High Performance Computing for Eigenvalue Solver in Density-Matrix Renormalization Group Method: Parallelization of the Hamiltonian Matrix-Vector Multiplication

Abstract

Abstract. The Density Matrix Renormalization Group (DMRG) method is widely used by computational physicists as a high accuracy tool to obtain the ground state of large quantum lattice models. Since the DMRG method has been originally developed for 1-D models, many extended method to a 2-D model have been proposed. However, some of them have issues in term of their accuracy. It is expected that the accuracy of the DMRG method extended directly to 2-D models is excellent. The direct extension DMRG method demands an enormous memory space. Therefore, we parallelize the matrix-vector multiplication in iterative methods for solving the eigenvalue problem, which is the most time-and memoryconsuming operation. We find that the parallel efficiency of the direct extension DMRG method shows a good one as the number of states kept increases.

Parallelization design on multi-core platforms in density matrix renormalization group toward 2-D quantum strongly-correlated systems

“…In this paper, we focus on the promising DMRG method and present its parallelization design on multi-core platforms to break the 1-D limitation and access 2-D models. So far, there have been only a few literature reporting high-performance computing techniques on DMRG [8,9,10]. The reason is that DMRG includes the diagonalization of a huge non-uniform sparse matrix, which is generally difficult issue for parallel computing.…”

Implementation trade-offs of the density matrix renormalization group algorithm on kilo-processor architectures

“…(Lanczos method) 7: Renormalize the basis of the block while keeping states with high eigenvalues. 8…”

“…In the past various works have been carried out to accelerate the DMRG algorithm [5]- [8], however, none of them took advantage of recent kilo-processor architectures such as the graphical processing unit (GPU).…”

Empirical Study for Communication Cost of Parallel Conjugate Gradient on a Star-Based Network

“…Similar studies have been made to parallelize CG in the objective to increase its performance ( [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]). [2] introduces applied fine-grained multithreading model on CG matrix-vector multiplication and inner-product calculations.…”

“…[2] introduces applied fine-grained multithreading model on CG matrix-vector multiplication and inner-product calculations. [3] parallelizes the matrix-vector multiplication. However, these works did not consider the CG communication cost.…”