Decompositions of higher-order tensor networks has been paid high attention in efficient description of certain many-body states and have applications in numerical algorithms. Since tensors are multidimensional generalizations of matrices, common decompositions like singular value decomposition (SVD) and QR decomposition known from linear algebra can be applied to tensors by mapping a tensor with more than two degrees into a matrix. For symmetric invariant tensors with quantum number preservation, the mapped matrices of them can be transformed into a special block-diagonal-similar (BDS) form and is usually sparse. Most coding packages provide decompositions on sparse matrices, but this block-diagonal similarity is usually not considered.
 
In this work, blockwise SVD and QR decomposition of quantum number preserving tensors are developed with consideration of this transformed BDS matrices.
A framework with a specific data structure for quantum number preserving tensors is implemented, which also allows for basic tensor manipulations, such as contraction, permutation, fusion of legs. The focus is the implementation of blockwise SVD and QR decomposition with algorithms for transformation between a quantum number preserving tensor and a BDS matrix, and blockwise decompositions of a BDS matrix with the correct mapping back to the original tensor. To this end, scaling studies were carried out on randomly generated tensors to compare blockwise decompositions with non-blockwise decompositions. The benchmarks show that blockwise decompositions, including both SVD and QR decomposition, are faster than common sparse decompositions using base functions provided by the selected package with the integration of LAPACK and BLAS. The results also show that the over-performance varies with the variables dimensions and sparsity of the matrix.