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The practical success of polynomial-time tensor network methods for computingground states of certain quantum local Hamiltonians has recently been given asound theoretical basis by Arad, Landau, Vazirani, and Vidick. The convergenceproof, however, relies on "rigorous renormalization group" (RRG) techniqueswhich differ fundamentally from existing algorithms. We introduce an efficientimplementation of the theoretical RRG procedure which finds MPS ansatzapproximations to the ground spaces and low-lying excited spectra of localHamiltonians in situations of practical interest. In contrast to other schemes,RRG does not utilize variational methods on tensor networks. Rather, itoperates on subsets of the system Hilbert space by constructing approximationsto the global ground space in a tree-like manner. We evaluate the algorithmnumerically, finding similar performance to DMRG in the case of a gappednondegenerate Hamiltonian. Even in challenging situations of criticality, orlarge ground-state degeneracy, or long-range entanglement, RRG remains able toidentify candidate states having large overlap with ground and low-energyeigenstates, outperforming DMRG in some cases.

Implementation of rigorous renormalization group method for ground space and low-energy states of local Hamiltonians