The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case
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.
Keywords
Cite
@article{arxiv.2004.09002,
title = {The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case},
author = {Edward Farhi and David Gamarnik and Sam Gutmann},
journal= {arXiv preprint arXiv:2004.09002},
year = {2020}
}
Comments
19 pages, no figures