Quantum Computational Phase Transition in Combinatorial Problems

TL;DR

This paper investigates the empirical performance of QAOA on hard combinatorial problems, revealing a phase transition related to problem density where quantum advantages are most evident.

Contribution

It identifies a computational phase transition in QAOA performance linked to problem density and connects this to circuit controllability and complexity, highlighting regions where quantum advantage is observable.

Findings

QAOA exhibits a phase transition at a critical problem density.

In high-density regions, QAOA's approximation ratio decays slower than classical algorithms.

Quantum advantage is most detectable in the high problem density region.

Abstract

Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that $BQP\neq NP$, it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT -- random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithm lies. Then, we show that the high problem density region, which limits QAOA's performance in hard optimization problems ({\it reachability deficits}), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified.