Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state from a number of copies that scales polylogarithmically with the system's size and polynomially on the locality of the target observables. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols. We achieve our results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter allows us to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well we approximate the expectation value of a fixed set of few-body observables. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity that are of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states.

中文翻译：

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从几个副本中学习量子多体系统

从测量中估计量子态的物理性质是量子科学中最基本的任务之一。在这项工作中，我们确定了状态条件，在这些条件下，可以从多个副本中推断出状态的所有准局部可观测量的期望值，这些副本随系统大小进行多对数缩放，并根据目标可观测量的局部性进行多项式缩放。我们证明，与最先进的断层扫描协议相比，这构成了副本数量的指数级改进。我们通过将最大熵方法与经典阴影和量子最优传输等新兴领域的工具相结合来实现我们的结果。后者允许我们根据可观测值的局部性以及我们对一组固定的少体可观测值的期望值的近似程度来微调估计可观测值的期望值时所产生的误差。我们推测我们的条件适用于所有表现出某种形式的相关性衰减的状态，并为其几个子集建立它。其中包括广泛研究的状态类别，例如任意超图或浅电路输出上的局部交换哈密顿量的一维热和高温吉布斯状态。此外，我们展示了最大熵方法的改进，超出了独立感兴趣的样本复杂性。这些包括识别可以有效执行后处理的机制以及多体状态协方差矩阵的条件数的新界限。