We propose SnCQA, a set of hardware-efficient variational circuits of equivariant quantum convolutional circuits respective to permutation symmetries and spatial lattice symmetries with the number of qubits $n$. By exploiting permutation symmetries of the system, such as lattice Hamiltonians common to many quantum many-body and quantum chemistry problems, Our quantum neural networks are suitable for solving machine learning problems where permutation symmetries are present, which could lead to significant savings of computational costs. Aside from its theoretical novelty, we find our simulations perform well in practical instances of learning ground states in quantum computational chemistry, where we could achieve comparable performances to traditional methods with few tens of parameters. Compared to other traditional variational quantum circuits, such as the pure hardware-efficient ansatz (pHEA), we show that SnCQA is more scalable, accurate, and noise resilient (with $20\times$ better performance on $3 \times 4$ square lattice and $200\% - 1000\%$ resource savings in various lattice sizes and key criterions such as the number of layers, parameters, and times to converge in our cases), suggesting a potentially favorable experiment on near-time quantum devices.

翻译：我们提出SnCQA，这是一组硬件高效的变分电路，它们是对应于置换对称性和空间晶格对称性的等变量子卷积电路，其量子比特数为$n$。通过利用系统的置换对称性（例如许多量子多体和量子化学问题中常见的晶格哈密顿量），我们的量子神经网络适用于解决存在置换对称性的机器学习问题，这可能导致计算成本的大幅节省。除了理论上的新颖性之外，我们发现我们的模拟在量子计算化学中学习基态的实际案例中表现良好，只需几十个参数即可实现与传统方法相当的性能。与其他传统变分量子电路（例如纯硬件高效拟设（pHEA））相比，我们表明SnCQA更具可扩展性、准确性更高且对噪声更具韧性（在$3 \times 4$方形晶格上性能提升20倍，在不同晶格尺寸和关键指标（如层数、参数数量及我们的案例中的收敛时间）上实现200%至1000%的资源节省），这表明它在近期量子设备上可能具有实验优势。