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**Motivation.** A promising approach to combinatorial optimization problems (COPs) is QAOA, which can potentially outperform classical methods when unlimited circuit depth is assumed. However, this requirement is impractical. Considering the practical utility of QAOA, where quantum resources are constrained by limited circuit depth, noise, and qubit connectivity, many variants of QAOA have been proposed to enhance its performance within these hardware constraints [Chandarana et al., 2022; Zhu et al., 2022; Yu et al., 2022; Herrman et al., 2022; Bartschi et al., 2020; Sauvage et al., 2022]. However, these alternatives often require deep domain expertise and lack generalizability across different tasks and circuit depths.
In our work, we consider **the allowable circuit depth as the primary hardware constraint, a common issue for both early fault-tolerant and NISQ devices.** We release all freedom on the mixer Hamiltonian, including operator type and parameter grouping. Our goal is to dynamically adjust the mixer Hamiltonian according to the given problem and the allowable circuit depth of a quantum device, thereby **enhancing the performance of QAOA while ensuring compatibility with the available quantum resources**.
**Contributions.** To fully exploit the potential of a QAOA circuit with arbitrarily limited circuit depth, we first theoretically analyze the convergence of QAOA:
- **Theoretical contributions.** Existing literature studied the convergence of VQAs from two aspects, namely the quantum neural tangent kernel [Liu et al., 2022; Liu et al., 2023] and gradient flow [You et al., 2022; You et al., 2023]. In particular, when analyzing the convergence rate of QNNs with symmetric ansatz, both of these two techniques involve utilizing the tools of effective dimension for quantifying the effect of symmetry degree of ansatz design on the convergence rate. While existing literature has explored the convergence theory of VQAs with symmetric ansatz well, **how concrete strategies for ansatz design affect the symmetry degree remains an open question**. In this study, we **initiate the first attempt to qualitatively analyze the effects of various parameter grouping strategies on the convergence of QAOA**.
- **Technical contributions.** We rigorously show that **fully or partially grouping the parameters according to the spatial symmetry of problem Hamiltonians could reduce the effective dimension** (Refer to Lemma B.6 in Appendix B), where the concrete reduction amount depends on the symmetric degree of problem Hamiltonians. Moreover, combined with the existing results on the effective dimension-based convergence analysis [You et al., 2022], we reach our main theoretical results (Theorem 3.1) regarding the convergence rate of QAOA with various symmetric structures in the overparameterization regime.
- **Implications.** Qualitatively, when $p$ is large enough to reach the overparameterization regime, a suitable parameters grouping strategy should be adopted to reduce the effective dimension for better convergence. Conversely, for a small $p$, the parameters should not be grouped to obtain as large a representation space as possible for better convergence. However, **it is difficult to quantitatively determine the effective dimension, the critical point for the overparameterization regime for general problem Hamiltonians**, as they are determined by the specific graph structure of the problem Hamiltonian which is case-by-case and hard to analyze. This inspires us to utilize the power of deep learning to automatically tailor the circuit depth-aware ansatz design with the optimal parameter grouping strategy for better convergence.
Guided by the established theoretical results, we propose MG-Net for automatical circuit design：
- The QAOA circuit generated by the proposed MG-Net achieves **higher approximation ratios at various circuit depths** compared to other quantum and traditional methods, advancing the **practical utility** of QAOAs.
- MG-Net **greatly reduces the cost of collecting labeled training data**, making it possible to handle **larger-scale problems and more complicated mixer Hamiltonians**.
- MG-Net provides a flexible circuit-generation framework where the data encoder can introduce more hardware constraints, and the model components can be conveniently replaced by advanced techniques. Although our theoretical analysis is based on noiseless settings, our implementation does not impose limitations on device configuration. **MG-Net can be flexibly extended to other hardware constraints**, such as **qubit connectivity** and **hardware noise**.