Corrections to adiabatic behavior for long paths

TL;DR

This paper examines the limitations of traditional measures for adiabatic quantum state preparation, proposing that path length, rather than total time, better indicates computational difficulty, supported by theoretical insights and no-go theorems.

Contribution

It introduces a no-go theorem showing total time is not a reliable measure and suggests path length as a more accurate proxy for computational cost in adiabatic processes.

Findings

Total time is not a good measure for adiabatic complexity.

Path length correlates with computational difficulty.

Proxies based on path length grow with system size.

Abstract

The cost and the error of the adiabatic theorem for preparing the final eigenstate are discussed in terms of path length. Previous studies in terms of the norm of the Hamiltonian and its derivatives with the spectral gap are limited in their ability to describe the cost of adiabatic state preparation for certain physically large systems. We argue that total time is not a good measure for determining the computational difficulty of adiabatic quantum computation by developing a no-go theorem. From the result of time-periodic Hamiltonian cases, we suggest that there are proxies for computational cost which typically grow as path length increases when the error is kept fixed and small and consider possible conjectures on how general the behavior is.