Size dependence of the minimum excitation gap in the Quantum Adiabatic Algorithm

TL;DR

This paper investigates how the minimum energy gap in the Quantum Adiabatic Algorithm for the Exact Cover problem scales with problem size, revealing polynomial median complexity for sizes up to 128.

Contribution

It provides the first large-scale analysis showing polynomial median complexity of QAA for Exact Cover, identifying the avoided crossing as the main bottleneck.

Findings

QAA exhibits polynomial median complexity for N <= 128.

The bottleneck is an isolated avoided crossing of Landau-Zener type.

Classical algorithms show exponential median complexity for the same problem sizes.

Abstract

We study the typical (median) value of the minimum gap in the quantum version of the Exact Cover problem using Quantum Monte Carlo simulations, in order to understand the complexity of the quantum adiabatic algorithm (QAA) for much larger sizes than before. For a range of sizes, N <= 128, where the classical Davis-Putnam algorithm shows exponential median complexity, the QAA shows polynomial median complexity. The bottleneck of the algorithm is an isolated avoided crossing point of a Landau-Zener type (collision between the two lowest energy levels only).