Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms

TL;DR

This paper bridges the gap between two languages used in describing ab initio DMRG algorithms, providing efficient implementation methods and improvements like Hamiltonian compression and parallelism, enhancing computational efficiency.

Contribution

It offers a detailed translation between renormalization group and matrix product state/operator languages for ab initio DMRG, including implementation strategies and algorithmic improvements.

Findings

Efficient implementation of ab initio DMRG using matrix product operators.

Equivalence established between renormalized operators and matrix product state approaches.

Introduction of Hamiltonian compression and parallelism techniques for DMRG algorithms.

Abstract

Current descriptions of the ab initio DMRG algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab-initio DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past, and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.