Reachability Deficits in Quantum Approximate Optimization of Graph Problems

TL;DR

This paper investigates how the density of constraints in graph problems affects the performance of the quantum approximate optimization algorithm (QAOA), revealing that higher density leads to approximation deficits and scaling challenges.

Contribution

It introduces the density of problem constraints as a key performance indicator for QAOA and analyzes its impact on approximation quality and depth requirements.

Findings

Density correlates strongly with approximation inefficiency.

Required QAOA depth scales critically with problem density.

Experimental data shows a rapid decline in performance beyond intermediate density.

Abstract

The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary quantum applications development. Here we show that the \emph{density} of problem constraints versus problem variables acts as a performance indicator. Density is found to correlate strongly with approximation inefficiency for fixed depth QAOA applied to random graph minimization problem instances. Further, the required depth for accurate QAOA solution to graph problem instances scales critically with density. Motivated by Google's recent experimental realization of QAOA, we preform a reanalysis of the reported data reproduced in an ideal noiseless setting. We found that the reported capabilities of instances addressed experimentally by Google, approach a rapid fall-off region in approximation quality experienced beyond intermediate-density. Our findings offer new insight into performance analysis of contemporary quantum optimization algorithms and contradict recent speculation regarding low-depth QAOA performance benefits.