# The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model

arXiv: https://arxiv.org/abs/2110.14206 · HTML: https://mlsanity.com/paper/2110.14206/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Oct 2021.

## Built on by (most cited first, 6 of 6)

- [Agent-Q: Fine-Tuning Large Language Models for Quantum Circuit Generation and Optimization](https://mlsanity.com/paper/2504.11109/) (2025)
- [Graph neural network initialisation of quantum approximate optimisation](https://mlsanity.com/paper/2111.03016/) (2021)
- [FrozenQubits: Boosting Fidelity of QAOA by Skipping Hotspot Nodes](https://mlsanity.com/paper/2210.17037/) (2022)
- [Generalized Probabilistic Approximate Optimization Algorithm](https://mlsanity.com/paper/2507.07420/) (2025)
- [Practical advantage beyond the quadratic speedup limit with fully-quantum walks](https://mlsanity.com/paper/2607.22818/) (2026)
- [Probabilistic Algorithms for Ising Machines from Optimization to Generative AI](https://mlsanity.com/paper/2610.03972/) (2026)

## Abstract (arXiv)

The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth p. We apply the QAOA to MaxCut on large-girth D-regular graphs. We give an iterative formula to evaluate performance for any D at any depth p. Looking at random D-regular graphs, at optimal parameters and as D goes to infinity, we find that the p=11 QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these D-regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max-q-XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as O(p2 4p). This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to p=20. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as p goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.