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

Abstract

In the numerical analysis of strongly correlated quantum lattice models one of the leading algorithms developed to balance the size of the effective Hilbert space and the accuracy of the simulation is the density matrix renormalization group (DMRG) algorithm, in which the run-time is dominated by the iterative diagonalization of the Hamilton operator. As the most time-dominant step of the diagonalization can be expressed as a list of dense matrix operations, the DMRG is an appealing candidate to fully utilize the computing power residing in novel kilo-processor architectures.In the paper a smart hybrid CPU-GPU implementation is presented, which exploits the power of both CPU and GPU and tolerates problems exceeding the GPU memory size. Furthermore, a new CUDA kernel has been designed for asymmetric matrix-vector multiplication to accelerate the rest of the diagonalization. Besides the evaluation of the GPU implementation, the practical limits of an FPGA implementation are also discussed.

Tensor product methods and entanglement optimization for ab initio quantum chemistry

“…Quite recently, it has been investigated how the DMRG method can utilize the enormous computing capabilities of novel kilo‐processor architectures: GPU and field‐programmable gate array . In case of GPU, a smart hybrid CPU‐GPU acceleration has been presented, which tolerates problems exceeding the GPU memory size, consequently, supporting wide range of problems and GPU configurations.…”

Massively parallel quantum chemical density matrix renormalization group method

“…When the data are no longer needed, the allocated memory is freed. Regarding the sources of parallelism, we combine operator and symmetry sector parallelisms similarly to the simpler shared memory approach, 41,45 in order to generate a large‐enough number of tasks (dense matrix–matrix operations), which can be executed in parallel. All three main steps are task‐based parallelized.…”

“…So, for example instead of the aforementioned term , we have A ↑↑ ⊗ I ⊗ I ⊗ a ↑ a ↑ and the loop over rs is performed sequentionally during the task execution. It is organized this way to avoid fetching of small memory chunks and with the view of a further GPU acceleration in future [parallel execution of matrix–matrix multiplications ) performed on different slices of the same dense tensors] 45 …”

“…Extension of parallelization scheme of Ref. [41] to a smart hybrid CPU‐GPU implementation has also been presented, exploiting the power of both CPU and GPU tolerating problems exceeding the GPU memory size 45 . In DMRG, the iterative construction of the Hamiltonian is decomposed into several independent matrix operations and each of these are further decomposed into smaller independent tasks based on symmetries, thus diagonalization has been expressed as a single list of dense matrix operations.…”

“…This article is thus a first attempt to port the QC‐DMRG method to a supercomputer platform. Our parallel approach is similarly to the shared memory algorithm 41,45 based on merging of the operator and symmetry sector loops and employ the global memory (GM) model. It relies on a fast internode connection.…”