On the relevance of avoided crossings away from quantum critical point to the complexity of quantum adiabatic algorithm

TL;DR

This paper critically examines the role of avoided crossings away from quantum critical points in the complexity of quantum adiabatic algorithms, challenging recent claims that they cause failure at small transverse fields.

Contribution

It provides a corrected analysis showing avoided crossings at the spectrum edges require high transverse fields, which may lead to phase transitions, disputing prior mechanisms of algorithm failure.

Findings

Avoided crossings at spectrum edges need high transverse fields.

Large instances exhibit effects only at exponential sizes.

Perturbation theory may diverge near quantum phase transitions.

Abstract

Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N. The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02 N) >> 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity.