# Training Saturation in Layerwise Quantum Approximate Optimisation

arXiv: https://arxiv.org/abs/2106.13814 · HTML: https://mlsanity.com/paper/2106.13814/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 35 references that are in the map.
- First posted on arXiv 25 Jun 2021.

## Builds on (most cited first, 4 of 4)

- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- ['Zhores' -- Petaflops supercomputer for data-driven modeling, machine learning and artificial intelligence installed in Skolkovo Institute of Science and Technology](https://mlsanity.com/paper/1902.07490/) (2019)
- [Reachability Deficits in Quantum Approximate Optimization](https://mlsanity.com/paper/1906.11259/) (2019)
- [Noise-Resilient Variational Hybrid Quantum-Classical Optimization](https://mlsanity.com/paper/1912.06744/) (2019)

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

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Hybrid Quantum Classical Graph Neural Networks for Particle Track Reconstruction](https://mlsanity.com/paper/2109.12636/) (2021)
- [Quantum approximate optimization via learning-based adaptive optimization](https://mlsanity.com/paper/2303.14877/) (2023)
- [TITAN: A Trajectory-Informed Technique for Adaptive Parameter Freezing in Large-Scale VQE](https://mlsanity.com/paper/2509.15193/) (2025)
- [On Circuit Depth Scaling For Quantum Approximate Optimization](https://mlsanity.com/paper/2205.01698/) (2022)

## Abstract (arXiv)

Quantum Approximate Optimisation (QAOA) is the most studied gate based variational quantum algorithm today. We train QAOA one layer at a time to maximize overlap with an n qubit target state. Doing so we discovered that such training always saturates -- called textittraining saturation -- at some depth p^*, meaning that past a certain depth, overlap can not be improved by adding subsequent layers. We formulate necessary conditions for saturation. Numerically, we find layerwise QAOA reaches its maximum overlap at depth p^*=n. The addition of coherent dephasing errors to training removes saturation, recovering robustness to layerwise training. This study sheds new light on the performance limitations and prospects of QAOA.