Robust approximation of tensor networks: application to grid-free tensor factorization of the Coulomb interaction

TL;DR

This paper introduces a robust tensor network approximation method that significantly reduces computational complexity and improves accuracy in modeling Coulomb interactions, with applications to quantum chemistry calculations.

Contribution

It develops a robust approximation technique for tensor networks that enhances accuracy and efficiency in Coulomb tensor factorization, applicable to quantum chemistry methods.

Findings

Reduces Coulomb tensor approximation rank significantly.

Decreases computational complexity from O(N^6) to O(N^5).

Maintains negligible errors in energy calculations.

Abstract

Approximation of a tensor network by approximating (e.g., factorizing) one or more of its constituent tensors can be improved by canceling the leading-order error due to the constituents' approximation. The utility of such robust approximation is demonstrated for robust canonical polyadic (CP) approximation of a (density-fitting) factorized 2-particle Coulomb interaction tensor. The resulting algebraic (grid-free) approximation for the Coulomb tensor, closely related to the factorization appearing in pseudospectral and tensor hypercontraction approaches, is efficient and accurate, with significantly reduced rank compared to the naive (non-robust) approximation. Application of the robust approximation to the particle-particle ladder term in the coupled-cluster singles and doubles reduces the size complexity from $\mathcal{O}(N^6)$ to $\mathcal{O}(N^5)$ with robustness ensuring negligible errors in chemically-relevant energy differences using CP ranks approximately equal to the size of the density-fitting basis.