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.
Comments
12 pages, 8 figures, accepted to IEEE QCE 2026 (QALG Track). Replaced the schematic in Fig. 8 with results based on real data and expanded the accompanying discussion
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.
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) 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.
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.
To address the limited expressive capability of the standard quantum approximate optimization algorithm (QAOA) in shallow circuits and the increased noise sensitivity caused by deeper quantum circuits, an improved QAOA, termed MA-RY-QAOA, was proposed by integrating parameter multi-angleization and an RY-assisted structure. In the proposed algorithm, edge-parameter multi-angleization and RY expansion were incorporated into the problem layer, while qubit-parameter multi-angleization was introduced into the mixing layer. Simulation experiments were conducted on the Max-cut problem of random graphs, complete graphs, 2-regular graphs, and 4-regular graphs with circuit depth ( p=1 <?fx-imagestate width="7.45066643" height="2.87866688"?> <?fx-imagestate width="7.45066643" height="2.87866688"?> ) and node sizes ranging from 4 to 12. The average approximation ratio was adopted as the evaluation metric. The results showed that the average approximation ratios achieved by MA-RY-QAOA and MA-QAOA were generally superior to those obtained by the standard QAOA, RY-QAOA, and QAOA+ algorithms. In particular, MA-RY-QAOA exhibited better performance on 2-regular and 4-regular graphs and achieved higher approximation ratios for certain scales of random graphs. These findings indicate that MA-RY-QAOA improves optimization performance while maintaining a shallow circuit structure and provides a useful reference for the design of QAOA variants on noisy intermediate-scale quantum (NISQ) 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.
Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA uses a parameterized circuit with p layers, where higher p yields better solutions, but requires optimizing 2p independent parameters, which is challenging at large p. We present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing the work of Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and coefficient count until convergence, our method constructs high-quality schedules for large p. We provide theoretical justification using Jackson's theorem and Lipschitz continuity to bound the required number of basis coefficients for a given accuracy. Our approach achieves better performance with fewer optimization steps than existing methods across three benchmark problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. Additionally, we observe that a mild growth in QAOA depth suffices to solve the SK model exactly, a result of independent theoretical interest.