On the Ordering of Sites in the Density Matrix Renormalization Group using Quantum Mutual Information

TL;DR

This paper links the heuristic of orbital ordering in quantum chemistry DMRG to minimizing entanglement entropy, providing a theoretical foundation and suggesting new ordering strategies based on quantum mutual information.

Contribution

It establishes a rigorous theoretical connection between orbital ordering heuristics and entanglement entropy minimization in DMRG, supported by entropy subadditivity.

Findings

Heuristic orbital ordering relates to entanglement entropy minimization.

Quantum mutual information guides optimal site ordering.

Provides theoretical justification for existing ordering methods.

Abstract

The density matrix renormalization group (DMRG) of White 1992 remains to this day an integral component of many state-of-the-art methods for efficiently simulating strongly correlated quantum systems. In quantum chemistry, QC-DMRG became a powerful tool for ab initio calculations with the non-relativistic Schr\"odinger equation. An important issue in QC-DMRG is the so-called ordering problem -- the optimal ordering of DMRG sites corresponding to electronic orbitals that produces the most accurate results. To this end, a commonly used heuristic is the grouping of strongly correlated orbitals as measured via quantum mutual information. In this work, we show how such heuristics can be directly related to minimizing the entanglement entropy of matrix product states and, consequently, to the truncation error of a fixed bond dimension approximation. Key to establishing this link is the strong subadditivity of entropy. This provides rigorous theoretical justification for the orbital ordering methods and suggests alternate ordering criteria.