Tensor contractions represent the most compute- intensive core kernels in ab initio computational quantum chemistry and nuclear physics. Symmetries in these tensor contractions make them difficult to load balance and scale to large distributed systems. In this paper, we develop an efficient and scalable algorithm to contract symmetric tensors. We introduce a novel approach that avoids data redistribution during contraction of symmetric tensors while also bypassing redundant storage and maintaining load balance. We present experimental results on two parallel supercomputers for several symmetric contractions that appear in the coupled cluster singles and doubles (CCSD) quantum chemistry method. We also present a novel approach to tensor redistribution that can take advantage of parallel hyperplanes when the initial distribution has replicated dimensions, and use collective broadcast when the final distribution has replicated dimensions, making the algorithm very efficient.

Tensor contractions are extremely compute intensive generalized matrix multiplication operations encountered in many computational science fields, such as quantum chemistry and nuclear physics. Unlike distributed matrix multiplication, which has been extensively studied, limited work has been done in understanding distributed tensor contractions. In this paper, we characterize distributed tensor contraction algorithms on torus networks. We develop a framework with three fundamental communication operators to generate communication-efficient contraction algorithms for arbitrary tensor contractions. We show that for a given amount of memory per processor, the framework is communication optimal for all tensor contractions. We demonstrate performance and scalability of the framework on up to 262,144 cores on a Blue Gene/Q supercomputer.