Exponential Complexity of the Quantum Adiabatic Algorithm for certain Satisfiability Problems

TL;DR

This paper investigates the quantum adiabatic algorithm's complexity on certain satisfiability problems, revealing an exponential increase in difficulty with problem size and correlating it with classical algorithm hardness.

Contribution

It provides the first analysis of the quantum adiabatic algorithm's complexity on typical instances of specific satisfiability problems, showing exponential scaling.

Findings

Complexity grows exponentially with problem size N.

Quantum and classical hardness are correlated.

Results suggest limitations of the quantum adiabatic approach for these problems.

Abstract

We determine the complexity of several constraint satisfaction problems using the quantum adiabatic algorithm in its simplest implementation. We do so by studying the size dependence of the gap to the first excited state of "typical" instances. We find that at large sizes N, the complexity increases exponentially for all models that we study. We also compare our results against the complexity of the analogous classical algorithm WalkSAT and show that the harder the problem is for the classical algorithm the harder it is also for the quantum adiabatic algorithm.