Qu&Co comments on this publication:
Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm that has been heavily investigated due to its potential during the NISQ era. It is designed to find approximate solutions to combinatorial search problems and was first applied to the Max-Cut problem for d-regular graphs. The system is initially prepared in a product state and then p layers of unitaries U(C,γ) and U(B, β) are alternately applied; this can be seen as a Trotterized version of (non-adiabatic) quantum annealing with parametrized annealing pathway.
In these 2 papers by Farhi, Gamarnik & Gutmann posted on April 20 and May 18, the focus problem is to find a large independent set in a random graph of fixed average degree d for the problem of Maximum Independent Set (MIS) on random graphs. Generally, the performance of the QAOA can only improve with depth p, but it is shown that for MIS the algorithm will fail to pass a certain performance barrier if 2p is less than w*log(n)/log(d/ln(2)) for any w <1 with d big enough. The quantum algorithm consists of p unitaries that each respect the locality of the underlying graph. With a fixed average degree of d this means that each qubit typically has an influence sphere of roughly dp other qubits. For qubits further than 2p apart on the graph these influence spheres do not intersect and we can show that measurements of these qubits are uncorrelated, however if p is large enough that dp exceeds n our arguments do not apply and we have no indication that the QAOA will fail.
Overlap Gap Property states that for a given random graph the intersection of any two large independent sets is either big or small, there is no middle ground. Using OGP and the locality of the QAOA, it is shown 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.
The worst case performance circumstances can be easily created to exploit QAOA’s weaknesses. Through construction of operators C and B, QAOA is inherently a local quantum algorithm since when conjugating a single qubit operator produces an operator only involving that qubit and those connected to it on the graph, creating a shallow circuit. This can be exploited to construct examples where the QAOA’s performance is provably below optimal. For example, for Max-Cut when p is a small enough constant times log(n) we show that the approximation ratio is no better than ½ for large d, and for MIS the approximation ratio goes to 0 at large d.
This is an important result for the problem of MIS on random graphs that, although not directly generalizable to other problems, is still valuable for creating bounds for QAOA-applied problems. Knowing that on the average case for MIS, QAOA requires to see the whole graph, therefore require a large p, will change the way QAOA and its potential is viewed for certain problems, and these papers illustrate methods to quantify that.