Tackling instabilities of quantum Krylov subspace methods: an analysis of the numerical and statistical errors

TL;DR

This paper analyzes the stability and error sources of quantum Krylov subspace methods, revealing that statistical fluctuations, not ill-conditioning, limit their reliability in noisy quantum computations.

Contribution

It introduces new metrics, the imaginary and unitary filters, to assess solution reliability without knowing the true eigenspectrum.

Findings

In ideal simulations, the eigenvalue problem becomes unstable with larger Krylov subspaces.

In realistic noisy settings, statistical fluctuations dominate over ill-conditioning.

The proposed filters effectively evaluate the reliability of quantum Krylov solutions.

Abstract

Krylov subspace methods are among the most extensively studied early fault-tolerant quantum algorithms for estimating ground-state energies of quantum systems. However, the rapid onset of ill-conditioning might make accurate energies difficult or even impossible to retrieve. In this communication, we analyse the numerical stability and statistical problems of these methods using numerical simulations both in the presence and absence of sampling noise. While in ideal numerical simulations the generalized eigenvalue problem indeed becomes unstable with increased Krylov subspace size, we find that, in realistic noisy settings, these methods do not primarily suffer from ill-conditioning. Instead, statistical fluctuations dominate and can prevent reliable solution extraction unless appropriate regularization or filtering techniques are employed. We consequently introduce two new metrics, the imaginary and unitary filters, that successfully assess the reliability of the obtained solutions without any knowledge of the true eigenspectrum.