基态量子多体态的纠缠聚簇
Entanglement Clustering for ground-stateable quantum many-body states
摘要：
尽管多体局域量子哈密顿量的基态在决定系统的量子力学性质方面具有根本的重要性，但它们在多体希尔伯特空间中形成了一组零测度。因此，确定一个给定的多体量子态是否是基态是一个具有挑战性的任务。在这里，我们提出了一种无监督的机器学习方法，称为纠缠聚类（EntanCl），分离出基态波函数从那些必须是激发态波函数使用纠缠结构信息。EntanCl使用交换算子集合的快照作为输入，并将此高维数据投影到二维，使用均匀流形近似和投影（UMAP）保留与不同纠缠结构相关的数据的重要拓扑特征。然后使用K-means聚类和$k=2$对投影数据进行聚类。通过将EntanCl应用于一维带绝缘体和二维复曲面程序两个例子，我们证明了EntanCl可以成功地分离基态和激发态，具有很高的计算效率。EntanCl独立于哈密顿量和相关能量估计，以计算高效的方式提供了一种新的范式来处理量子多体波函数。
英文摘要：
Despite their fundamental importance in dictating the quantum mechanical properties of a system, ground states of many-body local quantum Hamiltonians form a set of measure zero in the many-body Hilbert space. Hence determining whether a given many-body quantum state is ground-stateable is a challenging task. Here we propose an unsupervised machine learning approach, dubbed the Entanglement Clustering ("EntanCl"), to separate out ground-stateable wave functions from those that must be excited state wave functions using entanglement structure information. EntanCl uses snapshots of an ensemble of swap operators as input and projects this high dimensional data to two-dimensions, preserving important topological features of the data associated with distinct entanglement structure using the uniform manifold approximation and projection (UMAP). The projected data is then clustered using K-means clustering with $k=2$. By applying EntanCl to two examples, a one-dimensional band insulator and the two-dimensional toric code, we demonstrate that EntanCl can successfully separate ground states from excited states with high computational efficiency. Being independent of a Hamiltonian and associated energy estimates, EntanCl offers a new paradigm for addressing quantum many-body wave functions in a computationally efficient manner.
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