Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms.

Area of Science:

• Quantum Many-Body Physics
• Computational Chemistry
• Condensed Matter Physics

Background:

• The ab initio density matrix renormalization group (DMRG) algorithm is described using two distinct formalisms: renormalized operators and matrix product states/operators.
• These different vocabularies can obscure the underlying algorithm and hinder efficient implementation.

Purpose of the Study:

• To provide a clear translation between the renormalized operator and matrix product state/operator formalisms for the ab initio DMRG algorithm.
• To demonstrate how insights from the matrix product state/operator language can improve the ab initio DMRG algorithm's efficiency and parallelization.

Main Methods:

• Implementation of the ab initio DMRG sweep using matrix product operator (MPO) based code and comparison with renormalized operator (RO) implementations.
• Integration of general MPO/matrix product state (MPS) algebra into a pure RO-based DMRG code.
• Development of Hamiltonian compression and a sum-over-operators representation for enhanced computational parallelism.

Main Results:

• Demonstrated equivalence between MPO-based and RO-based ab initio DMRG sweep implementations.
• Successfully implemented MPO/MPS algebra within an RO-based DMRG framework.
• Introduced Hamiltonian compression and a sum-over-operators approach, enabling perfect computational parallelism.

Conclusions:

• The presented translations facilitate a unified understanding of the ab initio DMRG algorithm across different formalisms.
• The developed improvements, motivated by the MPO language, enhance the efficiency and scalability of ab initio DMRG and related algorithms.