The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case

TL;DR

This paper demonstrates that the Quantum Approximate Optimization Algorithm's effectiveness on large graphs is limited when the circuit depth is too small to see the entire graph, especially in finding large independent sets.

Contribution

It provides a theoretical analysis showing the limitations of QAOA with shallow circuits on large random graphs, highlighting the importance of circuit depth for performance.

Findings

QAOA cannot outperform a certain threshold with shallow circuits on large graphs.

Performance limitations are linked to the circuit depth relative to graph size.

At higher depths, QAOA can potentially see the entire graph, with no current evidence of performance limits.

Abstract

The Quantum Approximate Optimization Algorithm can naturally be applied to combinatorial search problems on graphs. The quantum circuit has p applications of a unitary operator that respects the locality of the graph. On a graph with bounded degree, with p small enough, measurements of distant qubits in the state output by the QAOA give uncorrelated results. We focus on finding big independent sets in random graphs with dn/2 edges keeping d fixed and n large. Using the Overlap Gap Property of almost optimal independent sets in random graphs, and the locality of the QAOA, we are able to show that if p is less than a d-dependent constant times log n, the QAOA cannot do better than finding an independent set of size .854 times the optimal for d large. Because the logarithm is slowly growing, even at one million qubits we can only show that the algorithm is blocked if p is in single digits. At higher p the algorithm "sees" the whole graph and we have no indication that performance is limited.