IEEE International Conference on High Performance Computing and Communications; IEEE International Conference on Smart City; IEEE International Conference on Data Science and Systems (HPCC/SmartCity/DSS)
Efficient Computing for Reduced Density Matrix on GPUs
Wenhao Liang, Runfeng Jin, Yingqi Tian, Yingjin Ma, Zhong Jin
The reduced density matrix (RDM) plays an important role in quantum physics. With the RDM, we can accurately describe the energy of the quantum system. However, solving the RDM is a computationally intensive task where matrix multiplication accounts for more than 95% of the computation time, and exponential complexity, large memory consumption, and tensor sparsity make it challenging to achieve efficient computation. This paper proposes an efficient, scalable and multi-GPU solver for computing the RDM. The complexity of computing RDM is reduced to polynomial by using a tensor network approach in the form of matrix product state (MPS). The sparsity blocks are composed of contiguous memory locations using the list algorithm. The precomputation approach uses a small memory overhead to replace some redundant computations. Furthermore, the sparsity blocks are computed on the GPU using variable-size batch general matrix multiplication (GEMM). We benchmark the performance using a water molecule, and our results show that we achieve a 22.4 × speedup when using four MI50 GPUs compared to a 32-core CPU. Additionally, we demonstrate excellent strong scaling, achieving over 92% parallel efficiency when comparing the performance on 256 nodes versus 16 nodes.
The reduced density matrix (RDM) plays an important role in quantum physics. With the RDM, we can accurately describe the energy of the quantum system. However, solving the RDM is a computationally intensive task where matrix multiplication accounts for more than 95% of the computation time, and exponential complexity, large memory consumption, and tensor sparsity make it challenging to achieve efficient computation. This paper proposes an efficient, scalable and multi-GPU solver for computing the RDM. The complexity of computing RDM is reduced to polynomial by using a tensor network approach in the form of matrix product state (MPS). The sparsity blocks are composed of contiguous memory locations using the list algorithm. The precomputation approach uses a small memory overhead to replace some redundant computations. Furthermore, the sparsity blocks are computed on the GPU using variable-size batch general matrix multiplication (GEMM). We benchmark the performance using a water molecule, and our results show that we achieve a 22.4 × speedup when using four MI50 GPUs compared to a 32-core CPU. Additionally, we demonstrate excellent strong scaling, achieving over 92% parallel efficiency when comparing the performance on 256 nodes versus 16 nodes.
The density matrix renormalization group (DMRG) algorithm is a powerful tool for solving eigenvalue problems to model quantum systems. DMRG relies on tensor contractions and dense linear algebra to compute properties of condensed matter physics systems. However, its efficient parallel implementation is challenging due to limited concurrency, large memory footprint, and tensor sparsity. We mitigate these problems by implementing two new parallel approaches that handle block sparsity arising in DMRG, via Cyclops, a distributed memory tensor contraction library. We benchmark their performance on two physical systems using the Blue Waters and Stampede2 supercomputers. Our DMRG performance is improved by up to 5.9X in runtime and 99X in processing rate over ITensor, at roughly comparable computational resource use. This enables higher accuracy calculations via larger tensors for quantum state approximation. We demonstrate that despite having limited concurrency, DMRG is weakly scalable with the use of efficient parallel tensor contraction mechanisms.
The Density Matrix Renormalization Group (DMRG) algorithm is a powerful tool for solving eigenvalue problems to model quantum systems. DMRG relies on tensor contractions and dense linear algebra to compute properties of condensed matter physics systems. However, its efficient parallel implementation is challenging due to limited concurrency, large memory footprint, and tensor sparsity. We mitigate these problems by implementing two new parallel approaches that handle block sparsity arising in DMRG, via Cyclops, a distributed memory tensor contraction library. We benchmark their performance on two physical systems using the Blue Waters and Stampede2 supercomputers. Our DMRG performance is improved by up to 5.9X in runtime and 99X in processing rate over ITensor, at roughly comparable computational resource use. This enables higher accuracy calculations via larger tensors for quantum state approximation. We demonstrate that despite having limited concurrency, DMRG is weakly scalable with the use of efficient parallel tensor contraction mechanisms.
