等距张量网络的黎曼优化
Riemannian optimization of isometric tensor networks
摘要：
几个张量网络由等距张量构建，即满足$W^\dagger W = \mathrm{I}$的张量。突出的例子包括标准形式的矩阵乘积态（MPS），多尺度纠缠重正化（MERA）和一般的量子线路，例如那些在状态准备和量子变分本征解算器中所需要的。我们展示了如何在黎曼流形上基于梯度的优化方法可以用来优化张量网络的等距表示，例如基态的一维量子哈密顿。我们讨论了几何的格拉斯曼和Stiefel流形，黎曼流形的等距张量，并审查如何国家的最先进的优化方法，如非线性共轭梯度和拟牛顿算法可以在这种情况下实施。我们将这些方法应用于无限MPS和MERA的背景下，并显示基准结果，其中它们优于先前已知的最佳优化方法，这些方法是为这些特定的变分类量身定制的。我们还提供我们的算法的开源实现。
英文摘要：
Several tensor networks are built of isometric tensors, i.e. tensors satisfying $W^\dagger W = \mathrm{I}$. Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.
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