Classical machine learning (ML) provides a potentially powerful approach to
solving challenging quantum many-body problems in physics and chemistry.
However, the advantages of ML over more traditional methods have not been
firmly established. In this work, we prove that classical ML algorithms can
efficiently predict ground state properties of gapped Hamiltonians in finite
spatial dimensions, after learning from data obtained by measuring other
Hamiltonians in the same quantum phase of matter. In contrast, under widely
accepted complexity theory assumptions, classical algorithms that do not learn
from data cannot achieve the same guarantee. We also prove that classical ML
algorithms can efficiently classify a wide range of quantum phases of matter.
Our arguments are based on the concept of a classical shadow, a succinct
classical description of a many-body quantum state that can be constructed in
feasible quantum experiments and be used to predict many properties of the
state. Extensive numerical experiments corroborate our theoretical results in a
variety of scenarios, including Rydberg atom systems, 2D random Heisenberg
models, symmetry-protected topological phases, and topologically ordered
phases.

这篇论文证明了经典机器学习算法可以有效地预测有限空间维度内间隙哈密顿量的基态性质及其它相类哈密顿量的数据，并可以有效地分类一系列的量子相，在量子实验中可以通过构建经典影像来预测系统的多种属性。

可证明高效的量子多体问题机器学习

Provably efficient machine learning for quantum many-body problems

The challenge of quantum many-body problems comes from the difficulty to
represent large-scale quantum states, which in general requires an
exponentially large number of parameters. Recently, a connection has been made
between quantum many-body states and the neural network representation
(\textit{arXiv:1606.02318}). An important open question is what characterizes
the representational power of deep and shallow neural networks, which is of
fundamental interest due to popularity of the deep learning methods. Here, we
give a rigorous proof that a deep neural network can efficiently represent most
physical states, including those generated by any polynomial size quantum
circuits or ground states of many body Hamiltonians with polynomial-size gaps,
while a shallow network through a restricted Boltzmann machine cannot
efficiently represent those states unless the polynomial hierarchy in
computational complexity theory collapses.

深度神经网络具有很高的表示能力，能够高效地表示大多数物理态，然而浅层神经网络在计算复杂度理论中的一些较高层次上不能够高效地表示这些态。