Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

TL;DR

This paper introduces efficient algorithms combining trace estimators and Krylov methods to approximate the von Neumann entropy of large sparse matrices, with proven error bounds and demonstrated effectiveness on network density matrices.

Contribution

It develops novel Krylov-based algorithms with error analysis for computing von Neumann entropy of large matrices, integrating randomized and probing trace estimation techniques.

Findings

Algorithms achieve accurate entropy approximations

Methods outperform traditional approaches in large-scale scenarios

Numerical results confirm efficiency on network density matrices

Abstract

We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.