论文标题
通过痕量估计器和合理的Krylov方法计算大型矩阵的von Neumann熵
Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods
论文作者
论文摘要
我们考虑了近似大型，稀疏，对称的半fin矩阵$ a $的von Neumann熵的问题，该问题定义为$ \ operatatorName {tr}（f（a））$，其中$ f（x）= - x）= -x \ log log x $。在建立此矩阵函数的一些有用属性后，我们考虑在两种类型的近似方法中同时使用多项式和有理的Krylov子空间算法，即基于图形颜色的随机痕量估计器和探测技术。我们开发了用于实施算法的错误范围和启发式方法。对不同类型网络的密度矩阵的数值实验说明了方法的性能。
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.
