神经网络量子态、弦键态和手征拓扑态
Neural-Network Quantum States, String-Bond States, and Chiral Topological States
摘要：
神经网络量子态最近被引入作为描述量子多体系统波函数的一种近似。我们证明了在任意维的限制玻尔兹曼机形式的神经网络量子态和某些类张量网络态之间存在着很强的联系。特别是，我们证明了短程限制玻尔兹曼机纠缠Plaquette状态，而完全连接的限制玻尔兹曼机的弦键状态与非局部几何和低键尺寸。这些结果揭示了限制玻尔兹曼机的基本结构和它们在表示多体量子态方面的效率。弦键态也提供了一种增强神经网络量子态能力的通用方法，并自然推广到具有更大局部希尔伯特空间的系统。我们比较了这些不同类别的状态的优点和缺点，并提出了一种方法将它们结合在一起。这使我们能够受益于张量网络的纠缠结构和神经网络量子态的效率，使其成为一个能够瞄准强相关系统的波函数的单个Anonymous。虽然用传统的张量网络来描述具有手征拓扑序的态仍然是一个挑战，但我们证明了神经网络量子态及其弦键态扩展可以精确地描述晶格分数量子霍尔态。此外，我们提供的数值证据表明，神经网络量子态可以近似手征自旋液体比纠缠Plaquette态和本地弦键态更好的精度。我们的研究结果表明，神经网络的效率来描述复杂的量子波函数，并铺平了道路，使用弦键态作为一种工具，在更传统的机器学习应用。
英文摘要：
Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dimensions. In particular we demonstrate that short-range Restricted Boltzmann Machines are Entangled Plaquette States, while fully connected Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of Restricted Boltzmann Machines and their efficiency at representing many-body quantum states. String-Bond States also provide a generic way of enhancing the power of Neural-Network Quantum States and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of Tensor Networks and the efficiency of Neural-Network Quantum States into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional Tensor Networks, we show that Neural-Network Quantum States and their String-Bond States extension can describe a lattice Fractional Quantum Hall state exactly. In addition, we provide numerical evidence that Neural-Network Quantum States can approximate a chiral spin liquid with better accuracy than Entangled Plaquette States and local String-Bond States. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of String-Bond States as a tool in more traditional machine-learning applications.
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