In this work, we consider a fundamental task in quantum many-body physics - finding and learning ground states of quantum Hamiltonians and their properties. Recent works have studied the task of predicting the ground state expectation value of sums of geometrically local observables by learning from data. For short-range gapped Hamiltonians, a sample complexity that is logarithmic in the number of qubits and quasipolynomial in the error was obtained. Here we extend these results beyond the local requirements on both Hamiltonians and observables, motivated by the relevance of long-range interactions in molecular and atomic systems. For interactions decaying as a power law with exponent greater than twice the dimension of the system, we recover the same efficient logarithmic scaling with respect to the number of qubits, but the dependence on the error worsens to exponential. Further, we show that learning algorithms equivariant under the automorphism group of the interaction hypergraph achieve a sample complexity reduction, leading in particular to a constant number of samples for learning sums of local observables in systems with periodic boundary conditions. We demonstrate the efficient scaling in practice by learning from DMRG simulations of $1$D long-range and disordered systems with up to $128$ qubits. Finally, we provide an analysis of the concentration of expectation values of global observables stemming from the central limit theorem, resulting in increased prediction accuracy.

翻译：在这项工作中，我们考虑了量子多体物理中的一个基本任务——寻找和学习量子哈密顿量的基态及其性质。近期研究通过从数据中学习，探讨了预测几何局域可观测量之和的基态期望值这一任务。对于短程能隙哈密顿量，已有工作实现了与量子比特数呈对数关系、与误差呈拟多项式关系的样本复杂度。在此，我们受分子和原子系统中长程相互作用相关性的启发，将上述结果推广至哈密顿量和可观测量均无需局域性的情形。对于以幂律形式衰减（指数大于系统维度的两倍）的相互作用，我们恢复了与量子比特数相同的对数高效标度，但误差依赖关系恶化至指数级。进一步，我们证明了在相互作用超图的自同构群下具有等变性的学习算法能够降低样本复杂度，特别地，对于周期边界系统中局域可观测量之和的学习，仅需常数个样本。我们通过对最多128个量子比特的一维长程和无序系统进行DMRM模拟，从实践角度验证了这种高效标度。最后，我们基于中心极限定理给出了全局可观测量期望值的集中性分析，从而提升了预测精度。