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 central limit theorem, resulting in increased prediction accuracy.

翻译：本文研究量子多体物理中的基本任务——寻找和学习量子哈密顿量的基态及其性质。近期工作通过数据驱动方法，研究了预测几何局域可观测量之和的基态期望值问题。对于短程有能隙哈密顿量，已有研究获得了与量子比特数呈对数关系、与误差呈准多项式关系的样本复杂度。受分子和原子系统中长程相互作用实际意义的启发，我们将这些结果推广至超越局域性要求的哈密顿量与可观测量。对于幂律衰减（指数大于系统维度两倍）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，但误差依赖关系退化为指数形式。进一步表明，在相互作用超图的自同构群下具有等变性的学习算法能实现样本复杂度降低，特别在周期边界条件下，学习局域可观测量之和仅需常数个样本。通过对至多128量子比特的一维长程及无序系统进行DMRM模拟学习，我们验证了实际中的高效标度。最后，基于中心极限定理提供的全局可观测量期望值集中性分析，我们实现了预测精度提升。