  The practical success of polynomial-time tensor network methods for computing
ground states of certain quantum local Hamiltonians has recently been given a
sound theoretical basis by Arad, Landau, Vazirani, and Vidick. The convergence
proof, however, relies on "rigorous renormalization group" (RRG) techniques
which differ fundamentally from existing algorithms. We introduce an efficient
implementation of the theoretical RRG procedure which finds MPS ansatz
approximations to the ground spaces and low-lying excited spectra of local
Hamiltonians in situations of practical interest. In contrast to other schemes,
RRG does not utilize variational methods on tensor networks. Rather, it
operates on subsets of the system Hilbert space by constructing approximations
to the global ground space in a tree-like manner. We evaluate the algorithm
numerically, finding similar performance to DMRG in the case of a gapped
nondegenerate Hamiltonian. Even in challenging situations of criticality, or
large ground-state degeneracy, or long-range entanglement, RRG remains able to
identify candidate states having large overlap with ground and low-energy
eigenstates, outperforming DMRG in some cases.
