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The design of fast algorithms for combinatorial optimization greatlycontributes to a plethora of domains such as logistics, finance, and chemistry.Quantum approximate optimization algorithms (QAOAs), which utilize the power ofquantum machines and inherit the spirit of adiabatic evolution, are novelapproaches to tackle combinatorial problems with potential runtime speedups.However, hurdled by the limited quantum resources nowadays, QAOAs areinfeasible to manipulate large-scale problems. To address this issue, here werevisit the MaxCut problem via the divide-and-conquer heuristic: seek thesolutions of subgraphs in parallel and then merge these solutions to obtain theglobal solution. Due to the $\mathbb{Z}_2$ symmetry in MaxCut, we prove thatthe merging process can be further cast into a new MaxCut problem and thus beaddressed by QAOAs or other MaxCut solvers. With this regard, we proposeQAOA-in-QAOA ($\text{QAOA}^2$) to solve arbitrary large-scale MaxCut problemsusing small quantum machines. We also prove that the approximation ratio of$\text{QAOA}^2$ is lower bounded by 1/2. Experiment results illustrate thatunder different graph settings, $\text{QAOA}^2$ attains a competitive or evenbetter performance over the best known classical algorithms when the node countis around 2000. Our method can be seamlessly embedded into other advancedstrategies to enhance the capability of QAOAs in large-scale combinatorialoptimization problems.

QAOA-in-QAOA: solving large-scale MaxCut problems on small quantum machines