神经网络态中的量子纠缠
Quantum Entanglement in Neural Network States
摘要：
机器学习是当今发展最快的跨学科领域之一，为解决复杂的量子多体问题提供了前所未有的视角。理解代表性人工神经网络状态的物理方面最近在机器学习技术应用于量子多体物理中变得非常可取。在这里，我们研究神经网络状态的量子纠缠特性，重点是限制玻尔兹曼机（RBM）架构。我们证明了在任意维数和二分几何条件下，所有短程RBM态的纠缠满足面积定律。对于长程RBM态，我们通过精确构造证明了这种态可以表现出体积律纠缠，这意味着RBM在有效地表示具有大质量纠缠的量子态方面具有显著的能力。我们进一步研究了具有随机权重参数的通用RBM状态。我们发现它们的平均纠缠熵服从体积律标度，同时也严重偏离了完全随机纯态的Page熵。结果表明，它们的纠缠谱不具有随机矩阵理论的普适部分，具有Poisson型能级统计。使用强化学习，我们证明了RBM是能够找到基态（与幂律纠缠）的模型哈密顿与长程相互作用。此外，我们通过一个具体的例子，一维的纠缠保护的拓扑簇态，RBM表示也可以被用来作为一种工具来解析计算纠缠谱。我们的研究结果揭示了人工神经网络在表示量子多体状态方面无与伦比的能力，这为基于计算机科学的机器学习技术与杰出的量子凝聚态物理问题之间的桥梁铺平了新的道路。
英文摘要：
Machine learning, one of today's most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states is recently becoming highly desirable in the applications of machine learning techniques to quantum many-body physics. Here, we study the quantum entanglement properties of neural-network states, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing efficiently quantum states with massive entanglement. We further examine generic RBM states with random weight parameters. We find that their averaged entanglement entropy obeys volume-law scaling and meantime strongly deviates from the Page-entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with long-range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry-protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states, which paves a novel way to bridge computer science based machine learning techniques to outstanding quantum condensed matter physics problems.
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