Loschmidt echo approach to Krylov-subspace approximation error estimation

TL;DR

This paper introduces a novel method to estimate the error in Krylov subspace approximations of quantum evolution by relating it to Loschmidt echo, providing efficient and accurate error bounds.

Contribution

It presents a new approach linking Krylov approximation error to Loschmidt echo, enabling simple physical interpretation and computationally cheap, precise error estimation.

Findings

Error bounds closely match actual approximation errors

Different time-regimes explained by physical intuition

Method applicable to large quantum systems

Abstract

The Krylov subspace method is a standard approach to approximate quantum evolution, allowing to treat systems with large Hilbert spaces. Although its application is general, and suitable for many-body systems, estimation of the committed error is involved. This makes it difficult to automate its use. In this paper, we solve this problem by realizing that such error can be regarded as a Loschmidt echo in a tight-binding Hamiltonian. We show that the different time-regimes of the approximation can be understood using simple physical ideas. More importantly, we obtain computationally cheap error bounds that describe with high precision the actual error in the approximation.