We present a hybrid numerical approach to simulate quantum many-body problems on two spatial dimensional quantum lattice models via the non-Abelian ab initio version of the density matrix renormalization group method on state-of-the-art high-performance computing infrastructures. We demonstrate that for the two-dimensional spinless fermion model and for the Hubbard model on torus geometry, altogether several orders of magnitude in computational time can be saved by performing calculations on an optimized basis and by utilizing hybrid CPU-multiGPU parallelization. At least an order of magnitude reduction in computational complexity results from mode optimization, while a further order of reduction in wall time is achieved by massive parallelization. Our results are measured directly in the number of floating point operations and seconds. A detailed scaling analysis of the obtained performance as a function of matrix ranks and as a function of system size up to $12\ifmmode\times\else\texttimes\fi{}12$ lattice topology is discussed. Our CPU-multiGPU model also tremendously accelerates the calculation of the one- and two-particle reduced density matrices, which can be used to construct various order parameters and trace quantum phase transitions with high fidelity.
We present a hybrid numerical approach to simulate quantum many-body problems on two spatial dimensional quantum lattice models via the non-Abelian ab initio version of the density matrix renormalization group method on state-of-the-art high-performance computing infrastructures. We demonstrate that for the two-dimensional spinless fermion model and for the Hubbard model on torus geometry, altogether several orders of magnitude in computational time can be saved by performing calculations on an optimized basis and by utilizing hybrid CPU-multiGPU parallelization. At least an order of magnitude reduction in computational complexity results from mode optimization, while a further order of reduction in wall time is achieved by massive parallelization. Our results are measured directly in the number of floating point operations and seconds. A detailed scaling analysis of the obtained performance as a function of matrix ranks and as a function of system size up to 12\ifmmode×\else\texttimes\fi12 lattice topology is discussed. Our CPU-multiGPU model also tremendously accelerates the calculation of the one- and two-particle reduced density matrices, which can be used to construct various order parameters and trace quantum phase transitions with high fidelity.
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.
We propose a new heterogeneous parallel strategy for the density matrix renormalization group (DMRG) method in the hybrid architecture with both central processing unit (CPU) and graphics processing unit (GPU). Focusing on the two most time-consuming sections in the finite DMRG sweeps, i.e., the diagonalization of superblock and the truncation of subblock, we optimize our previous hybrid algorithm to achieve better performance. For the former, we adopt OpenMP application programming interface on CPU and use our own subroutines with higher bandwidth on GPU. For the later, we use GPU to accelerate matrix and vector operations involving the reduced density matrix. Applying the parallel scheme to the Hubbard model with next-nearest hopping on the 4-leg ladder, we compute the ground state of the system and obtain the charge stripe pattern which is usually observed in high temperature superconductors. Based on simulations with different numbers of DMRG kept states, we show significant performance improvement and computational time reduction with the optimized parallel algorithm. Our hybrid parallel strategy with superiority in solving the ground state of quasi-two dimensional lattices is also expected to be useful for other DMRG applications with large numbers of kept states, e.g., the time dependent DMRG algorithms.
We demonstrate how to parallelize the density matrix renormalization group (DMRG) algorithm in real space through a straightforward modification of serial DMRG. This makes it possible to apply at least an order of magnitude more computational power to challenging simulations, greatly accelerating investigations of two-dimensional systems and large parameter spaces. We discuss details of the algorithm and present benchmark results including a study of valence-bond-solid order within the square-lattice Q2 model and N\'eel order within the triangular lattice Heisenberg model. The parallel DMRG algorithm also motivates an alternative canonical form for matrix product states.
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).
High Resolution Image Download MS PowerPoint Slide The interplay of quantum and classical simulation and the delicate divide between them is in the focus of massively parallelized tensor network state (TNS) algorithms designed for high performance computing (HPC). In this contribution, we present novel algorithmic solutions together with implementation details to extend current limits of TNS algorithms on HPC infrastructure building on state-of-the-art hardware and software technologies. Benchmark results obtained via large-scale density matrix renormalization group (DMRG) simulations on single node multiGPU NVIDIA A100 system are presented for selected strongly correlated molecular systems addressing problems on Hilbert space dimensions up to 4.17 × 10 35 .
This document aims to describe the methodology of quantum lattice gas algorithm based quantum physics simulations with details on implementation using graphics processing units. A variety of examples illustrating the general application of these methods are included. These examples are used to highlight topics such as scaling, potentials, momentums, multiple particle and dimensions, and various Hamiltonians in quantum simulations. Although intended to fullfill the doctor of philosophy degree requirement, this document is well suited as an introduction to this approach to modeling physics. In recent times parallelization has significantly improved the available computational power. Graphics processing units (GPUs) are a shining example of this, with thousands of compute cores and continually improved general computational capabilities, these processors can outshine their central processing unit (CPU) counterparts by orders of magnitude. Presented here are a number of examples of efficient computational methods for simulating physics using GPUs. Our quantum simulation method involves the quantum lattice gas model of quantum computation [59, 68, 64], however much of the parallel technology implementation is applicable to other numerical approaches. The quantum lattice gas model of computation is based on a unitary stream-collide quantum algorithm applied repeatedly over time upon an ordered array of qubits [53, 56, 68, 61, 63, 67]. The stream-collide quantum algorithm represents local quantum evolution of the system of quantum particles whose motion is constrained to reside on a spacetime lattice. Each particle in the system undergoes a quantum walk, interacting locally with other particles in the system that may reside on the same point. We will first mathematically describe the basic idea of quantum lattice gas algorithms. Next we will discuss GPUs and the basic features needed for implementation. We will then describe several examples of quantum lattice gas algorithm approach to illustrate the various applications and implementation challenges. Of particular interest is a look at modeling time-dependent many-body quantum mechanics using quantum lattice gas methods. For practical purposes we focus on the case of two fermions governed by the nonrelativistic Schrödinger equation. Numerical comparisons to analytic prediction show the high accuracy of these methods. Quantum lattice gas methods are chosen as a unitary algorithm for time dependent quantum simulation as they are efficient, accurate, and avoid the Fermi sign problem. Our findings from two-particle simulations under various conditions include the physics of the one and two particle systems diverging in their respective time-dependent behavior with increasing entanglement. We also confirm the multiple particle extension of the Ehrenfest theorem. Using the examples discussed throughout this document, we will then describe our specific computational implementation utilizing the computational power of GPUs through Nvidia’s compute unified device architecture (CUDA) language. Here we will discuss the open source software our group is developing based on these methods, along with some implementation verification and debugging concepts. Education is another application peripherally connected to this work and our software. The applications of this type of modeling as a learning tool is not fully explored yet, but the opportunities are promising.