Using Matrix-Free Tensor-Network Optimizations to Construct a Reduced-Scaling and Robust Second-Order M{\o}ller-Plesset Theory

TL;DR

This paper introduces a novel matrix-free tensor-network optimization combining CPD and THC to develop a reduced-scaling, accurate second-order M{ }ller-Plesset theory for electronic structure calculations, improving efficiency and resource use.

Contribution

It presents a new low-cost CPD solver leveraging THC, enabling efficient tensor factorizations and the development of a reduced-scaling LT MP2 method.

Findings

Efficient generation of CPD factorizations for two-electron integrals.

Reduced computational complexity of LT MP2 while maintaining accuracy.

Performance improvements over canonical LT MP2 in time and memory.

Abstract

We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver which leverages a precomputed THC factorization of an order-$4$ tensor to efficiently optimize the order-$4$ CPD with $\mathcal{O}(NR^2)$ scaling. With the matrix-free THC-based optimization strategy in hand we can: efficiently generate CPD factorizations of the order-4 two-electron integral tensors; and develop novel electronic structure methods which take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order M{\o}ller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore we take advantage of the robust fitting approximation to eliminate leading order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD+THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.