Adiabatic optimization without local minima

TL;DR

This paper investigates the fundamental limits of adiabatic quantum optimization, providing a counterexample where success is exponentially unlikely despite a simple energy landscape, and establishing conditions for polynomial scaling.

Contribution

It presents a counterexample showing exponential difficulty in adiabatic optimization even without local minima, and proves single-peaked ground states ensure better spectral gap scaling.

Findings

Counterexample with exponential eigenvalue gap decay

Single-peaked ground states guarantee polynomial gap scaling

Insights into the fundamental limitations of adiabatic quantum algorithms

Abstract

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two "lobes" separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.