Effect of Local Minima on Adiabatic Quantum Optimization

TL;DR

This paper introduces a perturbative approach to estimate spectral gaps in adiabatic quantum optimization, revealing that problems with many local minima near the global minimum have exponentially small gaps, limiting quantum advantage.

Contribution

It provides a new method to estimate spectral gaps and demonstrates the limitations of adiabatic quantum optimization for complex energy landscapes.

Findings

Spectral gap estimation can be approximated perturbatively.

Problems with many local minima have exponentially small gaps.

Adiabatic quantum advantage is limited for complex landscapes.

Abstract

We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. We show that for problems that have exponentially large number of local minima close to the global minimum, the gap becomes exponentially small making the computation time exponentially long. The quantum advantage of adiabatic quantum computation may then be accessed only via the local adiabatic evolution, which requires phase coherence throughout the evolution and knowledge of the spectrum. Such problems, therefore, are not suitable for adiabatic quantum computation.