Solving the ground state and the ground-state properties of quantum many-body systems is generically a hard task for classical algorithms. For a family of Hamiltonians defined on an $m$-dimensional space of physical parameters, the ground state and its properties at an arbitrary parameter configuration can be predicted via a machine learning protocol up to a prescribed prediction error $\varepsilon$, provided that a sample set (of size $N$) of the states can be efficiently prepared and measured. In a recent work [Huang et al., Science 377, eabk3333 (2022)], a rigorous guarantee for such an generalization was proved. Unfortunately, an exponential scaling, $N = m^{ {\cal{O}} \left(\frac{1}{\varepsilon} \right) }$, was found to be universal for generic gapped Hamiltonians. This result applies to the situation where the dimension of the parameter space is large while the scaling with the accuracy is not an urgent factor, not entering the realm of more precise learning and prediction. In this work, we consider an alternative scenario, where $m$ is a finite, not necessarily large constant while the scaling with the prediction error becomes the central concern. By exploiting physical constraints and positive good kernels for predicting the density matrix, we rigorously obtain an exponentially improved sample complexity, $N = \mathrm{poly} \left(\varepsilon^{-1}, n, \log \frac{1}{\delta}\right)$, where $\mathrm{poly}$ denotes a polynomial function; $n$ is the number of qubits in the system, and ($1-\delta$) is the probability of success. Moreover, if restricted to learning ground-state properties with strong locality assumptions, the number of samples can be further reduced to $N = \mathrm{poly} \left(\varepsilon^{-1}, \log \frac{n}{\delta}\right)$. This provably rigorous result represents a significant improvement and an indispensable extension of the existing work.

翻译：求解量子多体系统的基态及其基态性质通常对经典算法而言是一项艰巨任务。对于定义在$m$维物理参数空间中的哈密顿量族，若能够高效制备并测量一组样本（规模为$N$），则可通过机器学习协议预测任意参数配置下的基态及其性质，预测误差不超过预设的$\varepsilon$。近期工作[Huang et al., Science 377, eabk3333 (2022)]为此类泛化给出了严格证明。然而，针对一般有能隙哈密顿量，普遍存在指数级标度关系$N = m^{ {\cal{O}} \left(\frac{1}{\varepsilon} \right) }$。该结果适用于参数空间维度较大而精度标度非紧迫因素的情形，未涉及更精确的学习与预测领域。本文考虑另一场景：$m$为有限（不必很大）常数，而预测误差的标度成为核心关切。通过利用物理约束及用于预测密度矩阵的正定好核，我们严格获得了指数级改进的样本复杂度$N = \mathrm{poly} \left(\varepsilon^{-1}, n, \log \frac{1}{\delta}\right)$，其中$\mathrm{poly}$表示多项式函数；$n$为系统量子比特数，$(1-\delta)$为成功概率。进一步，若限制于学习具有强局域性假设的基态性质，样本数可降至$N = \mathrm{poly} \left(\varepsilon^{-1}, \log \frac{n}{\delta}\right)$。这一可证明的严格结果标志着对现有工作的显著改进与不可或缺的拓展。