Entropic barriers as a reason for hardness in both classical and quantum algorithms

TL;DR

This paper investigates the difficulty of solving 3-XORSAT problems on 3-regular graphs, revealing that entropic barriers significantly hinder both classical and quantum algorithms, leading to exponential runtimes.

Contribution

The study introduces a quasi-greedy algorithm and demonstrates that entropic barriers are a primary source of problem hardness for classical and quantum methods.

Findings

Entropic barriers cause exponential difficulty in classical local algorithms.

Quantum annealing is similarly affected by entropic barriers, not just tunnelling.

The identified difficulty is distinct from tunnelling barriers but still results in exponential times.

Abstract

We study both classical and quantum algorithms to solve a hard optimization problem, namely 3-XORSAT on 3-regular random graphs. By introducing a new quasi-greedy algorithm that is not allowed to jump over large energy barriers, we show that the problem hardness is mainly due to entropic barriers. We study, both analytically and numerically, several optimization algorithms, finding that entropic barriers affect in a similar way classical local algorithms and quantum annealing. For the adiabatic algorithm, the difficulty we identify is distinct from that of tunnelling under large barriers, but does, nonetheless, give rise to exponential running (annealing) times.