A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithms for learning ground states. Specifically, [Huang et al. Science 2022], introduced an approach for learning properties of the ground state of an $n$-qubit gapped local Hamiltonian $H$ from only $n^{\mathcal{O}(1)}$ data points sampled from Hamiltonians in the same phase of matter. This was subsequently improved by [Lewis et al. Nature Communications 2024], to $\mathcal{O}(\log n)$ samples when the geometry of the $n$-qubit system is known. In this work, we introduce two approaches that achieve a constant sample complexity, independent of system size $n$, for learning ground state properties. Our first algorithm consists of a simple modification of the ML model used by Lewis et al. and applies to a property of interest known beforehand. Our second algorithm, which applies even if a description of the property is not known, is a deep neural network model. While empirical results showing the performance of neural networks have been demonstrated, to our knowledge, this is the first rigorous sample complexity bound on a neural network model for predicting ground state properties. We also perform numerical experiments that confirm the improved scaling of our approach compared to earlier results.

翻译：量子多体物理学中的一个基本问题是寻找局部哈密顿量的基态。近期多项研究提出了可证明高效的机器学习算法用于学习基态性质。具体而言，[Huang等人，《科学》2022年]提出了一种方法，仅需从相同物质相哈密顿量中采样$n^{\mathcal{O}(1)}$个数据点，即可学习$n$量子比特有能隙局部哈密顿量$H$的基态性质。随后[Lewis等人，《自然·通讯》2024年]将所需样本改进至$\mathcal{O}(\log n)$个（当$n$量子比特系统的几何结构已知时）。本工作提出了两种实现恒定样本复杂度（与系统尺寸$n$无关）的学习基态性质的方法。我们的第一种算法通过对Lewis等人所用机器学习模型进行简单修改实现，适用于预先已知的目标性质。第二种算法采用深度神经网络模型，即使目标性质的数学描述未知时仍可适用。尽管神经网络的经验性能已有实证展示，但据我们所知，这是首个关于神经网络预测基态性质的严格样本复杂度界证明。我们还通过数值实验验证了本方法相较于早期成果的改进尺度特性。