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arXiv · 2004.09002

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

Edward Farhi·David Gamarnik·Sam Gutmann
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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.

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Edward Farhi, David Gamarnik, Sam Gutmann. 2020-04-20. The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case. https://arxiv.org/abs/2004.09002

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