# On Circuit Depth Scaling For Quantum Approximate Optimization

arXiv: https://arxiv.org/abs/2205.01698 · HTML: https://mlsanity.com/paper/2205.01698/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 34 references that are in the map.
- First posted on arXiv 3 May 2022.

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

- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Reachability Deficits in Quantum Approximate Optimization](https://mlsanity.com/paper/1906.11259/) (2019)
- [Training Saturation in Layerwise Quantum Approximate Optimisation](https://mlsanity.com/paper/2106.13814/) (2021)

## Built on by (most cited first, 1 of 1)

- [Quantum Approaches to Urban Logistics: From Core QAOA to Clustered Scalability](https://mlsanity.com/paper/2512.10813/) (2025)

## Abstract (arXiv)

Variational quantum algorithms are the centerpiece of modern quantum programming. These algorithms involve training parameterized quantum circuits using a classical co-processor, an approach adapted partly from classical machine learning. An important subclass of these algorithms, designed for combinatorial optimization on currrent quantum hardware, is the quantum approximate optimization algorithm (QAOA). It is known that problem density - a problem constraint to variable ratio - induces under-parametrization in fixed depth QAOA. Density dependent performance has been reported in the literature, yet the circuit depth required to achieve fixed performance (henceforth called critical depth) remained unknown. Here, we propose a predictive model, based on a logistic saturation conjecture for critical depth scaling with respect to density. Focusing on random instances of MAX-2-SAT, we test our predictive model against simulated data with up to 15 qubits. We report the average critical depth, required to attain a success probability of 0.7, saturates at a value of 10 for densities beyond 4. We observe the predictive model to describe the simulated data within a 3σ confidence interval. Furthermore, based on the model, a linear trend for the critical depth with respect problem size is recovered for the range of 5 to 15 qubits.