Improved Error Bounds for the Adiabatic Approximation

TL;DR

This paper provides rigorous bounds on the error of the adiabatic approximation in quantum systems, establishing conditions for its validity and addressing counterexamples like the Marzlin--Sanders case.

Contribution

It introduces elementary, asymptotically tight bounds on adiabatic error and refines criteria to determine when the approximation holds, excluding known counterexamples.

Findings

Bounds are often asymptotically tight for slow evolution.

Sufficient conditions are established for the validity of the adiabatic approximation.

Counterexamples like Marzlin--Sanders are excluded by the new criteria.

Abstract

Since the discovery of adiabatic quantum computing, a need has arisen for rigorously proven bounds for the error in the adiabatic approximation. We present in this paper, a rigorous and elementary derivation of upper and lower bounds on the error incurred from using the adiabatic approximation for quantum systems. Our bounds are often asymptotically tight in the limit of slow evolution for fixed Hamiltonians, and are used to provide sufficient conditions for the application of the adiabatic approximation. We show that our sufficiency criteria exclude the Marzlin--Sanders counterexample from the class of Hamiltonians that obey the adiabatic approximation. Finally, we demonstrate the existence of classes of Hamiltonians that resemble the Marzlin--Sanders counterexample Hamiltonian, but also obey the adiabatic approximation.