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arXiv:1309.5571 · Computer Physics Communications

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

Csaba Nemes, Gergely Barcza, Zoltán Nagy, Örs Legeza, Péter Szolgay

Published Feb 26, 2014• 22 citations
arXivHTMLPDF
DOI
10.1016/j.cpc.2014.02.021

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Numerical analysis of strongly correlated quantum lattice models has a great importance in quantum physics. The exponentially growing size of the Hilbert space makes these computations difficult, however sophisticated algorithms have been developed to balance the size of the effective Hilbert space and the accuracy of the simulation. One of these methods is the density matrix renormalization group (DMRG) algorithm which has become the leading numerical tool in the study of low dimensional lattice problems of current interest. In the algorithm a high computational problem can be translated to a list of dense matrix operations, which makes it an ideal application to fully utilize the computing power residing in both current multi-core processors and novel kilo-processor architectures.
The performance of scientific software often determines the scale of problems that can be solved in practice. As multiple implementations of the same algorithm emerge, systematic evaluation is needed to compare their strengths and limitations. The density matrix renormalization group (DMRG) algorithm, widely used to study quantum systems, has over 50 software implementations. These implementations vary in multiple aspects that can strongly affect performance. However, despite the need, performance evaluations of these implementations are scarce and lack a consistent standard; many existing evaluations are either too incomplete to enable meaningful comparisons or focus on objectives other than direct performance comparisons, thereby limiting understanding of how the implementations compare. Here, we present a performance-oriented benchmarking framework to facilitate meaningful comparisons of DMRG implementations, and we apply it to quantify the performance of eight implementations, highlighting similarities and differences among them. Furthermore, we examine multiple parameter settings, optimization strategies, and implementation-specific features to demonstrate how parameter configuration can affect performance and how systematic evaluation can reveal non-obvious trade-offs. The results show significant performance differences, up to two orders of magnitude in some cases, not only between different implementations when aligning parameters, but also within the same implementation when comparing different parameter configurations. Hence, our results demonstrate the significant value and insight that can be gained from conducting rigorous performance evaluations. Using our results and framework as a starting point, more rigorous benchmarking will ultimately help users and developers make informed decisions and support future development efforts to build better, more efficient software.