Calculating elements of matrix functions using divided differences

• URL: http://arxiv.org/abs/2107.14124v2
• Date: Tue, 16 Nov 2021 13:34:57 GMT
• Title: Calculating elements of matrix functions using divided differences
• Authors: Lev Barash, Stefan G\"uttel, Itay Hen
• Abstract summary: We introduce a method for calculating individual elements of matrix functions. We showcase our approach by calculating the matrix elements of the exponential of a transverse-field Ising model. We discuss practical applications of our method.
• Score: 0.3437656066916039
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We showcase our approach by calculating the matrix elements of the exponential of a transverse-field Ising model and evaluating quantum transition amplitudes for large many-body Hamiltonians of sizes up to $2^{64} \times 2^{64}$ on a single workstation. We also discuss the application of the method to matrix inverses. We relate and compare our method to the state-of-the-art and demonstrate its advantages. We also discuss practical applications of our method.

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