Asymptotic errors in adiabatic evolution

TL;DR

This paper analyzes the errors in adiabatic quantum evolution, distinguishing regimes where errors depend on the entire evolution or mainly on endpoints, and introduces a typical error measure with asymptotic properties.

Contribution

It identifies two regimes of adiabatic errors and introduces a new typical error metric that depends only on endpoints, advancing understanding of non-ideal adiabatic processes.

Findings

Errors depend on the evolution regime (adiabatic or hyperadiabatic).

The typical error depends only on endpoints and has an asymptotic series expansion.

Error coefficients involve contributions from both endpoints with phase factors.

Abstract

The adiabatic theorem in quantum mechanics implies that if a system is in a discrete eigenstate of a Hamiltonian and the Hamiltonian evolves in time arbitrarily slowly, the system will remain in the corresponding eigenstate of the evolved Hamiltonian. Understanding corrections to the adiabatic result that arise when the evolution of the Hamiltonian is slow -- but not arbitrarily slow -- has become increasingly important, especially since adiabatic evolution has been proposed as a method of state preparation in quantum computing. This paper identifies two regimes, an adiabatic regime in which corrections are generically small and can depend on details of the evolution throughout the path, and a hyperadiabatic regime in which the error is given by a form similar to an asymptotic expansion in the inverse of the evolution time with the coefficients depending principally on the behavior at the endpoints. However, the error in this hyperadiabatic regime is neither given by a true asymptotic series nor solely dependent on the endpoints: the coefficients combine the contributions from both endpoints, with relative phase factors that depend on the average spectral gaps along the trajectory, multiplied by the evolution time. The central result of this paper is to identify a quantity, referred to as the typical error, which is obtained by appropriately averaging the error over evolution times that are small compared to the evolution time itself. This typical error is characterized by an asymptotic series and depends solely on the endpoints of the evolution, remaining independent of the details of the intermediate evolution.