In the search for quantum advantage in real-world problems, one promising avenue is to use a quantum algorithm to improve on the solution found using an efficient classical algorithm. The quantum approximate optimization algorithm (QAOA) is particularly well adapted for such a “warm start” approach and can be combined with the powerful classical Goemans-Williamson (GW) algorithms based on semidefinite programing. Nonetheless, the best way to leverage the power of the QAOA remains an open question. Here we propose a general model that describes a class of QAOA variants and use it to explore routes to quantum advantages in a canonical optimization problem, MaxCut. For these algorithms we derive analytic expectation values of the cost Hamiltonian for the MaxCut problem in the level-1 case. Using these analytic results we obtain reliable averages over many instances for fairly large numbers of qubits. We find that the warm start adaptive-bias QAOA (WS-ab-QAOA) initialized by the GW algorithm outperforms previously proposed warm start variants on problems with 40 to 180 qubits. To assess whether a quantum advantage exists with this algorithm, we did numerical simulations with up to 1000 qubits to see whether the level-1 WS-ab-QAOA can improve the GW solution for three-regular graphs. In fact the improvement in the 1000-qubit case even in level 1 can only be matched by the GW algorithm after about ${10}^{5.5}$ random projections performed after the semidefinite program stage. Thus this work gives numerical evidence that the final stage of optimization after an efficient classical algorithm has produced an approximate solution may be a place where quantum advantages can be realized.

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• Quantum algorithms & computation
• Quantum computation