Limitations of quantum approximate optimization in solving generic higher-order constraint-satisfaction problems
Thorge Müller, Ajainderpal Singh, Frank K. Wilhelm, Tim Bode · Physical Review Research · 2025
The ability of the quantum approximate optimization algorithm (QAOA) to deliver a quantum advantage on combinatorial optimization problems is still unclear. Recently, a scaling advantage over a classical solver was postulated to exist for random 8-SAT at the satisfiability threshold. At the same time, the viability of quantum error mitigation for deep circuits on near-term devices has been put into doubt. Here we analyze the QAOA's performance on random Max- k XOR as a function of k and the clause-to-variable ratio. As a classical benchmark, we use the mean-field approximate optimization algorithm and find that it performs better than or equal to the QAOA on average. Still, for large k and numbers of layers p , there may remain a window of opportunity for the QAOA. However, by extrapolating our numerical results, we find that reaching high levels of satisfaction would require extremely large p , which must be considered rather difficult both in the variational context and on near-term devices.