Necessary Condition for the Quantum Adiabatic Approximation

Understand

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.