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

Abstract

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