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Constructing matrix product operators (MPO) is at the core of the moderndensity matrix renormalization group (DMRG) and its time dependent formulation.For DMRG to be conveniently used in different problems described by differentHamiltonians, in this work we propose a new generic algorithm to construct theMPO of an arbitrary operator with a sum-of-products form based on the bipartitegraph theory. We show that the method has the following advantages: (i) It isautomatic in that only the definition of the operator is required; (ii) It issymbolic thus free of any numerical error; (iii) The complementary operatortechnique can be fully employed so that the resulting MPO is globally optimalfor any given order of degrees of freedom; (iv) The symmetry of the systemcould be fully employed to reduce the dimension of MPO. To demonstrate theeffectiveness of the new algorithm, the MPOs of Hamiltonians ranging from theprototypical spin-boson model and Holstein model to the more complicated abinitio electronic Hamiltonian and the anharmonic vibrational Hamiltonian withsextic force field are constructed. It is found that for the former threecases, our automatic algorithm can reproduce exactly the same MPOs as theoptimally hand-crafted ones already known in the literature.

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