The quantum approximate optimization algorithm (QAOA) achieves monotonically improving performance with circuit depth $p$, yet the study of the high-depth regime has been obstructed by the exponential in $p$ cost of existing exact evaluation techniques. In this Letter, we prove that, in the infinite-size limit, the depth-$p$ QAOA state for the Sherrington-Kirkpatrick (SK) model converges to the state of a spin coupled to $p$ bosonic modes. We simulate the spin-boson system using matrix product states and provide numerical evidence that QAOA obtains a ($1-\epsilon$) approximation to the optimal energy of the SK model with circuit depth $\mathcal{O}(n/{\epsilon }^{1.13})$ in the average case. The modest computational cost of our approach allows us to optimize QAOA parameters and observe that QAOA achieves $ϵ\lesssim 2.2\%$ at $p=160$ in the infinite-size limit, extending far beyond $p\le 20$ accessible to prior exact methods. Our mapping provides a many-body route to study and optimize high-depth QAOA in regimes previously inaccessible to exact evaluation.

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• Quantum algorithms & computation