Anderson localization casts clouds over adiabatic quantum optimization

TL;DR

This paper demonstrates that Anderson localization causes exponentially small spectral gaps in adiabatic quantum optimization, leading to failure in solving large NP-complete problems due to system trapping.

Contribution

It introduces a novel analysis method for spectral gaps using concepts from quantum disordered systems, revealing localization effects in adiabatic quantum algorithms.

Findings

Exponential small gaps occur near the end of the adiabatic process.

Anderson localization causes system trapping in local minima.

Adiabatic quantum optimization is ineffective for large random NP-complete instances.

Abstract

Understanding NP-complete problems is a central topic in computer science. This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer's Hamiltonian. We show that the statistics of the gaps can be analyzed in a novel way, borrowed from the study of quantum disordered systems in statistical mechanics. It turns out that due to a phenomenon similar to Anderson localization, exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. This implies that unfortunately, adiabatic quantum optimization fails: the system gets trapped in one of the numerous local minima.