# Pre-processing the nuclear many-body problem: Importance truncation versus tensor factorization techniques

## TL;DR

This paper compares importance truncation and tensor factorization as pre-processing methods to reduce the complexity of nuclear many-body calculations, showing importance truncation's potential for high-accuracy, low-cost computations.

## Contribution

It introduces and benchmarks importance truncation against tensor factorization for nuclear many-body problem pre-processing, highlighting importance truncation's practical advantages.

## Key findings

- Importance truncation shows great potential for reducing computational costs.
- Tensor factorization requires significant numerical development for large spaces.
- Both methods perform well in small model space benchmarks.

## Abstract

The solution of the nuclear A-body problem encounters severe limitations from the size of many-body operators. These limitations are typically related to both the (iterative) storing of the associated tensors and to the computational time related to their multiple contractions in the calculation of various quantities of interest. However, not all the degrees of freedom encapsulated into these tensors equally contribute to the description of many-body observables. Identifying systematic and dominating patterns, a relevant objective is to achieve an \emph{a priori} reduction to the most relevant degrees of freedom via a pre-processing of the A-body problem. The present paper is dedicated to the analysis of two different paradigms to do so. The factorization of tensors in terms of lower-rank ones, whose know-how has been recently transferred to the realm of nuclear structure, is compared to a reduction of the tensors' index size based on an importance truncation. While the objective is to eventually utilize these pre-processing tools in the context of non-perturbative many-body methods, benchmark calculations are presently performed within the frame of perturbation theory. More specifically, we employ the recently introduced Bogoliubov many-body perturbation theory that is systematically applicable to open-shell nuclei displaying strong correlations. This extended perturbation theory serves as a jumpstart for non-perturbative Bogoliubov coupled cluster and Gorkov self-consistent Green's function theories. Results obtained in "small" model spaces are equally encouraging for tensor factorization and importance truncation techniques. While the former requires significant numerical developments to be applied in large model spaces, the latter is presently applied in this context and demonstrates great potential to enable high-accuracy calculations at a much reduced computational cost.