Elimination of Perturbative Crossings in Adiabatic Quantum Optimization

TL;DR

This paper presents a polynomial-time method to eliminate all perturbative anticrossings in adiabatic quantum optimization paths for Ising models, challenging the assumption that such anticrossings cause exponential runtimes.

Contribution

It provides a simple, deterministic construction to remove all perturbative anticrossings in polynomial time for Ising models with polynomial gaps.

Findings

All perturbative anticrossings can be eliminated in polynomial time.

Adiabatic quantum optimization may not require exponential time if anticrossings are avoided.

Some other factors must cause exponential runtimes in NP-complete problems.

Abstract

It was recently shown that, for solving NP-complete problems, adiabatic paths always exist without finite-order perturbative crossings between local and global minima, which could lead to anticrossings with exponentially small energy gaps if present. However, it was not shown whether such a path could be found easily. Here, we give a simple construction that deterministically eliminates all such anticrossings in polynomial time, space, and energy, for any Ising models with polynomial final gap. Thus, in order for adiabatic quantum optimization to require exponential time to solve any NP-complete problem, some quality other than this type of anticrossing must be unavoidable and necessitate exponentially long runtimes.