  A novel reduced-scaling, general-order coupled-cluster approach is formulated
by exploiting hierarchical representations of many-body tensors, combined with
the recently suggested formalism of scale-adaptive tensor algebra. Inspired by
the hierarchical techniques from the renormalization group approach,
H/H2-matrix algebra and fast multipole method, the computational scaling
reduction in our formalism is achieved via coarsening of quantum many-body
interactions at larger interaction scales, thus imposing a hierarchical
structure on many-body tensors of coupled-cluster theory. In our approach, the
interaction scale can be defined on any appropriate Euclidean domain (spatial
domain, momentum-space domain, energy domain, etc.). We show that the
hierarchically resolved many-body tensors reduce the storage requirements to
O(N), where N is the number of simulated quantum particles. Subsequently, we
prove that any connected many-body diagram with arbitrary-order tensors, e.g.,
an arbitrary coupled-cluster diagram, can be evaluated in O(NlogN)
floating-point operations. On top of that, we elaborate an additional
approximation to further reduce the computational complexity of higher-order
coupled-cluster equations, i.e., equations involving higher than double
excitations, which otherwise would introduce a large prefactor into formal
O(NlogN) scaling.
