论文标题
等距张量网络的Riemannian优化
Riemannian optimization of isometric tensor networks
论文作者
论文摘要
使用等距张量，即满足$ W^\ Dagger W = \ Mathrm {i} $的张量。突出的例子包括以规范形式的矩阵乘积状态（MPS），多尺度纠缠重新归一化ANSATZ（MERA）和一般的量子电路，例如状态制备中所需的和量子变异的特征粒子。我们展示了如何使用基于梯度的优化方法来优化异构体的张量网络以表示例如1d量子哈密顿人的基态。我们讨论了格拉斯曼和斯蒂夫尔流形的几何形状，等距张量的riemannian歧管，并回顾了如何在这种情况下实现诸如非线性共轭梯度和准牛顿算法之类的最先进的优化方法。我们在无限MP和MERA的背景下应用这些方法，并显示基准结果，在这些结果中，它们的表现优于以前最著名的优化方法，这些方法是针对这些特定变异类别量身定制的。我们还提供算法的开源实现。
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.
