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

Abstract

One of the most fascinating issues in modern condensed matter physics is to understand highly-correlated electronic structures and propose their novel device designs toward the reduced carbon-dioxide future. Among various developed numerical approaches for highly-correlated electrons, the density matrix renormalization group (DMRG) has been widely accepted as the most promising numerical scheme compared to Monte Carlo and exact diagonalization in terms of accuracy and accessible system size. In fact, DMRG almost perfectly resolves one-dimensional chain like long quantum systems. In this paper, we suggest its extended approach toward higher-dimensional systems by high-performance computing techniques. The computing target in DMRG is a huge non-uniform sparse matrix diagonalization. In order to efficiently parallelize the part, we implement communication step doubling together with reuse of the mid-point data between the doubled two steps to avoid severe bottleneck of all-to-all communications essential for the diagonalization. The technique is successful even for clusters composed of more than 1000 cores and offers a trustworthy exploration way for two-dimensional highly-correlated systems.

Tensor product methods and entanglement optimization for ab initio quantum chemistry

“…The original DMRG algorithm, introduced by S. R. White, was formulated as a single threaded algorithm 249 . In the past various works have been carried out to accelerate the DMRG algorithm on shared 10,90 and distributed memory 35,114,200,267 architectures. One of the first parallelizations was converting the projection operation to matrix-matrix multiplications and accelerating them via OpenMP interface 90 .…”

The density matrix renormalization group algorithm on kilo-processor architectures: Implementation and trade-offs

“…The original DMRG algorithm [1] was introduced in 1992 by Steven R. White and was formulated as a single threaded algorithm. In the past various works have been carried out to accelerate the DMRG algorithm on shared [8] [9] and distributed memory [10,11,12,13] architectures, however, none of them took advantage of recent kiloprocessor architectures: graphical processing unit (GPU) and field-programmable gate array (FPGA).…”

“…One of the first parallelizations was [8] converting the projection operation to matrix-matrix multiplications and accelerating them via OpenMP interface. In [12] a similar approach was presented for distributed memory environment (up-to 1024 cores) optimizing the communication between the cores, while in [13] the acceleration of the computation of correlation function had been investigated. Recently, [9] presented an acceleration on shared memory architectures exploiting SU(2) symmetries, while [14] proposed a novel direction for paralellization via a modification of the original serial DMRG algorithm.…”

Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

“…The second strategy, on which we focus in this work, lies in covering the two-dimensional lattice by a one-dimensional curve and then applying one-dimensional algorithms to the resulting effective chain. The drawback at this approach is that even if the original model contains only nearestneighbor interactions, the resulting model on the effective one-dimensional chain shows long-range interactions, that have an influence on the numerical approach efficiency [54].…”