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

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