The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: Worst Case Examples

TL;DR

This paper demonstrates that the Quantum Approximate Optimization Algorithm's performance on certain graph problems is limited when it only considers local neighborhoods, especially on large random regular graphs, showing fundamental worst-case constraints.

Contribution

The paper provides a worst-case analysis of QAOA, revealing its limitations when restricted to local graph neighborhoods, and establishes bounds on its approximation ratios for Max-Cut and Independent Set.

Findings

QAOA's performance depends only on local neighborhoods in certain graphs.

QAOA achieves at most 1/2 approximation ratio for Max-Cut on large bipartite regular graphs.

Approximation ratio for Independent Set diminishes as degree increases.

Abstract

The Quantum Approximate Optimization Algorithm can be applied to search problems on graphs with a cost function that is a sum of terms corresponding to the edges. When conjugating an edge term, the QAOA unitary at depth p produces an operator that depends only on the subgraph consisting of edges that are at most p away from the edge in question. On random d-regular graphs, with d fixed and with p a small constant time log n, these neighborhoods are almost all trees and so the performance of the QAOA is determined only by how it acts on an edge in the middle of tree. Both bipartite random d-regular graphs and general random d-regular graphs locally are trees so the QAOA's performance is the same on these two ensembles. Using this we can show that the QAOA with $(d-1)^{2p} < n^A$ for any $A<1$, can only achieve an approximation ratio of 1/2 for Max-Cut on bipartite random d-regular graphs for d large. For Maximum Independent Set, in the same setting, the best approximation ratio is a d-dependent constant that goes to 0 as d gets big.