Many fundamental properties of a quantum system are captured by its Hamiltonian and ground state. Despite the significance of ground states preparation (GSP), this task is classically intractable for large-scale Hamiltonians. Quantum neural networks (QNNs), which exert the power of modern quantum machines, have emerged as a leading protocol to conquer this issue. As such, how to enhance the performance of QNNs becomes a crucial topic in GSP. Empirical evidence showed that QNNs with handcraft symmetric ansatzes generally experience better trainability than those with asymmetric ansatzes, while theoretical explanations have not been explored. To fill this knowledge gap, here we propose the effective quantum neural tangent kernel (EQNTK) and connect this concept with over-parameterization theory to quantify the convergence of QNNs towards the global optima. We uncover that the advance of symmetric ansatzes attributes to their large EQNTK value with low effective dimension, which requests few parameters and quantum circuit depth to reach the over-parameterization regime permitting a benign loss landscape and fast convergence. Guided by EQNTK, we further devise a symmetric pruning (SP) scheme to automatically tailor a symmetric ansatz from an over-parameterized and asymmetric one to greatly improve the performance of QNNs when the explicit symmetry information of Hamiltonian is unavailable. Extensive numerical simulations are conducted to validate the analytical results of EQNTK and the effectiveness of SP.

翻译：量子系统的基本性质通常由其哈密顿量和基态描述。尽管基态制备（GSP）至关重要，但对于大规模哈密顿量的计算，经典方法难以处理。借助现代量子计算能力的量子神经网络（QNN）已成为解决该问题的领先方案。因此，如何提升QNN的性能成为基态制备领域的核心课题。实验证据表明，采用人工设计的对称拟设的QNN通常比非对称拟设具有更好的可训练性，但相关理论解释尚未明确。为填补这一理论空白，本文提出有效量子神经正切核（EQNTK），并将其与过参数化理论关联，以量化QNN收敛至全局最优解的能力。我们发现，对称拟设的优势在于其具有低有效维度的大EQNTK值，这使得仅需少量参数和量子电路深度即可进入过参数化区域，从而获得良好的损失景观和快速收敛性。基于EQNTK理论，我们进一步设计了对称剪枝（SP）方案，当哈密顿量的显式对称信息不可用时，该方案可自动从过参数化非对称拟设中裁剪出对称拟设，显著提升QNN性能。通过大规模数值模拟验证了EQNTK分析结果的理论正确性及SP方案的有效性。