Matrix Product States and Density Matrix Renormalization Group Algorithm

Abstract

The density matrix renormalization group (DMRG) is reviewed from a quantum chemistry perspective. Starting from a general many-body wavefunction, a new representation of the wavefunction called the matrix product state (MPS) representation is introduced. An explicit form of an MPS wavefunction is obtained from recursive decompositions of the original wavefunction. Based on the MPS wavefunction, overlap and expectation values are efficiently evaluated with algorithms having polynomial cost. A polynomial cost ansatz for the ground-state energy and wavefunction is therefore formulated by applying the variational principle to an MPS wavefunction. As a result, the DMRG algorithm is naturally obtained from general quantum mechanics as a variational ansatz for a many-body wavefunction. It is also important that the MPS representation leads to a site-based mean-field theory from an analogy to Hartree–Fock theory, which is a particle-based mean-field theory. In this sense, a sweep algorithm in the original DMRG is understood as a self-consistent field procedure. Advanced algorithms, such as that using complementary operators, that incorporating sparsity due to symmetry, and the two-site algorithm, are also described to achieve the optimal computational complexity for the DMRG algorithm in quantum chemistry.