A General Automatic Method for Optimal Construction of Matrix Product Operators Using Bipartite Graph Theory

TL;DR

This paper introduces a universal, error-free algorithm based on bipartite graph theory for automatically constructing optimal matrix product operators (MPOs) for various Hamiltonians, enhancing DMRG methods.

Contribution

The paper presents a generic, symbolic algorithm that automatically constructs globally optimal MPOs for arbitrary operators, utilizing bipartite graph theory and symmetry considerations.

Findings

Reproduces known MPOs for spin-boson, Holstein, and electronic Hamiltonians.

Demonstrates effectiveness on complex vibrational Hamiltonians.

Provides a fully automatic and error-free MPO construction method.

Abstract

Constructing matrix product operators (MPO) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For DMRG to be conveniently used in different problems described by different Hamiltonians, in this work we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) It is automatic in that only the definition of the operator is required; (ii) It is symbolic thus free of any numerical error; (iii) The complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; (iv) The symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.