Minimal Split Checkerboard Method for Exponentiating Sparse Matrices and Its Applications in Quantum Statistical Mechanics

Abstract

Multiplication of matrix exponentials is one of the computational kernels in simulations of quantum statistical mechanics, in which matrices are symmetric and sparse. Although the matrix is not particularly large, its multiplication needs to be performed millions of times and thus becomes a performance bottleneck. For a sparse symmetric matrix $A$, the checkerboard method splits $A=A_1+A_2+\cdots + A_k$ and approximates $e^A$ by $e^{A_1}e^{A_2}\cdots e^{A_k}$, in which each $e^{A_i}$ is sparse. When combined with sparse matrix techniques, the checkerboard method can significantly reduce the expense of storing and multiplying $e^A$. However, the accuracy of the checkerboard method degrades as the number of split matrices increases. In this paper, the Minimal Split Checkerboard method (MSCKB) is proposed to reduce the number of split matrices of the checkerboard method. In addition, the Block Checkerboard method (BlkCKB) and the Low-Rank Checkerboard method (LRCKB) are presented to further enhance the performance. All the ideas can be extended to exponentiate skew-symmetric matrices and general matrices. Experiments based on the simulations of quantum statistical mechanics demonstrate the effectiveness of the proposed methods.