A note on the switching adiabatic theorem

TL;DR

This paper derives a nearly optimal upper bound on the adiabatic theorem's running time for switching Hamiltonians in the Gevrey class, showing the error remains small for specific time scales related to the spectral gap.

Contribution

It provides a new upper bound on adiabatic evolution time for switching Hamiltonians with Gevrey class regularity, improving understanding of adiabatic approximation accuracy.

Findings

Error remains small for times of order g^{-2} |ln g|^{6α}.

Bound is nearly optimal for Gevrey class Hamiltonians.

Results relate the spectral gap to adiabatic evolution accuracy.

Abstract

We derive a nearly optimal upper bound on the running time in the adiabatic theorem for a switching family of Hamiltonians. We assume the switching Hamiltonian is in the Gevrey class $G^\alpha$ as a function of time, and we show that the error in adiabatic approximation remains small for running times of order $g^{-2}\,|\ln\,g\,|^{6\alpha}$. Here $g$ denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian.