A Lanczos approach to the Adiabatic Gauge Potential

TL;DR

This paper introduces a Lanczos algorithm-based method to efficiently evaluate the Adiabatic Gauge Potential (AGP) and its norm, facilitating the study of quantum eigenstates and chaos with improved computational techniques.

Contribution

It presents a novel Lanczos algorithm approach for constructing the AGP operator and norm, improving computational efficiency and providing new insights into quantum chaos and operator growth.

Findings

Lanczos algorithm effectively evaluates AGP in simple systems

Derived an integral transform relating AGP norm to autocorrelation functions

Modified variational method to approximate the regulated AGP norm

Abstract

The Adiabatic Gauge Potential (AGP) is the generator of adiabatic deformations between quantum eigenstates. There are many ways to construct the AGP operator and evaluate the AGP norm. Recently, it was proposed that a Gram-Schmidt-type algorithm can be used to explicitly evaluate the expression of the AGP. We employ a version of this approach by using the Lanczos algorithm to evaluate the AGP operator in terms of Krylov vectors and the AGP norm in terms of the Lanczos coefficients. It has the advantage of minimizing redundancies in evaluating the nested commutators in the analytic expression for the AGP operator. The algorithm is used to explicitly construct the AGP operator for some simple systems. We derive an integral transform relation between the AGP norm and the autocorrelation function of the deformation operator. We present a modification of the Variational approach to derive the regulated AGP norm. Using this, we approximate the AGP to varying degrees of success. Finally, we compare and contrast the quantum-chaos-probing capacities of the AGP and K-complexity in view of the Operator Growth Hypothesis.