Adiabatic quantum optimization fails for random instances of NP-complete problems

TL;DR

This paper demonstrates that adiabatic quantum optimization fails for large random instances of NP-complete problems due to an Anderson localization-like phenomenon causing exponentially small eigenvalue gaps, leading to exponential runtime requirements.

Contribution

It shows that random NP-complete problem instances cause exponential slowdowns in adiabatic quantum optimization, similar to worst-case hard instances, due to spectral localization effects.

Findings

Exponential eigenvalue gaps appear near the end of the adiabatic process for large random instances.

Adiabatic quantum optimization gets stuck in local minima for these instances.

The failure occurs unless the computation time is exponentially long.

Abstract

Adiabatic quantum optimization has attracted a lot of attention because small scale simulations gave hope that it would allow to solve NP-complete problems efficiently. Later, negative results proved the existence of specifically designed hard instances where adiabatic optimization requires exponential time. In spite of this, there was still hope that this would not happen for random instances of NP-complete problems. This is an important issue since random instances are a good model for hard instances that can not be solved by current classical solvers, for which an efficient quantum algorithm would therefore be desirable. Here, we will show that because of a phenomenon similar to Anderson localization, an exponentially small eigenvalue gap appears in the spectrum of the adiabatic Hamiltonian for large random instances, very close to the end of the algorithm. This implies that unfortunately, adiabatic quantum optimization also fails for these instances by getting stuck in a local minimum, unless the computation is exponentially long.