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arXiv:2601.20029 · Quantum Machine Intelligence

A Cyclic Layerwise QAOA Training

Enhyeok Jang, Zihan Chen, Dongho Ha, Seungwoo Choi, Yongju Lee, Jaewon Kwon, Eddy Z. Zhang, Yipeng Huang, Won Woo Ro

quant-phSubmitted Jan 27, 2026• Announced Jan 29, 2026• 1 citation
arXivHTMLPDF

Abstract

The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical algorithm for solving combinatorial optimization problems. Multi-angle QAOA (MA-QAOA), which assigns independent parameters to each Hamiltonian operator term, achieves superior approximation performance even with fewer layers than standard QAOA. Unfortunately, this increased expressibility can raise the classical computational cost due to a greater number of parameters. The recently proposed Layerwise MA-QAOA (LMA-QAOA) reduces this overhead by training one layer at a time, but it may suffer from obtaining the precise solution due to the previously fixed parameters. This work addresses two questions for efficient MA-QAOA training: (i) What is the optimal granularity for parameter updates per epoch, and (ii) How can we get precise final cost function results while only partially updating the parameters per epoch? Despite the benefit of reducing the parameters that update per epoch can reduce the classical computation overhead, too fine or coarse a granularity of Hamiltonian update can degrade the MA-QAOA training efficiency. We find that optimizing one complete layer per epoch is an efficient granularity. Moreover, selectively retraining each layer by tracking gradient variations can achieve a final cost function equivalent to the standard MA-QAOA while lowering the parameter update overhead. Based on these insights, we propose Orbit-QAOA, which cyclically revisits layers and selectively freezes stabilized parameters. Across diverse graph benchmarks, Orbit-QAOA reduces training steps by up to 81.8%, reduces approximation ratio error by up to 72x compared to the unified stop condition-applied enhanced LMA-QAOA, and achieves equivalent approximation performance compared to the standard MA-QAOA.

Journal reference
Quantum Mach. Intell. 8, 5 (2026)
DOI
10.1007/s42484-026-00357-w

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The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical algorithm for solving combinatorial optimization problems. Multi-angle QAOA (MA-QAOA), which assigns independent parameters to each Hamiltonian operator term, achieves superior approximation performance even with fewer layers than standard QAOA. Unfortunately, this increased expressibility can raise the classical computational cost due to a greater number of parameters. The recently proposed Layerwise MA-QAOA (LMA-QAOA) reduces this overhead by training one layer at a time, but it may suffer from obtaining the precise solution due to the previously fixed parameters. This work addresses two questions for efficient MA-QAOA training: (i) What is the optimal granularity for parameter updates per epoch, and (ii) How can we get precise final cost function results while only partially updating the parameters per epoch? Despite the benefit of reducing the parameters that update per epoch can reduce the classical computation overhead, too fine or coarse a granularity of Hamiltonian update can degrade the MA-QAOA training efficiency. We find that optimizing one complete layer per epoch is an efficient granularity. Moreover, selectively retraining each layer by tracking gradient variations can achieve a final cost function equivalent to the standard MA-QAOA while lowering the parameter update overhead. Based on these insights, we propose Orbit-QAOA, which cyclically revisits layers and selectively freezes stabilized parameters. Across diverse graph benchmarks, Orbit-QAOA reduces training steps by up to 81.8%, reduces approximation ratio error by up to 72x compared to the unified stop condition-applied enhanced LMA-QAOA, and achieves equivalent approximation performance compared to the standard MA-QAOA.
Multi-angle quantum approximate optimization algorithm (ma-QAOA) has been proposed as an alternative to regular QAOA for solving optimization problems using noisy intermediate scale quantum (NISQ) hardware. Here, we first implement QAOA and ma-QAOA algorithms to solve MaxCut on weighted and unweighted versions of the Erdős–Rényi (ER) and random cubic (RC) graphs of 6 and 10 nodes. Quantum circuit parameters are updated iteratively through classical feedback with the quantum circuit for forward pass modelled on Qiskit simulator without noise models. Then, the quantum circuit for the final run with updated parameters is implemented on Qiskit simulator without noise models, Qiskit simulator with experimentally benchmarked noise models (fake back-end), and also on actual experimental quantum hardware (IBM’s 127-qubit quantum processor ibm brisbane). We show that for most graph instances, ma-QAOA performs better than QAOA (in terms of approximation ratio (AR)) and needs less stages in the quantum circuit for all three aforementioned methods. Finally, we extend our results up to 14-node RC graphs while restricting ourselves to Qiskit simulation without noise (both for parameter update and final run) and show that for most instances, ma-QAOA again needs much less stages compared to QAOA to obtain reasonable AR.
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.
This paper introduces a novel variational quantum algorithm built upon the established Quantum Approximate Optimization Algorithm also known as QAOA. Since the known parameter fixing strategy imposes constraints on QAOA to enhance tractability at the cost of some expressive power, the proposed layerwise approach integrates it with the existing Multi-Angle QAOA technique, which is characterized in turn by height-ened expressiveness through an increased number of parameters, albeit with increased optimization challenges. Consequently, the proposed layerwise-Multi-Angle QAOA combines the strengths of one variant with the limitations of the other, striking a balance in algorithmic design. The effectiveness of the proposed algorithm is assessed through experimental evaluations on the Maximum Cut problem. This study reveals promising results in heuristic predictions, with robustness both in terms of approximation ratio and optimization capabilities.
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.
The multi-angle Quantum Approximate Optimization Algorithm (ma-QAOA) extends the Quantum Approximate Optimization Algorithm (QAOA) by assigning a larger number of independent variational parameters, thereby increasing expressivity and improving performance at low circuit depths. However, this larger parameterization makes training more difficult and requires repeated circuit evaluations for gradient-based optimization. In this work, we propose the Surrogate-Assisted and Fine-tuning Enhanced (SAFE) framework. SAFE first uses Low-Weight Pauli Propagation (LWPP) as a classical surrogate for pre-training ma-QAOA parameters before exact optimization. SAFE then applies parameter distillation, which removes angles that remain near zero after surrogate pre-training. Finally, SAFE performs exact fine-tuning by optimizing the remaining active parameters using the exact energy objective. We evaluate SAFE on instances of the Sherrington-Kirkpatrick model, two-dimensional square-lattice spin glass, and Max-Cut. SAFE with distillation provides the strongest overall results relative to exact-only: (i) a 64.3 percent reduction in active parameter count and (ii) a 94.5 percent reduction in estimated QPU workload. Within the SAFE workflow, adding distillation further reduces the optimizer steps to the near-optimal regime by 44.4 percent relative to without distillation. These results provide evidence that SAFE ma-QAOA can accelerate convergence to high-quality solutions while reducing the required quantum resources for exact fine-tuning, offering a resource-efficient route toward expressive ma-QAOA on NISQ hardware.
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.