Toward Using Matrix-free Tensor Decompositions to Systematically Improve Approximate Tensor-Networks

We
investigate a novel approach to approximate tensor-network contraction
via the exact, matrix-free decomposition of full tensor-networks.
We study this method as a means to eliminate the propagation of error
in the approximation of tensor-networks. Importantly, this decomposition-based
approach is generic, i.e., it does not depend on a specific tensor-network,
the tensor index (physical) ordering, or the choice of tensor decomposition.
Careful consideration should be made to determine the best decomposition
strategy. Furthermore, this method does not rely on robust cancellation
of errors (i.e., the Taylor expansion). As a means to study the effectiveness
of the approach, we replace the exact contraction of the particle–particle
ladder (PPL) tensor diagram in the popular coupled-cluster with single
and double excitation (CCSD) method with a low-rank tensor decomposition,
namely the canonical polyadic decomposition (CPD). With this approach,
we replace an O(N6) tensor contractions with
a potentially
reduced-scaling O(N4R) optimization
problem, where R is the CP rank, and we reduce the
computational storage of the PPL
tensor from O(N4) to O(NR), although we do not take advantage of this
compression in this study. To minimize the cost of the CPD optimization,
we utilize the iterative structure of CCSD to efficiently initialize
the CPD optimization. We show that accurate chemically relevant energy
values can be computed with an error of less than 1 kcal/mol using
a relatively low CP rank.