Contracting Symmetric Tensors Using Fewer Multiplications

We present more computationally-efficient algorithms for contracting symmetric tensors. Tensor contractions are reducible to matrix multiplication, but permutational symmetries of the data, which are expressed by the tensor representation, provide an opportunity for more efficient algorithms. Previously known methods have exploited only tensor symmetries that yield identical computations that are directly evident in the contraction expression. We present a new ‘symmetry preserving’ algorithm that uses an algebraic reorganization in order to exploit considerably more symmetry in the computation of the contraction than the conventional approach. The new algorithm requires fewer multiplications but more additions per multiplication than previous approaches. The applications of this result include the capability to multiply a symmetric matrix by a vector, as well as compute the rank-2 symmetric vector outer product in half the number of scalar multiplications, albeit with more additions. The symmetry preserving algorithm can also be adapted to perform the complex versions of these operations, namely the product of a Hermitian matrix and a vector and the rank-2 Hermitian vector outer product, in 3/4 of the overall operations. Consequently, the number of operations needed for the direct algorithm to compute the eigenvalues of a Hermitian matrix is reduced by the same factor. Our symmetry preserving tensor contraction algorithm can also be adapted to the antisymmetric case and is therefore applicable to the tensor-contraction computations employed in quantum chemistry. For these applications, notably the coupled-cluster method, our algorithm yields the highest potential speed-ups, since in many higher-order contractions the reduction in the number of multiplications achieved by our algorithm enables an equivalent reduction in overall contraction cost. We highlight that for three typical coupled-cluster contractions taken from methods of three different orders, our algorithm achieves 2X, 4X, and 9X improvements in arithmetic cost over the standard approach.