Determinant Quantum Monte Carlo (DQMC) simulation is one of few numerical methods that can explore the micro properties of fermions, which has many technically important applications in chemistry and material science. Conventionally, its parallelization relies on parallel Monte Carlo method, whose speedup is limited by the thermalization process and the underlying matrix computation. To achieve better performance, fine-grained parallelization on its numerical kernel is essential to utilize the massive parallel processing units, which are multicores and/or GPUs interconnected by high performance network. In this paper, we address the parallelization on one of the matrix kernel in the DQMC simulations: the multiplication of matrix exponentials. The matrix is derived from the kinetic Hamiltonian, which is highly sparse. We approximate its exponential by the checkerboard method, which decomposes the matrix exponential into a product of a sequence of block sparse matrices. We analyze the block sparse matrices of two common used lattice geometry: 2D torus and 3D cubic, and parallelize the computational kernel of multiplying them to a general matrix. The parallel algorithm is designed for multicore CPU and GPU. The results of experiments showed on a quad core processor, 3 times speedup can be observed in average, and on GPU, 145 times speedup is achievable.
Density matrix renormalization group (DMRG), as a numerical method of solving the ground state of one-dimensional strongly-correlated lattice model with very high accuracy, requires expensive computational and memory cost when applied to two- and quasi-two-dimensional problems. The number of DMRG kept states is generally very large to achieve a reliable accuracy for these applications, which results in numerous matrix and vector operations and unbearably consuming time in the absence of the proper parallelization. However, due to its sequential nature, the parallelization of DMRG algorithm is usually not straightforward. In this work, we propose a new hybrid parallelization strategy for the DMRG method. It takes advantage of the computing capability of both central processing unit (CPU) and graphics processing unit (GPU) of the computer. In order to achieve as many as DMRG kept states within a limited GPU memory, we adopt the four-block formulation of the Hamiltonian rather than the two-block formulation. The later consumes much more memories, which has been used in another pioneer work on the hybrid parallelization of the DMRG algorithm, and only a small number of DMRG kept states are available. Our parallel strategy focuses on the diagonalization of the Hamiltonian, which is the most time-consuming part of the whole DMRG procedure. A hybrid parallelization strategy of diagonalization method is implemented, in which the required data for diagonalization are distributed on both the host and GPU memory, and the data exchange between them is negligible in our data partitioning scheme. The matrix operations are also shared on both CPU and GPU when the Hamiltonian acts on a wave function, while the distribution of these operations is determined by a load balancing strategy. Taking fermionic Hubbard model for example, we examine the running performance of the hybrid parallelization strategy with different DMRG kept states and provide corresponding performance benchmark. On a 4-leg ladder, we employ the conserved quantities with U(1) symmetry of the model and a good-quantum number based task scheduling to further reduce the GPU memory cost. We manage to obtain a moderate speedup of the hybrid parallelization for a wide range of DMRG kept states. In our example, the ground state energy with high accuracy is obtained by the extrapolation of the results, with different numbers of states kept, and we show charge stripes which are usually experimentally observed in high-temperature superconductors. In this case, we keep 104 DMRG states and the GPU memory cost is less than 12 Gigabytes.
Density Matrix Renormalization Group (DMRG) method on current state-of-the-art NVIDIA DGX-H100 architectures. We evaluate the performance of the DMRG electronic structure calculations for the active compounds of the FeMoco, the primary cofactor of nitrogenase, and cytochrome P450 (CYP) enzymes with complete active space (CAS) sizes of up to 113 electrons in 76 orbitals [CAS(113, 76)] and 63 electrons in 58 orbitals [CAS(63, 58)], respectively. We achieve 246 teraFLOPS of sustained performance, an improvement of more than 2.5× compared to the performance achieved on the DGX-A100 architectures and an 80× acceleration compared to an OpenMP parallelized implementation on a 128-core CPU architecture. Our work highlights the ability of tensor network algorithms to efficiently utilize high-performance multi-GPU hardware and shows that the combination of tensor networks with modern large-scale GPU accelerators can pave the way toward solving some of the most challenging problems in quantum chemistry and beyond.
The presence of many degenerate d/f orbitals makes polynuclear transition-metal compounds, such as iron–sulfur clusters in nitrogenase, challenging for state-of-the-art quantum chemistry methods. To address this challenge, we present the first distributed multi-graphics processing unit (GPU) ab initio density matrix renormalization group (DMRG) algorithm suitable for modern high-performance computing (HPC) infrastructures. The central idea is to parallelize the most computationally intensive part─the multiplication of O ( K 2 ) operators with a trial wave function, where K is the number of spatial orbitals, by combining operator parallelism for distributing the workload with a batched algorithm for performing contractions on GPU. With this new implementation, we are able to reach an unprecedentedly large bond dimension D = 14,000 on 48 GPUs (NVIDIA A100 80 GB SXM) for an active space model (114 electrons in 73 active orbitals) of the P-cluster, which is nearly 3 times larger than the bond dimensions reported in previous DMRG calculations for the same system using only central processing units (CPUs).