A general automatic method for optimal construction of matrix product operators using bipartite graph theory

Area of Science:

• Computational Physics
• Quantum Chemistry
• Materials Science

Background:

• Matrix Product Operators (MPOs) are fundamental to the Density Matrix Renormalization Group (DMRG) and its time-dependent formulations.
• Efficient MPO construction is crucial for applying DMRG to diverse quantum many-body problems.

Purpose of the Study:

• To develop a generic, automatic algorithm for constructing MPOs of arbitrary operators in a sum-of-products form.
• To leverage bipartite graph theory for a robust and versatile MPO construction method.

Main Methods:

• The algorithm utilizes bipartite graph theory to represent and construct MPOs.
• It employs a symbolic approach, avoiding numerical errors.
• The complementary operator technique is integrated for global MPO optimality.
• Symmetry properties of the system are exploited to minimize MPO dimensions.

Main Results:

• The algorithm successfully constructs MPOs for various Hamiltonians, including the spin-boson, Holstein, ab initio electronic, and anharmonic vibrational models.
• For the spin-boson, Holstein, and ab initio electronic models, the generated MPOs precisely match existing, manually optimized MPOs.
• The method demonstrates advantages in automation, symbolic computation, optimality, and symmetry utilization.

Conclusions:

• The proposed algorithm provides an efficient, accurate, and generalizable method for MPO construction in DMRG.
• This advancement facilitates the application of DMRG to a wider range of complex quantum systems.
• The symbolic and automatic nature of the algorithm reduces human error and computational cost.