为高不均匀问题和少体问题定制张量网络技术以量化表示
Tailoring tensor network techniques to the quantics representation for highly inhomogeneous problems and few body problems
AI总结：
本文提出通过定制张量网络算法解决高不均匀和少体问题，利用多网格方法提升算法收敛性，展示了在二维、三维和四维中线性和特征值问题上的应用。
摘要：
张量网络技术正逐渐成为求解所谓量化表示中偏微分方程的热门工具。其受欢迎的原因在于其空间分辨率仅与网格点数量对数相关，使其在存在两个或多个显著长度尺度差异的情况下极具吸引力。第一代技术使用了张量网络工具包中的“即用型”算法（例如著名的密度矩阵乘积态（DMRG）算法）来解决这些问题。这些技术最初设计用于（例如量子磁性）不同自由度（例如自旋）扮演等同角色的情况。然而，在量化表示中，不同自由度对应于不同尺度的物理现象，因此扮演不等同的角色。本文展示，通过定制张量网络算法以适应这种情况，以多网格方法为精神，我们获得更快更稳健的算法收敛性。我们展示了该方法在二维、三维和四维中的线性（泊松方程）和特征值（薛定谔方程）问题上的应用。我们的模拟涉及多达2^80个网格点，我们认为这将对传统方法构成极大的挑战。
英文摘要：
Tensor network techniques are becoming increasingly popular tools to solve partial differential equations within the so-called quantics representation. Their popularity stems from the fact that their spatial resolution depends only logarithmically on the number of grid points, making them very tempting approaches in situations where two or more characteristic length scales are vastly different. A first generation of technique used ``out-of-the-box'' algorithms of the tensor network toolkit (e.g. the celebrated Density Matrix Product State (DMRG) algorithm) to solve these problems. These techniques were designed for situations (e.g. quantum magnetism) where the different degrees of freedom (e.g. spins) play equivalent roles. In the quantics representation, however, the different degrees of freedom correspond to the physics at different scales and therefore play inequivalent role. Here we show that by tailoring the tensor network algorithms to this particular case, in the spirit of the multigrid approach, we obtain faster and more robust convergence of the algorithms. We showcase the approach on linear (Poisson equation) and eigenvalue (Schrödinger equation) problems in two, three and four dimensions. Our simulations involve up to $2^{80}$ grid points and would represent, we argue, a very strong challenge for conventional approaches.
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