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.
DMRG is a variational numerical approach developed to treat low-dimensional interacting many-body quantum systems efficiently @cite_32 @cite_4 @cite_15 . In fact, it has become an exceptionally successful method to study the low energy physics of strongly correlated quantum systems which exhibit chain-like entanglement structure @cite_11 . For example, it can be applied to simulate properties of anisotropic materials, such as polymers @cite_25 , or to describe accurately the electronic structure of open @math shell molecules @cite_19 , which is beyond the capability of standard quantum chemical approaches. Additionally, the interacting system of atoms trapped in an optical lattice, proposed as physical implementation of quantum computer, is also tractable via DMRG @cite_2 .
@cite_4 @cite_32 @cite_19 @cite_2 @cite_15 @cite_25 @cite_11
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The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG. The density matrix renormalization group [1] (DMRG), devised to improve on the Wilson’s numerical renormalization group (NRG) [2], has become the method of choice to simulate one-dimensional quantum many-body systems at zero temperature and has found a large number of applications in the fields of the condensed matter physics, quantum chemistry and quantum information theory where it turned out that DMRG is essentially equivalent to simulating quantum systems in terms of matrix product states (MPS) [3]. In all these methods, a quantum many-body state is represented by associating matrices to local configurations at sites, and the coefficients in the expansion over the configuration states are given as products of the corresponding matrices. The DMRG can be used to calcu Abstract The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms. In the past few years, there has been an increasingly active exchange of ideas and methods between the formerly rather disjunct fields of quantum information and many-body physics. This has been due, on the one hand, to the growing sophisti- cation of methods and the increasing complexity of problems treated in quantum information theory, and, on the other, to the recognition that a number of central issues in many-body quantum systems can fruitfully be approached from the quan- tum information point of view. Nowhere has this been more evident than in the context of the family of numerical methods that go under the rubric density-matrix renormalization group. In particular, the concept of entanglement and its definition, measurement, and manipulation lies at the heart of much of quantum information theory (1). The density-matrix renormalization group (DMRG) methods use proper- ties of the entanglement of a bipartite system to build up an accurate approximation to particular many-body wave functions. The cross-fertilization between the two fields has led to improvements in the understanding of interacting quantum systems in general and the DMRG method in particular, has led to new algorithms related to and generalizing the DMRG, and has opened up the possibility of studying many new physical problems, ones of interest both for quantum information theory and for understanding the behavior of strongly correlated quantum systems (2). In this line, we discuss some relevant concepts in quantum information theory, including the relation between the DMRG and data compression and entanglement. As an application, we will use the quantum information entropy calculated with the DMRG to study quantum phase transitions, in particular in the bilinear-biquadratic spin-one chain and in the frustrated spin-1 2 Heisenberg chain.