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

## TL;DR

This paper presents an efficient implementation of the rigorous renormalization group method for approximating ground and low-energy states of local Hamiltonians, demonstrating competitive performance with DMRG, especially in challenging scenarios.

## Contribution

It introduces a practical, non-variational RRG algorithm that constructs ground space approximations via a tree-like Hilbert space approach, extending theoretical methods to real-world applications.

## Key findings

- RRG performs comparably to DMRG on gapped nondegenerate Hamiltonians.
- RRG effectively handles criticality, degeneracy, and long-range entanglement.
- In some cases, RRG outperforms DMRG in identifying low-energy states.

## Abstract

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.