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.
翻译:经典机器学习(ML)提供了解决物理和化学领域数量-体积问题的潜在有力方法。然而,ML相对于传统方法的优势尚未牢固确立。在这项工作中,我们证明古典ML算法在学习了测量其他汉密尔顿人在同一量级物质阶段获得的数据后,能够有效地预测有限空间空间空间空间范围内的缺陷汉密尔顿人的地面状态。相反,在广泛接受的复杂理论假设下,不从数据中学习的经典算法不能达到同样的保证。我们还证明古典ML算法能够有效地分类一系列广泛的量级物质。我们的论点基于古典阴影的概念,即对多种体积状态的简洁的典型描述,可以建立在可行的量级实验中,并用来预测状态的许多特性。广泛的数字实验证实了我们在各种情景中的理论结果,包括Rydberg 原子系统、 2D 随机海森堡模型、对称保护的表层阶段以及按地形顺序排列的阶段。