Predicting the properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a ‘classical shadow’, can be used to predict many different properties; order \({\mathrm{log}}\,(M)\) measurements suffice to accurately predict M different functions of the state with high success probability. The number of measurements is independent of the system size and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

预测复杂的大规模量子系统的性质对于开发量子技术至关重要。我们提出了一种使用很少的状态测量来构造量子状态的近似经典描述的有效方法。这种描述称为“经典阴影”，可用于预测许多不同的属性；例如，阶次\（{\ mathrm {log}} \，（M）\）测量足以准确预测M状态不同的功能具有很高的成功概率。测量次数与系统大小无关，并且会使信息理论的下限饱和。此外，可以在测量完成后选择要预测的目标属性。我们通过大量的数值实验来支持我们的理论发现。我们应用经典阴影来预测量子保真度，纠缠熵，两点相关函数，局部可观测值的期望值以及多体局部哈密顿量的能量方差。数值结果突出了相对于先前已知方法的经典阴影的优点。