Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparametrization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA), a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems: MAX-CUT and MAX-2-SAT. For MAX-CUT on 2-regular graphs we observe that overparametrization is both sufficient and necessary. To establish this, we analytically show that the optimal circuit depth for such problems scale as $n/2$ for even $n$. For a more general case of random graphs, the overparametrization is observed to be sufficient, yet necessary only for a statistically dominant fraction of instances. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.

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• Quantum algorithms
• Quantum algorithms & computation