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arXiv · 2212.09642

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

Michele Benzi·Michele Rinelli·Igor Simunec
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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.

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Michele Benzi, Michele Rinelli, Igor Simunec. 2022-12-19. Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods. https://arxiv.org/abs/2212.09642

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