The density-matrix renormalization group algorithm has emerged as a promising new method in ab initio quantum chemistry. However, many problems still need to be solved before this method can be applied routinely. At the start of such a calculation, the orbitals originating from a preceding quantum chemical calculation must be placed in a specific order on a one-dimensional lattice. This ordering affects the convergence of the density-matrix renormalization group iterations significantly. In this paper, we present two approaches to obtain optimized orderings of the orbitals. First, we use a genetic algorithm to optimize the ordering with respect to a low total electronic energy obtained at a predefined stage of the density-matrix renormalization group algorithm with a given number of total states kept. In addition to that, we derive orderings from the one- and two-electron integrals of our test system. This test molecule is the chromium dimer, which is known to possess a complicated electronic structure. For this molecule, we have carried out calculations for the various orbital orderings obtained. The convergence behavior of the density-matrix renormalization group iterations is discussed in detail.
In this study, the authors propose a new two‐stage decoding scheme for low‐density parity‐check codes to lower the error‐floor, which consists of the conventional belief propagation (BP) decoding algorithm as the first‐stage decoding and the re‐decodings with manipulated log‐likelihood ratios (LLRs) of variable nodes as the second‐stage decoding. In the first‐stage decoding, an early stopping criterion is proposed for early detection of decoding failure and the candidate set of the variable nodes are determined, which can be partly included in the small trapping sets. In the second‐stage decoding, the scores of the variable nodes in the candidate set are computed by the proposed unreliable path search algorithm and the variable nodes are sorted in ascending order by their scores for the re‐decoding trials. Each re‐decoding trial is performed by BP decoding algorithm with manipulated LLR of a selected variable node in the candidate set one at a time with the second early stopping criterion. The parallel unreliable path search algorithm is also proposed for practical application of the proposed algorithm. Numerical results show that the proposed early stopping criteria and the proposed decoding algorithms for the second‐stage decoding can correct most of the unsuccessfully decoded codewords by the first‐stage decoding in the error‐floor region.