Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices

TL;DR

This paper introduces three randomized algorithms for efficiently estimating the von Neumann entropy of large density matrices, leveraging recent advances in randomized linear algebra with proven accuracy and significant speed improvements.

Contribution

The paper presents novel randomized algorithms with theoretical guarantees for approximating von Neumann entropy, reducing computational costs for large matrices.

Findings

Algorithms achieve speedup over exact methods

Provable accuracy bounds established for each algorithm

Experimental results confirm theoretical predictions

Abstract

Thevon Neumann entropy, named after John von Neumann, is an extension of the classical concept of entropy to the field of quantum mechanics. From a numerical perspective, von Neumann entropy can be computed simply by computing all eigenvalues of a density matrix, an operation that could be prohibitively expensive for large-scale density matrices. We present and analyze three randomized algorithms to approximate von Neumann entropy of {real} density matrices: our algorithms leverage recent developments in the Randomized Numerical Linear Algebra (RandNLA) literature, such as randomized trace estimators, provable bounds for the power method, and the use of random projections to approximate the eigenvalues of a matrix. All three algorithms come with provable accuracy guarantees and our experimental evaluations support our theoretical findings showing considerable speedup with small loss in accuracy.