Limitations of Quantum Approximate Optimization in Solving Generic Higher-Order Constraint-Satisfaction Problems

• URL: http://arxiv.org/abs/2411.19388v1
• Date: Thu, 28 Nov 2024 21:39:58 GMT
• Title: Limitations of Quantum Approximate Optimization in Solving Generic Higher-Order Constraint-Satisfaction Problems
• Authors: Thorge Müller, Ajainderpal Singh, Frank K. Wilhelm, Tim Bode,
• Abstract summary: The ability of the Quantum Approximate Optimization Algorithm to deliver a quantum advantage on optimization problems is still unclear. We analyze the QAOA's performance on random Max-$k$XOR as a function of $k$ and the clause-to-variable ratio. 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.
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• Abstract: 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 in 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 (MF-AOA) 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.

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