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arXiv:2309.13552 · Physical Review A

Iterative layerwise training for the quantum approximate optimization algorithm

Xinwei Lee, Xinjian Yan, Ningyi Xie, Dongsheng Cai, Yoshiyuki Saito, Nobuyoshi Asai

Published May 2, 2024• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.109.052406

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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 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) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area of research, as bad initialization might deteriorate the quality of the results, especially at great circuit depths. We first discuss the patterns of optimal parameters in QAOA in two directions: the angle index and the circuit depth. Then, we discuss the symmetries and periodicity of the expectation that is used to determine the bounds of the search space. Based on the patterns in optimal parameters and the bounds restriction, we propose a strategy that predicts the new initial parameters by taking the difference between the previous optimal parameters. Unlike most other strategies, the strategy we propose does not require multiple trials to ensure success. It only requires one prediction when progressing to the next depth. We compare this strategy with our previously proposed strategy and the layerwise strategy for solving the Max-cut problem in terms of the approximation ratio and the optimization cost. We also address the non-optimality in previous parameters, which is seldom discussed in other works despite its importance in explaining the behavior of variational quantum algorithms.
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 variational quantum algorithm (VQA) ideal for noisy intermediate-scale quantum (NISQ) processors, and is highly successful in solving combinatorial optimization problems (COPs). It has been observed that the optimal parameters obtained from one instance of a COP can be transferred to another instance, resulting in generally good solutions for the latter. In this work, we propose a refinement scheme in which only a subset of QAOA layers is optimized following parameter transfer, with a focus on the Max-Cut problem. Our motivation is to reduce the complexity of the loss landscape when optimizing all the layers of high-depth QAOA circuits, as well as to reduce the optimization time. We investigate the potential hierarchical roles of different layers and analyze how the approximation ratio scales with increasing problem size. Our findings indicate that the selective layer optimization scheme offers a favorable trade-off between solution quality and computational time, and can be more beneficial than full optimization at a lower optimization time.
The quantum approximate optimization algorithm (QAOA) has numerous promising applications in solving the combinatorial optimization problems on near-term Noisy Intermediate Scalable Quantum (NISQ) devices. QAOA has a quantum-classical hybrid structure. Its quantum part consists of a parameterized alternating operator ansatz, and its classical part comprises an optimization algorithm, which optimizes the parameters to maximize the expectation value of the problem Hamiltonian. This expectation value depends highly on the parameters, this implies that a set of good parameters leads to an accurate solution. However, at large circuit depth of QAOA, it is difficult to achieve global optimization due to the multiple occurrences of local minima or maxima. In this paper, we propose a parameters fixing strategy which gives high approximation ratio on average, even at large circuit depths, by initializing QAOA with the optimal parameters obtained from the previous depths. We test our strategy on the Max-cut problem of certain classes of graphs such as the 3-regular graphs and the Erdös-Rényi graphs.
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.