# Necessary Condition for the Quantum Adiabatic Approximation

## TL;DR

This paper establishes a fundamental lower bound on the time required for adiabatic quantum processes to approximately prepare eigenstates, showing that the evolution time must scale with the eigenstate path length and inverse gap, and introduces a local property-based necessary condition.

## Contribution

It provides a rigorous lower bound on adiabatic process duration and introduces a local property-based necessary condition for the adiabatic approximation.

## Key findings

- Minimum time for adiabatic state preparation scales with path length and inverse gap.
- No adiabatic condition can guarantee a smaller evolution time than the derived lower bound.
- A local property-based necessary condition for the adiabatic approximation is proposed.

## Abstract

A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.