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Addressing Barren Plateaus in QAOA using layer-wise and adaptive optimization

Urvi Sharma

Published Jan 30, 2025• 0 citations
OpenAlex

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational quan- tum algorithm for solving combinatorial optimization problems. One major challenge with QAOA arises when the optimizer is unable to progress due to barren plateau problems. To address this, we propose a novel optimization technique called the Layered Quantum Approximate Optimization Al- gorithm (LQAOA), which uses warm-starting by fine-tuning parameters from previously optimized layers to initialize subsequent layers, training one layer at a time. Additionally, we incorporate adaptive layer-adding strategies, where layers are added based on a cost-convergence condition. The optimization approach, tested on Erdős–Rényi graphs for the Max-Cut problem with various hyperparameter settings, demonstrates improved performance over standard QAOA.

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The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational quan- tum algorithm for solving combinatorial optimization problems. One major challenge with QAOA arises when the optimizer is unable to progress due to barren plateau problems. To address this, we propose a novel optimization technique called the Layered Quantum Approximate Optimization Al- gorithm (LQAOA), which uses warm-starting by fine-tuning parameters from previously optimized layers to initialize subsequent layers, training one layer at a time. Additionally, we incorporate adaptive layer-adding strategies, where layers are added based on a cost-convergence condition. The optimization approach, tested on Erdős–Rényi graphs for the Max-Cut problem with various hyperparameter settings, demonstrates improved performance over standard QAOA.
The capability of the quantum approximate optimization algorithm (QAOA) in solving combinatorial optimization problems has been intensively studied in recent years due to its application in the quantum-classical hybrid regime. Despite having difficulties that are innate in the variational quantum algorithms (VQA), such as barren plateaus and the local minima problem, QAOA remains one of the applications that is suitable for the recent noisy intermediate scale quantum (NISQ) devices. Recent works have shown that the performance of QAOA largely depends on the initial parameters, which motivate parameter initialization strategies to obtain good initial points for the optimization of QAOA. On the other hand, optimization strategies focus on the optimization part of QAOA instead of the parameter initialization. Instead of having absolute advantages, these strategies usually impose trade-offs to the performance of the optimization problems. One such example is the layerwise optimization strategy, in which the QAOA parameters are optimized layer by layer instead of the full optimization. The layerwise strategy costs less in total compared to the full optimization, in exchange of lower approximation ratio. In this work, we propose the iterative layerwise optimization strategy and explore the possibility for the reduction of optimization cost in solving problems with QAOA. Using numerical simulations, we found that by combining the iterative layerwise strategy with proper initialization strategies, the optimization cost can be significantly reduced in exchange for a minor reduction in the approximation ratio. We also show that in some cases, the approximation ratio given by the iterative layerwise strategy is even higher than that given by the full optimization.
In the Noisy Intermediate-Scale Quantum (NISQ) era, the Quantum Approximate Optimization Algorithm (QAOA) has attracted significant attention as an algorithm for solving combinatorial optimization problems that are classically intractable. Various modifications and training strategies have been proposed to enhance the performance of QAOA. However, as the circuit depth increases, existing methods encounter greater training difficulty and reach a barren plateau in performance improvements. In this paper, we propose the Smart Insert Layerwise QAOA (SIL QAOA) to overcome these challenges. In SIL QAOA, at each stage, a new layer is inserted into all possible positions; each circuit is independently optimized, and the parameter set from the circuit yielding the highest expectation value is adopted as the initial parameter for the next stage. Applying the proposed method to the MaxCut problem, we achieved up to approximately 11% higher approximation ratios compared to conventional Layerwise learning QAOA, while successfully mitigating early performance saturation observed at shallow depths. These results demonstrate the superior performance improvement capability of SIL QAOA. Furthermore, we also discuss its potential applicability to more complex optimization problems and deeper variational quantum circuits in future work.
The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA ansatz is not optimal, there is no systematic approach for finding better ansätze. We address this problem by developing an iterative version of QAOA that is problem-tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, while simultaneously reducing the required number of CNOT gates and optimization parameters. We provide evidence that this speedup is connected to the concept of shortcuts to adiabaticity.
Graph-based combinatorial optimization problems are computationally challenging for classical optimization techniques due to their NP-hard nature. This paper proposes a novel Quantum-Assisted Hybrid Optimization Algorithm (QAHOA) that integrates the Quantum Approximate Optimization Algorithm (QAOA), spectral graph theory, and classical optimization strategies for graph-based combinatorial optimization. The developed framework uses spectral initialization via the graph Laplacian Fiedler vector to provide structurally informed starting variational parameters, aiming to stabilize the parameter search space and improve convergence trajectories under noisy conditions. The mathematical framework, alongside a convergence analysis to first-order stationary points, is provided to establish theoretical baseline properties. Evaluated on synthetic Erdős–Rényi random graph instances, the current study serves as a preliminary, small-to-medium-scale proof-of-concept. Controlled evaluations demonstrate that the proposed QAHOA achieves marginal yet stable and statistically significant improvements in mean cut values and approximation ratios over classical Greedy heuristics, Simulated Annealing, the standard QAOA, and Warm-Start QAOA. Formal paired hypothesis testing mathematically verifies the stability of the framework, while preliminary 4-qubit hardware executions demonstrate baseline architectural compatibility under physical NISQ noise channels.
Feedback-based adaptive quantum optimization (FALQON) is a promising approach for solving combinatorial problems on noisy intermediate-scale quantum (NISQ) devices, requiring only single circuit evaluations per layer. However, standard FALQON relies on fixed hyperparameters that severely limit convergence speed, requiring hundreds to thousands of layers for acceptable solutions. This paper proposes Optimal FALQON, an optimization-based formulation that treats the per-layer time step (δkδ_kδk​) and scaling factor (MkM_kMk​) as decision variables optimized via classical methods. We present a comprehensive empirical study on all 94 non-isomorphic 3-regular graphs with 12 vertices, comparing Optimal FALQON with standard FALQON and multiple QAOA variants. Results demonstrate statistically significant improvements in success probability, evaluation efficiency, and depth-normalized cost across the evaluated benchmarks. Furthermore, initializing QAOA with parameters from Optimal FALQON yields superior warm-start performance compared to fixed initialization.
The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA Ansatz is not optimal, there is no systematic approach for finding better Ans\"atze. We address this problem by developing an iterative version of QAOA that is problem tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, while simultaneously reducing the required number of CNOT gates and optimization parameters. We provide evidence that this speedup is connected to the concept of shortcuts to adiabaticity.