SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B108-B131, January 2021.
The tensor train (TT) approximation of electronic wave functions lies at the core of the quantum chemistry density matrix renormalization group (QC-DMRG) method, a recent state-of-the-art method for numerically solving the $N$-electron Schrödinger equation. It is well known that the accuracy of TT approximations is governed by the decay of the associated singular values, which in turn strongly depends on the ordering of the one-body basis. Here we find that the singular values $s_1\ge s_2\ge \cdots \ge s_d$ of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, $s_1s_d=s_2s_{d-1}$$=s_3s_{d-2}=\ldots,$ thus reducing the tail behavior to the leading singular value and a single hidden invariant, which moreover depends explicitly on the ordering of the basis. For correlated wave functions, we find that the tail is upper bounded by a suitable superposition of the invariants. Optimizing the invariants or their superposition thus provides a new ordering scheme for QC-DMRG. Numerical tests on simple examples, i.e., linear combinations of a few Slater determinants, show that the new scheme reduces the tail of the singular values by several orders of magnitude over existing methods, including the widely used Fiedler order.

中文翻译：

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量子化学的张量列车逼近中奇异值的反对称性和新的轨道排序方法

SIAM科学计算杂志，第43卷，第1期，第B108-B131页，2021年1月。
电子波函数的张量链（TT）逼近是量子化学密度矩阵重整化组（QC-DMRG）方法的核心，该方法是一种用于数值求解$ N $-电子Schrödinger的最新技术方程。众所周知，TT近似值的精度由相关奇异值的衰减控制，而衰减又反过来取决于单体的阶数。在这里我们发现表示非相互作用哈密顿量的基态的张量的奇异值$ s_1 \ ge s_2 \ ge \ cdots \ ge s_d $具有令人惊讶的反对称性，$ s_1s_d = s_2s_ {d-1} $$ = s_3s_ {d- 2} = \ ldots，$从而将尾部行为减少到前导的奇异值和一个单独的隐藏不变式，此外，它们明确地取决于基序。对于相关的波函数，我们发现尾巴的上界是不变量的适当叠加。优化不变量或其叠加从而为QC-DMRG提供了新的排序方案。对简单示例（即几个Slater行列式的线性组合）的数值测试表明，与现有方法（包括广泛使用的Fiedler阶）相比，该新方案将奇异值的尾部减少了几个数量级。