Calculating elements of matrix functions using divided differences

TL;DR

This paper presents a memory-efficient method for calculating individual elements of matrix functions, demonstrated on large quantum many-body Hamiltonians, with potential applications in quantum physics and large-scale matrix computations.

Contribution

The authors introduce a novel series expansion technique for matrix functions that is efficient for large matrices and applicable to various functions like exponentials and inverses.

Findings

Successfully computed matrix elements for Hamiltonians up to 2^64 size

Demonstrated advantages over existing methods in efficiency and memory usage

Applicable to quantum transition amplitudes and matrix inverses

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.