across spectra with varying incident energies, effectively reducing the computational complexity to that of linear spectroscopies. Both mathematical proofs and numerical benchmarks confirm that this algorithm substantially accelerates spectral simulations, achieving constant complexity independent of the number of incident energies, while ensuring accuracy and stability. This development provides a scalable, versatile framework for simulating advanced spectroscopies in quantum materials.