# Spin Summations: A High-Performance Perspective

## TL;DR

This paper develops optimized algorithms for spin summations in quantum chemistry, significantly improving computational speed and memory efficiency by leveraging hardware features and problem properties.

## Contribution

It introduces a novel, high-performance algorithm for spin summations that combines multiple optimization strategies, outperforming existing methods in speed and memory usage.

## Key findings

- Achieves 2.4x to 5.5x speedup over NCC software
- Performs in-place spin summations reducing memory footprint by 2x
- Exploits hardware and problem-specific properties for optimization

## Abstract

Besides tensor contractions, one of the most pronounced computational bottlenecks in the non-orthogonally spin-adapted forms of the quantum chemistry methods CCSDT and CCSDTQ, and their approximate forms---including CCSD(T) and CCSDT(Q)---are spin summations. At a first sight, spin summations are operations similar to tensor transpositions; a closer look instead reveals additional challenges to high-performance calculations, including temporal locality as well as scattered memory accesses. This publication explores a sequence of algorithmic solutions for spin summations, each exploiting individual properties of either the underlying hardware (e.g. caches, vectorization), or the problem itself (e.g. factorizability). The final algorithm combines the advantages of all the solutions, while avoiding their drawbacks; this algorithm, achieves high-performance through parallelization, vectorization, and by exploiting the temporal locality inherent to spin summations. Combined, these optimizations result in speedups between 2.4x and 5.5x over the NCC quantum chemistry software package. In addition to such a performance boost, our algorithm can perform the spin summations in-place, thus reducing the memory footprint by 2x over an out-of-place variant.