# Layer VQE: A Variational Approach for Combinatorial Optimization on Noisy Quantum Computers

arXiv: https://arxiv.org/abs/2102.05566 · HTML: https://mlsanity.com/paper/2102.05566/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2021.

## Built on by (most cited first, 14 of 14)

- [BEINIT: Avoiding Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2204.13751/) (2022)
- [A semi-agnostic ansatz with variable structure for quantum machine learning](https://mlsanity.com/paper/2103.06712/) (2021)
- [A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers](https://mlsanity.com/paper/2205.12481/) (2022)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Quantum Federated Learning: A Comprehensive Survey](https://mlsanity.com/paper/2508.15998/) (2025)
- [QAOA-GPT: Efficient Generation of Adaptive and Regular Quantum Approximate Optimization Algorithm Circuits](https://mlsanity.com/paper/2504.16350/) (2025)
- [QAdaPrune: Adaptive Parameter Pruning For Training Variational Quantum Circuits](https://mlsanity.com/paper/2408.13352/) (2024)
- [Quantum Approximate Optimization Algorithm applied to the binary perceptron](https://mlsanity.com/paper/2112.10219/) (2021)
- [Generative Quantum-inspired Kolmogorov-Arnold Eigensolver](https://mlsanity.com/paper/2605.04604/) (2026)
- [Towards Practical Explainability with Cluster Descriptors](https://mlsanity.com/paper/2210.10662/) (2022)
- [Quantum-Enhanced Forecasting: Leveraging Quantum Gramian Angular Field and CNNs for Stock Return Predictions](https://mlsanity.com/paper/2310.07427/) (2023)
- [Estimating the randomness of quantum circuit ensembles up to 50 qubits](https://mlsanity.com/paper/2205.09900/) (2022)
- [Noisy Tensor Ring approximation for computing gradients of Variational Quantum Eigensolver for Combinatorial Optimization](https://mlsanity.com/paper/2307.03884/) (2023)
- [Improving Parameter Training for VQEs by Sequential Hamiltonian Assembly](https://mlsanity.com/paper/2312.05552/) (2023)

## Abstract (arXiv)

Combinatorial optimization on near-term quantum devices is a promising path to demonstrating quantum advantage. However, the capabilities of these devices are constrained by high noise or error rates. In this paper, we propose an iterative Layer VQE (L-VQE) approach, inspired by the Variational Quantum Eigensolver (VQE). We present a large-scale numerical study, simulating circuits with up to 40 qubits and 352 parameters, that demonstrates the potential of the proposed approach. We evaluate quantum optimization heuristics on the problem of detecting multiple communities in networks, for which we introduce a novel qubit-frugal formulation. We numerically compare L-VQE with Quantum Approximate Optimization Algorithm (QAOA) and demonstrate that QAOA achieves lower approximation ratios while requiring significantly deeper circuits. We show that L-VQE is more robust to finite sampling errors and has a higher chance of finding the solution as compared with standard VQE approaches. Our simulation results show that L-VQE performs well under realistic hardware noise.