Automatic transformation of irreducible representations for efficient contraction of tensors with cyclic group symmetry

TL;DR

This paper introduces a novel technique for efficiently contracting tensors with cyclic group symmetry by transforming irreducible representations, reducing computational overhead and enabling scalable parallel computations in quantum chemistry and tensor network applications.

Contribution

The authors develop a general algorithm for group symmetric tensor contractions using only dense tensors, simplifying implementation and improving performance over existing block-sparse methods.

Findings

Achieves efficient tensor contractions with cyclic group symmetry using dense tensors.

Demonstrates scalability on supercomputers with up to 4096 cores.

Enables use of optimized dense matrix multiplication libraries like MKL.

Abstract

Tensor contractions are ubiquitous in computational chemistry and physics, where tensors generally represent states or operators and contractions express the algebra of these quantities. In this context, the states and operators often preserve physical conservation laws, which are manifested as group symmetries in the tensors. These group symmetries imply that each tensor has block sparsity and can be stored in a reduced form. For nontrivial contractions, the memory footprint and cost are lowered, respectively, by a linear and a quadratic factor in the number of symmetry sectors. State-of-the-art tensor contraction software libraries exploit this opportunity by iterating over blocks or using general block-sparse tensor representations. Both approaches entail overhead in performance and code complexity. With intuition aided by tensor diagrams, we present a technique, irreducible representation alignment, which enables efficient handling of Abelian group symmetries via only dense tensors, by using contraction-specific reduced forms. This technique yields a general algorithm for arbitrary group symmetric contractions, which we implement in Python and apply to a variety of representative contractions from quantum chemistry and tensor network methods. As a consequence of relying on only dense tensor contractions, we can easily make use of efficient batched matrix multiplication via Intel's MKL and distributed tensor contraction via the Cyclops library, achieving good efficiency and parallel scalability on up to 4096 Knights Landing cores of a supercomputer.